Technology is an essential tool for learning mathematics in the 21st century, and all schools must ensure that all their students have access to technology. Effective teachers
Technology is an essential tool for learning mathematics in the 21st century, and all schools must ensure that all their students have access to technology. Effective teachers maximize the potential of technology to develop students’ understanding, stimulate their interest, and increase their proficiency in mathematics. When technology is used strategically, it can provide access to mathematics for all students.
for all students.
increase their proficiency in mathematics. When technology is used strategically, it can provide access to mathematics for all students.
Calculators and other technological tools, such as computer algebra systems, interactive geometry software, applets, spreadsheets, and interactive presentation devices, are vital components of a high-quality mathematics education. With guidance from effective mathematics teachers, students at different levels can use these tools to support and extend mathematical reasoning and sense making, gain access to mathematical content and problem-solving contexts, and enhance computational fluency. In a well-articulated mathematics program, students can use these tools for computation, construction, and representation as they explore problems. The use of technology also contributes to mathematical reflection, problem identification, and decision making.
The use of technology cannot replace conceptual understanding, computational fluency, or problem-solving skills. In a balanced mathematics program, the strategic use of technology enhances mathematics teaching and learning. Teachers must be knowledgeable decision makers in determining when and how their students can use technology most effectively. All schools and mathematics programs should provide students and teachers with access to instructional technology, including appropriate calculators, computers with mathematical software, Internet connectivity, handheld data-collection devices, and sensing probes. Curricula and courses of study should incorporate instructional technology in learning outcomes, lesson plans, and assessments of students’ progress.
Programs in teacher education and professional development must continually update practitioners’ knowledge of technology and its classroom applications. Such programs should include the development of mathematics lessons that take advantage of technology-rich environments and the integration of technology in day-to-day instruction, instilling an appreciation for the power of technological tools and their potential impact on students’ learning and use of mathematics. All teachers must remain open to learning new technologies, implementing them effectively in a coherent and balanced instructional program
Calculators and other technological tools, such as computer algebra systems, interactive geometry software, applets, spreadsheets, and interactive presentation devices, are vital components of a high-quality mathematics education. With guidance from effective mathematics teachers, students at different levels can use these tools to support and extend mathematical reasoning and sense making, gain access to mathematical content and problem-solving contexts, and enhance computational fluency. In a well-articulated mathematics program, students can use these tools for computation, construction, and representation as they explore problems. The use of technology also contributes to mathematical reflection, problem identification, and decision making.
The use of technology cannot replace conceptual understanding, computational fluency, or problem-solving skills. In a balanced mathematics program, the strategic use of technology enhances mathematics teaching and learning. Teachers must be knowledgeable decision makers in determining when and how their students can use technology most effectively. All schools and mathematics programs should provide students and teachers with access to instructional technology, including appropriate calculators, computers with mathematical software, Internet connectivity, handheld data-collection devices, and sensing probes. Curricula and courses of study should incorporate instructional technology in learning outcomes, lesson plans, and assessments of students’ progress.
Programs in teacher education and professional development must continually update practitioners’ knowledge of technology and its classroom applications. Such programs should include the development of mathematics lessons that take advantage of technology-rich environments and the integration of technology in day-to-day instruction, instilling an appreciation for the power of technological tools and their potential impact on students’ learning and use of mathematics. All teachers must remain open to learning new technologies, implementing them effectively in a coherent and balanced instructional program. These tools, including those used specifically for teaching and learning mathematics, not only complement mathematics teaching and learning but also prepare all students for their future lives, which technology will influence every day.
. These tools, including those used specifically for teaching and learning mathematics, not only complement mathematics teaching and learning but also prepare all students for their future lives, which technology will influence every day.
Monday, October 13, 2008
Assignment
Marvelino M. Niem July 23, 2008
BSE 4m
SEMTECH
Assignment:
Choose two programs.
In what particular lesson can you use those programs?
Explain details how you integrate it.
What makes it effective compared to traditional method?
Answer:
LINEAR ALGEBRA 2
· This program can be use in solving matrices, finding solution of linear system of equations adding matrices, scalar and vector multiplication and other topics under linear algebra.
· Using this program, instead of solving it manually using paper and pen, it takes less time because you will take less time and effort because you will just input the data and then the answer is just one click away.
· It is very effective compared to traditional method aside from the time spent and the effort gave, the answer is surely accurate. But then the disadvantage of this is the students will be more dependent on it to the extent that they will miss the step-by-step or the procedures on how to solve it manually. Like in scalar and vector multiplication, Cramers rule, reducing to row echelon form, etc.
VISUAL SHAPES
· This program can be use in geometry class specifically in topics involving shapes and polygon.
· Like in geometry for example the topic is about polygon, I will show some of the shapes like triangle, square, pentagon, etc. which are input in the software and then I will ask them to arrange it. Say for example triangle and then after arranging it I will discuss some information with regard to the shape presented to them.
· It is effective because the students visualize the actual shapes and then manipulate it. It is also helps the students to think critically how the shape or the pattern of the shapes will be. Lastly, they will enjoy what they doing.
REACTION PAPER
Technology is an essential tool for learning mathematics in the 21st century, and all schools must ensure that all their students have access to technology. Effective teachers maximize the potential of technology to develop students’ understanding, stimulate their interest, and increase their proficiency in mathematics. When technology is used strategically, it can provide access to mathematics for all students.
Calculators and other technological tools, such as computer algebra systems, interactive geometry software, applets, spreadsheets, and interactive presentation devices, are vital components of a high-quality mathematics education. With guidance from effective mathematics teachers, students at different levels can use these tools to support and extend mathematical reasoning and sense making, gain access to mathematical content and problem-solving contexts, and enhance computational fluency. In a well-articulated mathematics program, students can use these tools for computation, construction, and representation as they explore problems. The use of technology also contributes to mathematical reflection, problem identification, and decision making.
The use of technology cannot replace conceptual understanding, computational fluency, or problem-solving skills. In a balanced mathematics program, the strategic use of technology enhances mathematics teaching and learning. Teachers must be knowledgeable decision makers in determining when and how their students can use technology most effectively. All schools and mathematics programs should provide students and teachers with access to instructional technology, including appropriate calculators, computers with mathematical software, Internet connectivity, handheld data-collection devices, and sensing probes. Curricula and courses of study should incorporate instructional technology in learning outcomes, lesson plans, and assessments of students’ progress.
Programs in teacher education and professional development must continually update practitioners’ knowledge of technology and its classroom applications. Such programs should include the development of mathematics lessons that take advantage of technology-rich environments and the integration of technology in day-to-day instruction, instilling an appreciation for the power of technological tools and their potential impact on students’ learning and use of mathematics. All teachers must remain open to learning new technologies, implementing them effectively in a coherent and balanced instructional program. These tools, including those used specifically for teaching and learning mathematics, not only complement mathematics teaching and learning but also prepare all students for their future lives, which technology will influence every day.
These tools, including those used specifically for teaching and learning mathematics, not only complement mathematics teaching and learning but also prepare all students for their future lives, which technology will influence every day.
- (NCTM)
In this year, we are in a computer generation world that’s why it is a must for a teacher today to be a computer literate teacher.
Technologies as link to new knowledge, resources and high order thinking skills have entered classrooms and schools worldwide. Personal computers, CD-ROMS, on line services, the World Wide Web and other innovative technologies have enriched curricula and altered the types of teaching have available in the classroom. Schools’ access to technology is increasing steadily everyday and most of these newer technologies are now even used in traditional classrooms (Salandan, G. et al., 2006).
The technology utilization in the classroom specifically in some private schools should take a balance approach. The use of technology should not maximize because of its limitations and we must look on other hand the traditional approach.
On the other hand, there are some roles of technology in achieving the goal of learning with understanding (Goldman, S, Williams, R. et al, 1996). They are the following:
· Technology provides support to the solution of meaningful problems.
· Technology acts as cognitive support.
· Technology promotes collaboration as well as independent learning.
Because of some roles of technology, it is also the goal of the teacher to teach the students and mold them to be better than the teacher, so the utilization and integration of technology together with the traditional approach will help the students fully in their learning and I think this is the right combination of strategies.
Lastly, I formulate the formula in learning. This is the equqtion:
Tecnology + Traditional Approach = Learnings
(computers, calculators) + (chalk and board) = (n_n)
BSE 4m
SEMTECH
Assignment:
Choose two programs.
In what particular lesson can you use those programs?
Explain details how you integrate it.
What makes it effective compared to traditional method?
Answer:
LINEAR ALGEBRA 2
· This program can be use in solving matrices, finding solution of linear system of equations adding matrices, scalar and vector multiplication and other topics under linear algebra.
· Using this program, instead of solving it manually using paper and pen, it takes less time because you will take less time and effort because you will just input the data and then the answer is just one click away.
· It is very effective compared to traditional method aside from the time spent and the effort gave, the answer is surely accurate. But then the disadvantage of this is the students will be more dependent on it to the extent that they will miss the step-by-step or the procedures on how to solve it manually. Like in scalar and vector multiplication, Cramers rule, reducing to row echelon form, etc.
VISUAL SHAPES
· This program can be use in geometry class specifically in topics involving shapes and polygon.
· Like in geometry for example the topic is about polygon, I will show some of the shapes like triangle, square, pentagon, etc. which are input in the software and then I will ask them to arrange it. Say for example triangle and then after arranging it I will discuss some information with regard to the shape presented to them.
· It is effective because the students visualize the actual shapes and then manipulate it. It is also helps the students to think critically how the shape or the pattern of the shapes will be. Lastly, they will enjoy what they doing.
REACTION PAPER
Technology is an essential tool for learning mathematics in the 21st century, and all schools must ensure that all their students have access to technology. Effective teachers maximize the potential of technology to develop students’ understanding, stimulate their interest, and increase their proficiency in mathematics. When technology is used strategically, it can provide access to mathematics for all students.
Calculators and other technological tools, such as computer algebra systems, interactive geometry software, applets, spreadsheets, and interactive presentation devices, are vital components of a high-quality mathematics education. With guidance from effective mathematics teachers, students at different levels can use these tools to support and extend mathematical reasoning and sense making, gain access to mathematical content and problem-solving contexts, and enhance computational fluency. In a well-articulated mathematics program, students can use these tools for computation, construction, and representation as they explore problems. The use of technology also contributes to mathematical reflection, problem identification, and decision making.
The use of technology cannot replace conceptual understanding, computational fluency, or problem-solving skills. In a balanced mathematics program, the strategic use of technology enhances mathematics teaching and learning. Teachers must be knowledgeable decision makers in determining when and how their students can use technology most effectively. All schools and mathematics programs should provide students and teachers with access to instructional technology, including appropriate calculators, computers with mathematical software, Internet connectivity, handheld data-collection devices, and sensing probes. Curricula and courses of study should incorporate instructional technology in learning outcomes, lesson plans, and assessments of students’ progress.
Programs in teacher education and professional development must continually update practitioners’ knowledge of technology and its classroom applications. Such programs should include the development of mathematics lessons that take advantage of technology-rich environments and the integration of technology in day-to-day instruction, instilling an appreciation for the power of technological tools and their potential impact on students’ learning and use of mathematics. All teachers must remain open to learning new technologies, implementing them effectively in a coherent and balanced instructional program. These tools, including those used specifically for teaching and learning mathematics, not only complement mathematics teaching and learning but also prepare all students for their future lives, which technology will influence every day.
These tools, including those used specifically for teaching and learning mathematics, not only complement mathematics teaching and learning but also prepare all students for their future lives, which technology will influence every day.
- (NCTM)
In this year, we are in a computer generation world that’s why it is a must for a teacher today to be a computer literate teacher.
Technologies as link to new knowledge, resources and high order thinking skills have entered classrooms and schools worldwide. Personal computers, CD-ROMS, on line services, the World Wide Web and other innovative technologies have enriched curricula and altered the types of teaching have available in the classroom. Schools’ access to technology is increasing steadily everyday and most of these newer technologies are now even used in traditional classrooms (Salandan, G. et al., 2006).
The technology utilization in the classroom specifically in some private schools should take a balance approach. The use of technology should not maximize because of its limitations and we must look on other hand the traditional approach.
On the other hand, there are some roles of technology in achieving the goal of learning with understanding (Goldman, S, Williams, R. et al, 1996). They are the following:
· Technology provides support to the solution of meaningful problems.
· Technology acts as cognitive support.
· Technology promotes collaboration as well as independent learning.
Because of some roles of technology, it is also the goal of the teacher to teach the students and mold them to be better than the teacher, so the utilization and integration of technology together with the traditional approach will help the students fully in their learning and I think this is the right combination of strategies.
Lastly, I formulate the formula in learning. This is the equqtion:
Tecnology + Traditional Approach = Learnings
(computers, calculators) + (chalk and board) = (n_n)
Promoting Appropriate Uses of Technology in Mathematics Teacher Preparation
In the Principles and Standards of School Mathematics the National Council of Teachers of Mathematics (NCTM) identified the "Technology Principle" as one of six principles of high quality mathematics education (NCTM, 2000). This principle states: "Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students' learning" (p. 24). There is widespread agreement that mathematics teachers, not technological tools, are the key change agents to bringing about reform in mathematics teaching with technology (Kaput, 1992; NCTM 1991, 2000). Yet, preparing teachers to use technology appropriately is a complex task for teacher educators (Mergendoller, 1994). Waits and Demana (2000) argue that adoption of technology by teachers requires professional development that focuses on both conceptual and pedagogical issues, ongoing support in terms of "intensive start-up assistance and regular follow-up activities" and a desire to change from within the profession (p. 53). In addition, studies of teachers' implementation of educational technology document that at least three to five years are needed for teachers to become competent and confident in teaching with technology (Dwyer, Ringstaff, & Sandholtz, 1991; Means & Olson, 1994).
The Curry Center for Technology and Teacher Education at the University of Virginia is developing materials to help preservice secondary mathematics, social studies, and science teachers (PSTs) learn to incorporate technology appropriately into their teaching. The focus of the mathematics team is to devise activities that will prepare secondary teachers to use technology to enhance and extend their students' learning of mathematics. In this article we discuss the approach to developing and using materials for this purpose.
Use of Technology in Teacher Education
Technology is being incorporated into teacher education in numerous ways. Not surprisingly, there are different ways to categorize the various approaches taken by teacher educators to bring technology into their programs. One way to categorize these approaches is according to the primary user or controller of the technology—the teacher educator, the teacher, or the student. In some uses of technology in teacher education, the teacher educator is the primary user of the technology. For example, some teacher educators use multi-media case studies of rich teaching episodes to help PSTs analyze teaching and learning environments, and some use technology to present information or to demonstrate explorations. In many teacher education programs the teacher is being prepared to be the primary user of technology. For example, PSTs are being prepared to use technology productivity tools for word processing, grade and record keeping, web page production, and presentations. Also, many PSTs are using subject-specific software and websites to create presentations, lectures, lessons, and assessments. A third approach to incorporating technology in teacher education is to prepare PSTs to have their future students use technology to investigate concepts and solve meaningful problems in the content areas. For example, in the area of mathematics, PSTs are learning how to guide their students to use technologies such as spreadsheets, graphing calculators, dynamic geometry programs, and playable websites to explore mathematics concepts and use mathematics to solve problems in applied contexts.
The three uses of technology in teacher education presented above are connected with different purposes and all can lead to better teacher effectiveness and improved student learning. Thus, all are important. However, it has been our experience that the most direct and effective way to use technology to bring about enhanced student learning of mathematics is to prepare PSTs to incorporate into their teaching an array of activities that engage students in mathematical thinking facilitated by technological tools. Hence, in our preparation of secondary PSTs we emphasize the third use, in which ultimately the student is the primary user, and to some degree, the second use, in which the teacher is the primary user. Our materials reflect these emphases and thus are being developed around significant mathematical activities for school students.
In our classes, PSTs complete activities that, with some modification, are appropriate for secondary mathematics courses. We then use these completed activities to anchor class discussions of issues connected with secondary curriculum and instruction, national and state standards, sequencing of topics, the role of technology, and assessment. In the course of completing these activities, PSTs not only learn how to use the technology, but also how to incorporate technology into their teaching.
Guidelines for Technology-Based Activity Development
In the early phase of our work, we devised a set of guidelines to shape our development of mathematics activities and materials (Garofalo, Shockey, Harper, & Drier, 1999). The five guidelines below reflect what we believe to be appropriate uses of technology in mathematics teaching:
introduce technology in context
address worthwhile mathematics with appropriate pedagogy
take advantage of technology
connect mathematics topics
incorporate multiple representations
Each of these guidelines is discussed below and illustrated with one or more of our activities.
Introduce Technology in Context
Features of technology, whether mathematics-specific or more generic, should be introduced and illustrated in the context of meaningful content-based activities. Teaching a set of technology or software-based skills and then trying to find mathematical topics for which they might be useful is comparable to teaching a set of procedural mathematical skills and then giving a collection of "word problems" to solve using the procedures. Such an approach can obscure the purpose of learning and using technology, make mathematics appear as an afterthought, and lead to contrived activities. The use of technology in mathematics teaching is not for the purpose of teaching about technology, but for the purpose of enhancing mathematics teaching and learning with technology. Furthermore, in our experience, teachers who learn to use technology while exploring relevant mathematics topics are more likely to see its potential benefits and use it in their subsequent teaching. This guideline is in accord with the first recommendation of the President's Committee of Advisors on Science and Technology, Panel on Educational Technology (1997): "Focus on learning with technology, not about technology" (p. 7).
Example: Simulating Freefall With Parametric Quadratic Equations
In this activity connecting quadratic equations and projectile motion, PSTs are introduced to the parametric graphing features of graphing calculators. PSTs are asked to derive an expression for the height of an object dropped from 500m above the surface of the Earth, as a function of time. They are then asked to construct a graph of this relationship, first with paper and pencil and then with graphing calculators. Our PSTs are able to derive a correct equation and generate a graph similar to those in Figure 1. (Click on the caption of the screenshot of the graph in the figure to see the graph being drawn.)
Figure 1. Casio 9850 Plus graphing calculator screenshots of freefall equation and graph
The graph in Figure 1 is appropriate, considering that the x -axis represents time and the y -axis represents the height of the object. However, many students often fail to fully attend to axis variables and sometimes interpret such a graph as a picture of the situation being represented. They interpret this graph as implying that the path of the object is outward and downward rather than as straight downward. Such a misinterpretation is referred to as an iconic interpretation (Kerslake, 1977; Leinhardt, Zaslavsky, & Stein, 1990) and is prevalent with secondary school students. We challenge our PSTs to generate a graph that simulates the actual path of a freefalling object. This involves use of parametric equations, which all of our PSTs have studied in calculus, but most have forgotten or have not considered for high school use. Rather than teach PSTs to use the graphing calculator parametric features ahead of time and apply here, we introduce these features in this context, where PSTs can see its direct applicability and usefulness. Figure 2 shows the parametric equation and graph that simulates the object's path. (Click on the caption of the graph screenshot in the figure to see the graph being drawn.)
Figure 2. Calculator screenshots of parametric freefall equation and graph
PSTs then compare and contrast the graphs. Subsequently, we use the graphs as springboards to discuss various aspects of visual representations (e.g., connected versus plot graphing), iconic misrepresentations, incorporation of parametric equations in the curriulum, and use of parametric equations to enrich the treatment of other school mathematics topics. PSTs then apply these features to tasks involving horizontal motion and angular projectile motion.
Address Worthwhile Mathematics with Appropriate Pedagogy
Content-based activities using technology should address worthwhile mathematics concepts, procedures, and strategies, and should reflect the nature and spirit of mathematics. Activities should support sound mathematical curricular goals and should not be developed merely because technology makes them possible. Indeed, the use of technology in mathematics teaching should support and facilitate conceptual development, exploration, reasoning and problem solving, as described by the NCTM (1989, 1991, 2000).
Technology should not be used to carry out procedures without appropriate mathematical and technological understanding (e.g., inserting rote formulas into a spreadsheet to demonstrate population growth). Nor should it be used in ways that can distract from the underlying mathematics (e.g., adding so many bells and whistles into a Power Point slideshow that the mathematics gets lost). In other words, mathematical content and pedagogy should not be compromised.
Another way to prevent technology use from compromising mathematics is to encourage users to connect their experiential findings to more formal aspects of mathematics. For example, students using software to explore geometric shapes and relationships should be asked to use previously proved theorems to validate their empirical results, or use their new findings to propose new conjectures. Mathematical notions of "proof " and "rigor" need to be addressed as well. In other words, technology should not influence students to take things at face value or to become what Schoenfeld (1985) referred to as "naïve empiricists." This guideline is in accord with the second recommendation of the President's Committee of Advisors on Science and Technology, Panel on Educational Technology (1997): "Emphasize content and pedagogy, and not just hardware" (p. 7).
Example: Exploring the Pythagorean Theorem
On way we explore issues regarding appropriate pedagogy with technology with our PSTs is with the following activity. The emphasis of the activity is more on the principles of teaching the mathematics than on the mathematics itself. PSTs discuss how this topic is traditionally taught to students, as a rote memorization of a 2 + b 2 = c 2 without any conceptual understanding that a 2 , b 2 and c 2 represent areas of squares with sides length a , b , and c . We then discuss how technology could be used to enhance the students' understanding of the theorem, and guide them through a model lesson of how the Pythagorean theorem could be taught using The Geometer's Sketchpad (Jackiw, 1997).
PSTs are first asked to use the Sketchpad to construct a right triangle, measure each side, and numerically confirm the a 2 + b 2 = c 2 relationship. They then learn how to use Sketchpad features to script a construction of a square. Next the PSTs play back their scripts to place a square on each of the sides of the right triangle and add measurements to create a construction similar to the one in Figure 3. The dynamic Sketchpad environment allows the PSTs to drag the triangle's vertices or sides to manipulate the construction, keeping the characteristics of the geometric figures intact. As the construction changes, the sum of the area of the squares on the legs of the right triangle always remain equal to the area of the square on the hypotenuse of the right triangle. (Click on Figure 3 caption to see an animation of this construction.)
Figure 3. Sketchpad file of the Pythagorean theorem
PSTs then discuss connections between the different representations of the Pythagorean Theorem, advantages of each representation in teaching this topic, and benefits of using the Sketchpad to create and manipulate the constructions. This activity illustrates a point made in the Technology Principle of the NCTM Principles and Standards : "Technology also provides a focus as students discuss with one another and with their teacher the objects on the screen and the effects of the various dynamic transformations that technology allows (NCTM, 2000, p. 24)."
We typically pose the following question to our PSTs: Does the manipulation of this construction constitute a mathematical proof of the Pythagorean theorem? One semester we received four types of responses: "I think it's a proof;" "I don't know if this is a proof;" "I hope it's a proof;" and "Of course this is not a proof!" These responses gave us the perfect opportunity to discuss the notion of an "informal" geometry proof, the role of technology in an informal proof, and the necessity of a formal proof. It is important for PSTs to engage in such discussions since they will be helping their future students "select and use various types of reasoning and methods of proof" (NCTM, 2000, p. 342). Subsequently, we investigate other dynamic constructions of geometric representations of the Pythagorean theorem and associated formal proofs to help PSTs see other representations of the mathematical structure of the Pythagorean theorem.
Take Advantage of Technology
Activities should take advantage of the capabilities of technology, and hence should extend beyond or significantly enhance what could be done without technology. Technology enables users to explore topics in more depth (e.g., interconnect mathematics topics, write programs, devise multiple proofs and solutions) and in more interactive ways (e.g., simulations, data collection with probes). Technology also makes accessible the study of mathematics topics that were previously impractical, such as recursion and regression, by removing computational constraints.
Using technology to teach the same mathematical topics in fundamentally the same ways that could be taught without technology does not strengthen students' learning of mathematics and belies the usefulness of technology. Furthermore, using technology to perform tasks that are just as easily or even better carried out without technology may actually be a hindrance to learning. Such uses of technology may convince teachers and administrators that preparing teachers to use technology is not worth the considerable effort and expense necessary to do so.
This guideline supports the Technology Principle of the NCTM Principles and Standards for School Mathematics . "Teachers should use technology to enhance their students learning opportunities by selecting or creating mathematical tasks that take advantage of what technology can do efficiently and well—graphing, visualizing, and computing" (NCTM, 2000, p. 25).
Example: Exploring Sierpinski Polygons
The following activity introduces our PSTs to Sierpinski Triangles and related fractals . Sierpinski's Triangle, first introduced by Waclaw Sierpinski in 1916, can be constructed in several ways. One way is to start with an equilateral triangle and a randomly chosen point inside, on, or outside the triangle. Next, randomly choose a vertex of the triangle and place a new point half way between the initial point and the randomly chosen vertex. Then, again randomly choose a vertex of the triangle and place a third point half way between the second point and the randomly chosen vertex. The triangle is constructed by continuing this recursive process, that is, by randomly choosing a vertex of the triangle and placing a point half way between that vertex and the previous point.
We ask our PSTs to construct a Sierpinski Triangle of 20 points using a triangle template and tossing one die to randomly determine vertices, observe their results, and predict what the triangle would look like if they were able to continue the process for a total of 5,000 points. Clearly, plotting 5,000 points is impractical to do by hand, so we developed our Sierpinski Polygon program in MicroWorlds (Logo Computer Systems, Inc., 1997) to carry out this process. Figure 4 shows Sierpinski's triangle constructed with 7,525 points. (Click on the figure caption to see an animation of the construction.
Figure 4. Sierpinski's Triangle
We ask our PSTs to analyze this triangle and comment on their observations. Next, we ask them to make predictions about and use the program to investigate various "What if" questions (e.g., What if we started with a hexagon? What if we used a ratio of one third instead of one-half? What if we used ratios greater than one?). They then explore various shapes and ratios, and determine the ratio, as a function of the number of sides of a polygon, which gives the "best" Sierpinski polygon. Figure 5 shows a triangle constructed with a ratio of 1.5. (Click on the figure caption to see an animation of the construction.
Figure 5. Sierpinski Triangle with r = 1.5
We use this Sierpinski activity as a springboard to engage PSTs in mathematical discussions of constrained randomness, "best" polygons, and fractals and self-similarity. Our pedagogical discussions focus on the role of fractal geometry in the mathematics curriculum, how this activity connects to local and national curriculum standards, connections between aesthetics and mathematics, and the usefulness and place of "What if" questions in mathematics instruction to help students appreciate how mathematicians advance their discipline.
Connect Mathematics Topics
Technology-augmented activities should facilitate mathematical connections in two ways: (a) interconnect mathematics topics and (b) connect mathematics to real-world phenomena. Technology "blurs some of the artificial separations among some topics in algebra, geometry and data analysis by allowing students to use ideas from one area of mathematics to better understand another area of mathematics" (NCTM, 2000, p. 26). Many school mathematics topics can be used to model and resolve situations arising in the physical, biological, environmental, social, and managerial sciences. Many mathematics topics can be connected to the arts and humanities as well. Appropriate use of technology can facilitate such applications by providing ready access to real data and information, by making the inclusion of mathematics topics useful for applications more practical (e.g., regression and recursion), and by making it easier for teachers and students to bring together multiple representations of mathematics topics. This guideline supports the curriculum standards of the NCTM (1989, 2000).
Example: Connecting Infinite Series and Geometric Constructions
The Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989) stated, "The interplay between geometry and algebra strengthens students' abilities to formulate and analyze problems from situations both with in and outside mathematics" (p. 161). One way students can investigate the connections between geometric and algebraic representations is with infinite series. In this activity, PSTs explore connections between convergent infinite geometric series and recursive processes of subdividing regular polygonal regions. These recursive processes result in the construction of Baravelle Spirals. In this context PSTs learn to construct recursive scripts with the Sketchpad. Figure 6 illustrates the recursive process of (a) constructing the midpoints of the sides of an innermost equilateral triangle, (b) connecting these midpoints to generate four small congruent equilateral triangles (each one fourth the area of the larger triangle), and (c) coloring a sequence of successively smaller triangles in a clockwise rotation. PSTs use this process to visualize and subsequently prove that:
(Click on the Figure 6 caption to see an animation of this construction.)
Figure 6. Sketchpad sketch of a Baravelle spiral generated from an equilateral triangle
PSTs explore infinite series by computing several partial sums initially and then following with more formal calculations based on mathematical limits. They discuss the role and advantages of geometric representations, instructional strategies for teaching infinite series, partial sums and convergent series, and how this activity connects to both state and national algebra and geometry standards.
Example: Connect Mathematics With Social Studies
To illustrate how technology can facilitate connections between mathematics and other disciplines we present our PSTs with the following information from a story in our local newspaper:
On Sunday, March 12, 2000 the Charlottesville Daily Progress contained an article reporting the results of a Washington Post-ABC News Poll. The article reported "Gore leads Bush by 48 percent to 45 percent, a statistical tie." It also reported, "A total of 1,218 adults, including 999 self-described registered voters, were interviewed" and the "margin of error for the overall results is plus or minus 3 percentage points, and slightly larger for results based on only the registered voters."
We ask PSTs to interpret, in their own words, what is meant by a "margin of error of plus or minus 3 percentage points" and what is meant by "a statistical tie." They are then asked to discuss the appropriate statistical test for such a poll, assess the accuracy of the newspapers' reporting, and comment on the appropriateness of the sample size. Using their graphing calculators, PSTs check the numbers in the article, run similar numbers for several different sample sizes, and make informed comments about the questions presented to them. Figure 9 shows that the newspaper reporting was accurate, with a confidence interval ranging from approximately .45 to .51 (.48 ± .03).
Figure 7. 95% Confidence Interval for p = .48.
Next we suggest they consider the algebraic representation of the standard error s as a function of sample size n , s ( n )= Ö ( p (1- p )/ n ), and graph this relationship for this particular sample proportion (p = .48) and confidence interval (.95) on their graphing calculators. By tracing the graph they observe numerically the margins of error ( y -coordinate) associated with various sample sizes ( x -coordinate). Furthermore, they see from the graphical representation that there is a point of diminishing returns. The animation below shows the margins of error for sample sizes of 100, 500 and 1000. (Click on the Figure 8 caption to view the animation.)
Figure 8. Graph of the 95% Standard Error.
The PSTs then discuss the trade-offs involved with various sample sizes and determine reasonable samples for presidential and other polls (e.g., Does it make sense to poll a sample of 10,000?).
Initially, many PSTs struggle with the above questions, even though they have recently studied the underlying statistical ideas (e.g., sampling distribution and confidence interval) in mathematical statistics courses. They often fail to connect the abstract mathematics they are learning in class with real-world uses. This presents us with a "pedagogical moment" —a time to discuss why they were able to operate successfully with these concepts in a mathematics class, yet struggle with applying them outside of class. Our PSTs then analyze and discuss why their mathematics education did not prepare them to apply mathematics concepts to real situations and devise instructional experiences to help their students to see how mathematics is connected to other disciplines.
Other class discussions center around how such an activity is consistent with state and national mathematics standards, how such activities should be sequenced in a unit on confidence intervals, and what value the technology has in such an activity. PSTs see that the technology facilitates this exploration by reducing the effort and time associated with performing repeated calculations and graphing and, thus, frees students to observe and analyze results and patterns. The Technology Principle of the NCTM Principles and Standards for School Mathematics emphasizes this point: "The computational capacities of technological tools extends the ranges of problems accessible to students and also enables them to execute routine procedures quickly and accurately, thus allowing more time for conceptualization and modeling" (NCTM, 2000, p. 24).
Incorporate Multiple Representations
Activities should incorporate multiple representations of mathematical topics. Research shows that many students have difficulty connecting the verbal, graphical, numerical and algebraic representations of mathematical functions (Goldenberg, 1988; Leinhardt et al., 1990). Appropriate use of technology can be effective in helping students make such connections (e.g., connecting tabulated data to graphs and curves of best fit, generating sequences and series numerically, algebraically, and geometrically). "We, as mathematics educators, should make the best use of multiple representations, especially those enhanced by the use of technology, encourage and help our students to apply multiple approaches to mathematical problem solving and engage them in creative thinking" (Jiang & McClintock, 2000, p.19).
Example: Exploring Maximum Area
An example of using technology to explore multiple representations involves a popular problem in which students find the maximum area for a rectangular pigpen given a fixed amount of fencing. To begin the investigation in our methods course, PSTs are given a handful of square tiles (1" x 1"), asked to build all possible rectangular pigpens with a perimeter of 24 inches, record the dimensions and resulting area, and verbalize a possible relationship. As the PSTs note, the maximum area (36 in 2 ) is obtained with a 6"x 6" pigpen. We then pose the question: "With any given perimeter, does a square pigpen always provide the maximum area?"
The spreadsheet in Figure 9 allows students to explore this question by systematically varying the length of side X for a fixed perimeter and observing the subsequent numerical, geometrical, and graphical representations. The PSTs also change the value of the perimeter and then vary X to find the associated maximum. Through repeated experimentation within this Microsoft Excel (Microsoft, 2000) environment, PSTs recognize that the value of the parameter X resulting in the maximum area of the rectangle is one fourth of the given perimeter P (e.g., P = 60, X = 15: P = 100, X = 25). PSTs can extend this problem through calculus by using algebraic formulas for perimeter and area, symbolic manipulation, and the first derivative of the equation Area = X ( P- 2 X )/2 with respect to X . (Click on the Figure 9 caption to see an animation of this exploration.)
Figure 9. Linked multiple representations of the maximum area problem.
The common "pigpen" problem allows PSTs to investigate and deepen their understanding of the relationship between perimeter and area with a multitude of representations. They begin with a manipulative geometric exploration, tabular recordings and a verbal description of the relationship. They then move to a digital environment with numerical , graphical , and geometrical representations and finally use a paper-and-pencil algebraic representation and symbolic manipulations to confirm the relationship. This investigation is used to prompt discussions with PSTs about (a) the benefits and drawbacks of using multiple representations, including manipulatives, in understanding mathematical relationships and (b) the importance of connecting mathematical topics from middle school through calculus.
Illustrations in Action
Our five guidelines are not independent of one another. All the examples given within each of the guidelines above also illustrate one or more of the other guidelines. The examples used were intended to provide quick illustrations highlighting a particular guideline. A more in-depth look at two illustrations in action will highlight the interconnectedness of the guidelines and provide examples of rich investigations in our secondary methods course.
Illustration in Action 1: Linear Relationships and Regression
The study of linear functions has been a mainstay and an important building block in secondary school mathematics. The advent of powerful technology tools has extended the study of linear functions to include a data-analysis technique called linear regression, in which data points clustering in a linear fashion on a scatterplot can be "best fit" by a line to describe the relationship between the two variables. The most commonly used technique, least-squares regression, can be quickly performed on graphing calculators, computer graphing software, and standard spreadsheets. Although technology places this computationally intense topic within the reach of all students (Vonder Embse, 1997), it is important that PSTs and their future students have a conceptual understanding of what it means to perform a least-square regression on data points and how to interpret the results.
We use the following activity to introduce students to linear regression in the context of examining a data set on smoking and lung cancer. We first give the PSTs the data set and ask them to do a qualitative analysis and conjecture a relationship between the two variables. Because this data is clustered by occupational groups, PSTs can discuss a variety of social and environmental factors that could affect either variable (e.g., professionals tend to have a low smoking rate, furnace and mill workers have the highest rate of death from lung cancer). One conjecture that PSTs make is that, as the rate for smoking increases, the rate for death from lung cancer also seems to increase. To help analyze this conjecture, the PSTs learn how to sort data in ascending order and create scatterplots in a spreadsheet. This allows them to see trends and examples of data that are contrary to the conjectured trend. The PSTs learn how to change the scale on the axes to represent the data in favorable and unfavorable ways for the tobacco industry and discuss the implications from each representation. (See Figure 10.)
Figure 10. Data displayed (a) favorable and (b) unfavorable for the tobacco industry.
The focus of the lesson then turns to comparing their analyses with each representation (table and graph) and mathematically describing a possible relationship between the variables. Questions about the "strength" of relationship lead into a discussion about the Pearson product-moment correlation coefficient ( r ), which gives a mathematical index of the strength of the relationship between two variables. Having secondary students understand the correlation coefficient poses "a dilemma, unknown a generation ago, for mathematics teachers" (Coes, 1995, p. 758). Instead of rotely memorizing the meaning of r , we have our PSTs use the Correlation Visualized spreadsheet (Neuwirth, 1995) shown in Figure 11 to construct their own interpretation of the values of r, and discuss how they could use this visualization to help their students understand the meaning of a correlation coefficient. (Click on the Figure 11 caption to view an animation of the changes in the scatterplot.)
Figure 11. The Correlation Visualized spreadsheet
The PSTs use the Correlation Visualized spreadsheet to estimate the value of r for the smoking and lung cancer data and then use the Excel function CORREL( y range, x range) to calculate the exact value of r . We discuss how to interpret the value ( r = 0.71) in terms of the data and using the square of the correlation coefficient ( r 2 = 0.49) to describe the proportion of variance in one variable (smoking) that is associated with the variance in the other variable (death from lung cancer). With a moderately strong value for r , it is reasonable to consider predictive questions such as "If an occupational group has a smoking index of 120, what would you predict their mortality index to be?" The question of prediction encourages the PSTs to use the numerical and graphical representations of data to make an estimate for the predicted mortality index and leads to a discussion of using a line to approximate the relationship and facilitate the prediction process. After using the drawing tools to place an estimated line of best fit on the scatterplot and using algebraic techniques to determine an equation for the line, the PSTs learn how to add a linear regression trendline to a scatterplot in Excel . (See Figure 12). The PSTs then compare their estimated line and equation with the ones calculated through Excel , and compare the predictions made from both techniques.
Figure 12. Scatterplot including estimated and calculated linear trendlines
The comparison of the PSTs' estimated line and the one calculated through Excel prompts an investigation into why the calculated line is the "best." We have the PSTs use a multimedia activity from the ExploreMath.com website to investigate how to minimize the sum of the squared error (numerically and geometrically) in fitting a line to a set of data points. The website includes linked algebraic, numeric, and graphic representations that can change dynamically through direct manipulation with all of the representations. The visualization of the squared error represented geometrically is a powerful tool in facilitating conceptual understanding of the least-squares regression model. (Click on the Figure 13 caption to see an animation of the activity.)
Figure 13. ExploreMath file illustrating a least-squares linear regression line
This investigation of the relationship between smoking and lung cancer also includes discussions about residuals and the effect of outliers on the linear regression equation. The PSTs learn many mathematics concepts and technology skills and engage in meaningful discourse about their own learning in this activity and the pedagogical issues surrounding teaching these topics and using the technology to effectively promote conceptual understanding, rather than mindless "speedy" calculations.
This "illustration-in-action" is rich in mathematics and pedagogy discussions and uses a variety of technology tools to facilitate the teaching and learning process. The PSTs learn how to use the technology tools in the context of the investigation, when the needed technology skill is appropriate (e.g., adding a trendline to a scatterplot). By focusing on conceptual understanding of both correlation coefficient and least-squares regression, we are modeling appropriate pedagogy and sound mathematical practices. We take advantage of technology as a vehicle for both procedural and conceptual understanding through using multiple representations, as well as facilitating classroom discussions. In addition, the mathematics is connected to a real-world controversial issue and also connects the study of linear functions in algebra to statistical techniques of linear regression.
Illustration in Action 2: Trigonometric Graphs
The study of trigonometry, which translates verbatim as "triangle measurement," began more than 2,000 years ago, partially as a means to solving land surveying problems. Traditionally, trigonometry is introduced as the ratios of the lengths of sides of a right triangle. This is commonly the first time the students hear the "SOH-CAH-TOA" acronym to remember the three common trigonometric ratios:
Because most students are not introduced to these ratios as functions of angle measurement until precalculus courses, the connections between the "right triangle" trigonometry and the graphs of functions of sine, cosine and tangent are not apparent. The following description is an illustration of an activity we use in our secondary methods class to connect right triangle and functional representations of trigonometry.
Using The Geometer's Sketchpad , PSTs construct a unit circle centered at the origin. We ask them to investigate a group of right triangles where: (a) the length of the hypotenuse is equal to the radius of the circle, (b) the vertex of the right angle is confined to the x-axis, and (c) a vertex is at the origin. Working in pairs, the PSTs construct a right triangle that satisfies these three conditions, comparing their sketch to the one in Figure 14.
Figure 14. Constructed right triangle with reference < DAB
Our PSTs drag point B around the circle and focus their attention on the lengths of sides a and b . They record qualitative descriptions of the lengths of sides a and b, as point B is dragged around the circle. As they discover, it is difficult to describe specifics about lengths a and b without being able to pinpoint locations around the unit circle. Introducing the measurement of angle DAB helps them describe where point B is located on the circle. One point of conflict arises when the PSTs observe that The Geometer's Sketchpad displays angle measurement as a number between -180° and 180°, rather than between 0° and 360°, to which they are accustomed. We discuss the possible conceptual and instructional problems that could arise in a high school classroom from this aspect of the technology. The PSTs then drag point B around the circle to find the maximum and minimum lengths of sides a and b .
In their sketches, the PSTs notice the lengths of the sides of the triangle changing as point B is dragged around the circle and devise a plan to graph the length of side a versus the measure of angle DAB geometrically . Once they finish the construction, we have the PSTs predict the shape of the graph on paper before activating their sketch (See Figure 15), and then reconcile any differences between their predicted graph and the graph sketched by the Sketchpad . (Click on the Figure 15 caption to activate the sketch.)
Figure 15. Sketchpad file of a geometric construction of the sine function
As an in-class exercise or for homework, we have the PSTs construct a graph of the length of side b versus the measure of angle DAB to illustrate the graph of the cosine function. This geometric construction is similar to the previous construction; however it is more challenging. Once the PSTs successfully construct the graph of side b versus the measure of angle DAB , we qualitatively and quantitatively compare and contrast the sine and cosine functions using a sketch similar to Figure 16. This investigation of the relationship between the right triangle and functional representations of trigonometry extend to activities challenging the PSTs to develop a geometric plan to construct the graphs of the other four trigonometric functions using the Sketchpad .
Figure 16. Sketchpad file of a geometric construction of the sine and cosine functions
PSTs follow these constructions with discussions of the various representations of trigonometric functions and how they relate to secondary students' conceptions and misconceptions. They then discuss how such constructions can be integrated into instruction. Subsequently, we continue our investigation of trigonometric functions with a graphing calculator activity that illustrates how technology facilitates use of mathematics to model real-world phenomena, in this case connecting trigonometry to physical geography. Together, these activities provide opportunities for the PSTs to view "mathematical ideas from multiple perspectives" (NCTM, 2000, p.24).
The PSTs examine the average monthly temperature data given in Table 1 and describe and interpret patterns in the data over one year.
Table 1Average Monthly Temperatures
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Jan
Feb
Mar
Apr
Wash. DC
54
64
73
77
75
68
57
46
37
34
36
45
54
Verhoyansk
5
32
54
57
48
36
5
-35
-53
-57
-48
-25
5
Buenos Aries
63
55
48
50
52
55
59
66
72
73
73
70
63
Next, they plot the data on their graphing calculators and compare the shape of their scatterplots with their descriptions and interpretations. We discuss mathematical models of the data and the general form of the sine function, y = A sin( B ( x + C ))+ D . We ask our PSTs to determine a "best fit" sine curve for the Washington data incrementally , by determining one coefficient at a time and graphing the resulting curves. Click on the Figure 17 caption to see an animation of an incremental curve fit that first adjusts the amplitude ( A ), then the vertical shift ( D ), then the period ( B ), and finally the horizontal shift ( C).
Figure 17. Incremental curve fitting
Our PSTs next discuss the fit of this curve and then compare it with the least squares sine regression equation determined by using the graphing calculator feature. (See Figure 18.)
Figure 18. Calculator screenshot of the calculation of a sine regression
PSTs are then able to estimate from the scatterplots the coefficients for the Verhoyansk and Buenos Aires data and later verify their estimates. We then discuss the relationships between the coefficients for the best-fit equations for the three cities, and the geographical locations of these cities (e.g., latitude, proximity to an ocean). We discuss advantages of the various forms of representation and how they should be connected in secondary mathematics instruction.
PSTs are often "rusty" with trigonometric equations, and from this activity they develop better understandings of the effect of each of the coefficients on the graph of the sine function. We discuss the teaching of this topic and how real-world phenomena can be incorporated into instruction. This is followed by investigations of periodicity in other contexts. For example, we use calculator-based probes to collect and display data from physical experiments (e.g., a light intensity probe to investigate cycles per second with florescent lights and an amplifier to investigate cycles per second of notes generated by tuning forks).
This "illustration-in-action" solidifies what we believe to be an appropriate use of technology in mathematics teaching. The PSTs learn how to use the technology tools (e.g., the trace , animation and movement features of Sketchpad) in the context of the investigation. Focusing on relationships between the right triangle and functional representations of trigonometric functions demonstrates worthwhile mathematics with appropriate pedagogy . Furthermore, this activity takes advantage of technology to connect trigonometric topics traditionally taught in geometry to those addressed in the pre-calculus curriculum, and connect trigonometry to another discipline, in each case using multiple representations of trigonometric relationships.
Conclusion
The preparation of secondary mathematics teachers who are able to use technology to enhance students' learning of mathematics is not a trivial matter. PSTs need to develop technology skills, enhance and extend their knowledge of mathematics with technological tools, and become critical developers and users of technology-enabled pedagogy. The creation of guidelines and materials represents a starting point in our efforts to prepare PSTs to use technology appropriately
We believe that our guidelines for the use of technology in mathematics teaching reflect the views of most mathematics educators and that our materials have the potential to provide mathematics educators with concrete starting places for integrating meaningful technology activities in their teacher education programs. Internal and external field-testing of our materials indicate that mathematics educators have found the activities useful in mathematics content courses, mathematics methods courses, and courses designed specifically on technology and mathematics education. However, we are now striving to create models for best using such materials in teacher education programs to help PSTs use technology to enhance their students' learning of mathematics.
We invite readers to access all of our graphing calculator, Excel , Geometer's Sketchpad , and MicroWorlds materials at the Curry Center for Technology and Teacher Education mathematics web site. We encourage you to use our activities in your courses for preservice and inservice mathematics teachers and send us your feedback and suggestions.
The Curry Center for Technology and Teacher Education at the University of Virginia is developing materials to help preservice secondary mathematics, social studies, and science teachers (PSTs) learn to incorporate technology appropriately into their teaching. The focus of the mathematics team is to devise activities that will prepare secondary teachers to use technology to enhance and extend their students' learning of mathematics. In this article we discuss the approach to developing and using materials for this purpose.
Use of Technology in Teacher Education
Technology is being incorporated into teacher education in numerous ways. Not surprisingly, there are different ways to categorize the various approaches taken by teacher educators to bring technology into their programs. One way to categorize these approaches is according to the primary user or controller of the technology—the teacher educator, the teacher, or the student. In some uses of technology in teacher education, the teacher educator is the primary user of the technology. For example, some teacher educators use multi-media case studies of rich teaching episodes to help PSTs analyze teaching and learning environments, and some use technology to present information or to demonstrate explorations. In many teacher education programs the teacher is being prepared to be the primary user of technology. For example, PSTs are being prepared to use technology productivity tools for word processing, grade and record keeping, web page production, and presentations. Also, many PSTs are using subject-specific software and websites to create presentations, lectures, lessons, and assessments. A third approach to incorporating technology in teacher education is to prepare PSTs to have their future students use technology to investigate concepts and solve meaningful problems in the content areas. For example, in the area of mathematics, PSTs are learning how to guide their students to use technologies such as spreadsheets, graphing calculators, dynamic geometry programs, and playable websites to explore mathematics concepts and use mathematics to solve problems in applied contexts.
The three uses of technology in teacher education presented above are connected with different purposes and all can lead to better teacher effectiveness and improved student learning. Thus, all are important. However, it has been our experience that the most direct and effective way to use technology to bring about enhanced student learning of mathematics is to prepare PSTs to incorporate into their teaching an array of activities that engage students in mathematical thinking facilitated by technological tools. Hence, in our preparation of secondary PSTs we emphasize the third use, in which ultimately the student is the primary user, and to some degree, the second use, in which the teacher is the primary user. Our materials reflect these emphases and thus are being developed around significant mathematical activities for school students.
In our classes, PSTs complete activities that, with some modification, are appropriate for secondary mathematics courses. We then use these completed activities to anchor class discussions of issues connected with secondary curriculum and instruction, national and state standards, sequencing of topics, the role of technology, and assessment. In the course of completing these activities, PSTs not only learn how to use the technology, but also how to incorporate technology into their teaching.
Guidelines for Technology-Based Activity Development
In the early phase of our work, we devised a set of guidelines to shape our development of mathematics activities and materials (Garofalo, Shockey, Harper, & Drier, 1999). The five guidelines below reflect what we believe to be appropriate uses of technology in mathematics teaching:
introduce technology in context
address worthwhile mathematics with appropriate pedagogy
take advantage of technology
connect mathematics topics
incorporate multiple representations
Each of these guidelines is discussed below and illustrated with one or more of our activities.
Introduce Technology in Context
Features of technology, whether mathematics-specific or more generic, should be introduced and illustrated in the context of meaningful content-based activities. Teaching a set of technology or software-based skills and then trying to find mathematical topics for which they might be useful is comparable to teaching a set of procedural mathematical skills and then giving a collection of "word problems" to solve using the procedures. Such an approach can obscure the purpose of learning and using technology, make mathematics appear as an afterthought, and lead to contrived activities. The use of technology in mathematics teaching is not for the purpose of teaching about technology, but for the purpose of enhancing mathematics teaching and learning with technology. Furthermore, in our experience, teachers who learn to use technology while exploring relevant mathematics topics are more likely to see its potential benefits and use it in their subsequent teaching. This guideline is in accord with the first recommendation of the President's Committee of Advisors on Science and Technology, Panel on Educational Technology (1997): "Focus on learning with technology, not about technology" (p. 7).
Example: Simulating Freefall With Parametric Quadratic Equations
In this activity connecting quadratic equations and projectile motion, PSTs are introduced to the parametric graphing features of graphing calculators. PSTs are asked to derive an expression for the height of an object dropped from 500m above the surface of the Earth, as a function of time. They are then asked to construct a graph of this relationship, first with paper and pencil and then with graphing calculators. Our PSTs are able to derive a correct equation and generate a graph similar to those in Figure 1. (Click on the caption of the screenshot of the graph in the figure to see the graph being drawn.)
Figure 1. Casio 9850 Plus graphing calculator screenshots of freefall equation and graph
The graph in Figure 1 is appropriate, considering that the x -axis represents time and the y -axis represents the height of the object. However, many students often fail to fully attend to axis variables and sometimes interpret such a graph as a picture of the situation being represented. They interpret this graph as implying that the path of the object is outward and downward rather than as straight downward. Such a misinterpretation is referred to as an iconic interpretation (Kerslake, 1977; Leinhardt, Zaslavsky, & Stein, 1990) and is prevalent with secondary school students. We challenge our PSTs to generate a graph that simulates the actual path of a freefalling object. This involves use of parametric equations, which all of our PSTs have studied in calculus, but most have forgotten or have not considered for high school use. Rather than teach PSTs to use the graphing calculator parametric features ahead of time and apply here, we introduce these features in this context, where PSTs can see its direct applicability and usefulness. Figure 2 shows the parametric equation and graph that simulates the object's path. (Click on the caption of the graph screenshot in the figure to see the graph being drawn.)
Figure 2. Calculator screenshots of parametric freefall equation and graph
PSTs then compare and contrast the graphs. Subsequently, we use the graphs as springboards to discuss various aspects of visual representations (e.g., connected versus plot graphing), iconic misrepresentations, incorporation of parametric equations in the curriulum, and use of parametric equations to enrich the treatment of other school mathematics topics. PSTs then apply these features to tasks involving horizontal motion and angular projectile motion.
Address Worthwhile Mathematics with Appropriate Pedagogy
Content-based activities using technology should address worthwhile mathematics concepts, procedures, and strategies, and should reflect the nature and spirit of mathematics. Activities should support sound mathematical curricular goals and should not be developed merely because technology makes them possible. Indeed, the use of technology in mathematics teaching should support and facilitate conceptual development, exploration, reasoning and problem solving, as described by the NCTM (1989, 1991, 2000).
Technology should not be used to carry out procedures without appropriate mathematical and technological understanding (e.g., inserting rote formulas into a spreadsheet to demonstrate population growth). Nor should it be used in ways that can distract from the underlying mathematics (e.g., adding so many bells and whistles into a Power Point slideshow that the mathematics gets lost). In other words, mathematical content and pedagogy should not be compromised.
Another way to prevent technology use from compromising mathematics is to encourage users to connect their experiential findings to more formal aspects of mathematics. For example, students using software to explore geometric shapes and relationships should be asked to use previously proved theorems to validate their empirical results, or use their new findings to propose new conjectures. Mathematical notions of "proof " and "rigor" need to be addressed as well. In other words, technology should not influence students to take things at face value or to become what Schoenfeld (1985) referred to as "naïve empiricists." This guideline is in accord with the second recommendation of the President's Committee of Advisors on Science and Technology, Panel on Educational Technology (1997): "Emphasize content and pedagogy, and not just hardware" (p. 7).
Example: Exploring the Pythagorean Theorem
On way we explore issues regarding appropriate pedagogy with technology with our PSTs is with the following activity. The emphasis of the activity is more on the principles of teaching the mathematics than on the mathematics itself. PSTs discuss how this topic is traditionally taught to students, as a rote memorization of a 2 + b 2 = c 2 without any conceptual understanding that a 2 , b 2 and c 2 represent areas of squares with sides length a , b , and c . We then discuss how technology could be used to enhance the students' understanding of the theorem, and guide them through a model lesson of how the Pythagorean theorem could be taught using The Geometer's Sketchpad (Jackiw, 1997).
PSTs are first asked to use the Sketchpad to construct a right triangle, measure each side, and numerically confirm the a 2 + b 2 = c 2 relationship. They then learn how to use Sketchpad features to script a construction of a square. Next the PSTs play back their scripts to place a square on each of the sides of the right triangle and add measurements to create a construction similar to the one in Figure 3. The dynamic Sketchpad environment allows the PSTs to drag the triangle's vertices or sides to manipulate the construction, keeping the characteristics of the geometric figures intact. As the construction changes, the sum of the area of the squares on the legs of the right triangle always remain equal to the area of the square on the hypotenuse of the right triangle. (Click on Figure 3 caption to see an animation of this construction.)
Figure 3. Sketchpad file of the Pythagorean theorem
PSTs then discuss connections between the different representations of the Pythagorean Theorem, advantages of each representation in teaching this topic, and benefits of using the Sketchpad to create and manipulate the constructions. This activity illustrates a point made in the Technology Principle of the NCTM Principles and Standards : "Technology also provides a focus as students discuss with one another and with their teacher the objects on the screen and the effects of the various dynamic transformations that technology allows (NCTM, 2000, p. 24)."
We typically pose the following question to our PSTs: Does the manipulation of this construction constitute a mathematical proof of the Pythagorean theorem? One semester we received four types of responses: "I think it's a proof;" "I don't know if this is a proof;" "I hope it's a proof;" and "Of course this is not a proof!" These responses gave us the perfect opportunity to discuss the notion of an "informal" geometry proof, the role of technology in an informal proof, and the necessity of a formal proof. It is important for PSTs to engage in such discussions since they will be helping their future students "select and use various types of reasoning and methods of proof" (NCTM, 2000, p. 342). Subsequently, we investigate other dynamic constructions of geometric representations of the Pythagorean theorem and associated formal proofs to help PSTs see other representations of the mathematical structure of the Pythagorean theorem.
Take Advantage of Technology
Activities should take advantage of the capabilities of technology, and hence should extend beyond or significantly enhance what could be done without technology. Technology enables users to explore topics in more depth (e.g., interconnect mathematics topics, write programs, devise multiple proofs and solutions) and in more interactive ways (e.g., simulations, data collection with probes). Technology also makes accessible the study of mathematics topics that were previously impractical, such as recursion and regression, by removing computational constraints.
Using technology to teach the same mathematical topics in fundamentally the same ways that could be taught without technology does not strengthen students' learning of mathematics and belies the usefulness of technology. Furthermore, using technology to perform tasks that are just as easily or even better carried out without technology may actually be a hindrance to learning. Such uses of technology may convince teachers and administrators that preparing teachers to use technology is not worth the considerable effort and expense necessary to do so.
This guideline supports the Technology Principle of the NCTM Principles and Standards for School Mathematics . "Teachers should use technology to enhance their students learning opportunities by selecting or creating mathematical tasks that take advantage of what technology can do efficiently and well—graphing, visualizing, and computing" (NCTM, 2000, p. 25).
Example: Exploring Sierpinski Polygons
The following activity introduces our PSTs to Sierpinski Triangles and related fractals . Sierpinski's Triangle, first introduced by Waclaw Sierpinski in 1916, can be constructed in several ways. One way is to start with an equilateral triangle and a randomly chosen point inside, on, or outside the triangle. Next, randomly choose a vertex of the triangle and place a new point half way between the initial point and the randomly chosen vertex. Then, again randomly choose a vertex of the triangle and place a third point half way between the second point and the randomly chosen vertex. The triangle is constructed by continuing this recursive process, that is, by randomly choosing a vertex of the triangle and placing a point half way between that vertex and the previous point.
We ask our PSTs to construct a Sierpinski Triangle of 20 points using a triangle template and tossing one die to randomly determine vertices, observe their results, and predict what the triangle would look like if they were able to continue the process for a total of 5,000 points. Clearly, plotting 5,000 points is impractical to do by hand, so we developed our Sierpinski Polygon program in MicroWorlds (Logo Computer Systems, Inc., 1997) to carry out this process. Figure 4 shows Sierpinski's triangle constructed with 7,525 points. (Click on the figure caption to see an animation of the construction.
Figure 4. Sierpinski's Triangle
We ask our PSTs to analyze this triangle and comment on their observations. Next, we ask them to make predictions about and use the program to investigate various "What if" questions (e.g., What if we started with a hexagon? What if we used a ratio of one third instead of one-half? What if we used ratios greater than one?). They then explore various shapes and ratios, and determine the ratio, as a function of the number of sides of a polygon, which gives the "best" Sierpinski polygon. Figure 5 shows a triangle constructed with a ratio of 1.5. (Click on the figure caption to see an animation of the construction.
Figure 5. Sierpinski Triangle with r = 1.5
We use this Sierpinski activity as a springboard to engage PSTs in mathematical discussions of constrained randomness, "best" polygons, and fractals and self-similarity. Our pedagogical discussions focus on the role of fractal geometry in the mathematics curriculum, how this activity connects to local and national curriculum standards, connections between aesthetics and mathematics, and the usefulness and place of "What if" questions in mathematics instruction to help students appreciate how mathematicians advance their discipline.
Connect Mathematics Topics
Technology-augmented activities should facilitate mathematical connections in two ways: (a) interconnect mathematics topics and (b) connect mathematics to real-world phenomena. Technology "blurs some of the artificial separations among some topics in algebra, geometry and data analysis by allowing students to use ideas from one area of mathematics to better understand another area of mathematics" (NCTM, 2000, p. 26). Many school mathematics topics can be used to model and resolve situations arising in the physical, biological, environmental, social, and managerial sciences. Many mathematics topics can be connected to the arts and humanities as well. Appropriate use of technology can facilitate such applications by providing ready access to real data and information, by making the inclusion of mathematics topics useful for applications more practical (e.g., regression and recursion), and by making it easier for teachers and students to bring together multiple representations of mathematics topics. This guideline supports the curriculum standards of the NCTM (1989, 2000).
Example: Connecting Infinite Series and Geometric Constructions
The Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989) stated, "The interplay between geometry and algebra strengthens students' abilities to formulate and analyze problems from situations both with in and outside mathematics" (p. 161). One way students can investigate the connections between geometric and algebraic representations is with infinite series. In this activity, PSTs explore connections between convergent infinite geometric series and recursive processes of subdividing regular polygonal regions. These recursive processes result in the construction of Baravelle Spirals. In this context PSTs learn to construct recursive scripts with the Sketchpad. Figure 6 illustrates the recursive process of (a) constructing the midpoints of the sides of an innermost equilateral triangle, (b) connecting these midpoints to generate four small congruent equilateral triangles (each one fourth the area of the larger triangle), and (c) coloring a sequence of successively smaller triangles in a clockwise rotation. PSTs use this process to visualize and subsequently prove that:
(Click on the Figure 6 caption to see an animation of this construction.)
Figure 6. Sketchpad sketch of a Baravelle spiral generated from an equilateral triangle
PSTs explore infinite series by computing several partial sums initially and then following with more formal calculations based on mathematical limits. They discuss the role and advantages of geometric representations, instructional strategies for teaching infinite series, partial sums and convergent series, and how this activity connects to both state and national algebra and geometry standards.
Example: Connect Mathematics With Social Studies
To illustrate how technology can facilitate connections between mathematics and other disciplines we present our PSTs with the following information from a story in our local newspaper:
On Sunday, March 12, 2000 the Charlottesville Daily Progress contained an article reporting the results of a Washington Post-ABC News Poll. The article reported "Gore leads Bush by 48 percent to 45 percent, a statistical tie." It also reported, "A total of 1,218 adults, including 999 self-described registered voters, were interviewed" and the "margin of error for the overall results is plus or minus 3 percentage points, and slightly larger for results based on only the registered voters."
We ask PSTs to interpret, in their own words, what is meant by a "margin of error of plus or minus 3 percentage points" and what is meant by "a statistical tie." They are then asked to discuss the appropriate statistical test for such a poll, assess the accuracy of the newspapers' reporting, and comment on the appropriateness of the sample size. Using their graphing calculators, PSTs check the numbers in the article, run similar numbers for several different sample sizes, and make informed comments about the questions presented to them. Figure 9 shows that the newspaper reporting was accurate, with a confidence interval ranging from approximately .45 to .51 (.48 ± .03).
Figure 7. 95% Confidence Interval for p = .48.
Next we suggest they consider the algebraic representation of the standard error s as a function of sample size n , s ( n )= Ö ( p (1- p )/ n ), and graph this relationship for this particular sample proportion (p = .48) and confidence interval (.95) on their graphing calculators. By tracing the graph they observe numerically the margins of error ( y -coordinate) associated with various sample sizes ( x -coordinate). Furthermore, they see from the graphical representation that there is a point of diminishing returns. The animation below shows the margins of error for sample sizes of 100, 500 and 1000. (Click on the Figure 8 caption to view the animation.)
Figure 8. Graph of the 95% Standard Error.
The PSTs then discuss the trade-offs involved with various sample sizes and determine reasonable samples for presidential and other polls (e.g., Does it make sense to poll a sample of 10,000?).
Initially, many PSTs struggle with the above questions, even though they have recently studied the underlying statistical ideas (e.g., sampling distribution and confidence interval) in mathematical statistics courses. They often fail to connect the abstract mathematics they are learning in class with real-world uses. This presents us with a "pedagogical moment" —a time to discuss why they were able to operate successfully with these concepts in a mathematics class, yet struggle with applying them outside of class. Our PSTs then analyze and discuss why their mathematics education did not prepare them to apply mathematics concepts to real situations and devise instructional experiences to help their students to see how mathematics is connected to other disciplines.
Other class discussions center around how such an activity is consistent with state and national mathematics standards, how such activities should be sequenced in a unit on confidence intervals, and what value the technology has in such an activity. PSTs see that the technology facilitates this exploration by reducing the effort and time associated with performing repeated calculations and graphing and, thus, frees students to observe and analyze results and patterns. The Technology Principle of the NCTM Principles and Standards for School Mathematics emphasizes this point: "The computational capacities of technological tools extends the ranges of problems accessible to students and also enables them to execute routine procedures quickly and accurately, thus allowing more time for conceptualization and modeling" (NCTM, 2000, p. 24).
Incorporate Multiple Representations
Activities should incorporate multiple representations of mathematical topics. Research shows that many students have difficulty connecting the verbal, graphical, numerical and algebraic representations of mathematical functions (Goldenberg, 1988; Leinhardt et al., 1990). Appropriate use of technology can be effective in helping students make such connections (e.g., connecting tabulated data to graphs and curves of best fit, generating sequences and series numerically, algebraically, and geometrically). "We, as mathematics educators, should make the best use of multiple representations, especially those enhanced by the use of technology, encourage and help our students to apply multiple approaches to mathematical problem solving and engage them in creative thinking" (Jiang & McClintock, 2000, p.19).
Example: Exploring Maximum Area
An example of using technology to explore multiple representations involves a popular problem in which students find the maximum area for a rectangular pigpen given a fixed amount of fencing. To begin the investigation in our methods course, PSTs are given a handful of square tiles (1" x 1"), asked to build all possible rectangular pigpens with a perimeter of 24 inches, record the dimensions and resulting area, and verbalize a possible relationship. As the PSTs note, the maximum area (36 in 2 ) is obtained with a 6"x 6" pigpen. We then pose the question: "With any given perimeter, does a square pigpen always provide the maximum area?"
The spreadsheet in Figure 9 allows students to explore this question by systematically varying the length of side X for a fixed perimeter and observing the subsequent numerical, geometrical, and graphical representations. The PSTs also change the value of the perimeter and then vary X to find the associated maximum. Through repeated experimentation within this Microsoft Excel (Microsoft, 2000) environment, PSTs recognize that the value of the parameter X resulting in the maximum area of the rectangle is one fourth of the given perimeter P (e.g., P = 60, X = 15: P = 100, X = 25). PSTs can extend this problem through calculus by using algebraic formulas for perimeter and area, symbolic manipulation, and the first derivative of the equation Area = X ( P- 2 X )/2 with respect to X . (Click on the Figure 9 caption to see an animation of this exploration.)
Figure 9. Linked multiple representations of the maximum area problem.
The common "pigpen" problem allows PSTs to investigate and deepen their understanding of the relationship between perimeter and area with a multitude of representations. They begin with a manipulative geometric exploration, tabular recordings and a verbal description of the relationship. They then move to a digital environment with numerical , graphical , and geometrical representations and finally use a paper-and-pencil algebraic representation and symbolic manipulations to confirm the relationship. This investigation is used to prompt discussions with PSTs about (a) the benefits and drawbacks of using multiple representations, including manipulatives, in understanding mathematical relationships and (b) the importance of connecting mathematical topics from middle school through calculus.
Illustrations in Action
Our five guidelines are not independent of one another. All the examples given within each of the guidelines above also illustrate one or more of the other guidelines. The examples used were intended to provide quick illustrations highlighting a particular guideline. A more in-depth look at two illustrations in action will highlight the interconnectedness of the guidelines and provide examples of rich investigations in our secondary methods course.
Illustration in Action 1: Linear Relationships and Regression
The study of linear functions has been a mainstay and an important building block in secondary school mathematics. The advent of powerful technology tools has extended the study of linear functions to include a data-analysis technique called linear regression, in which data points clustering in a linear fashion on a scatterplot can be "best fit" by a line to describe the relationship between the two variables. The most commonly used technique, least-squares regression, can be quickly performed on graphing calculators, computer graphing software, and standard spreadsheets. Although technology places this computationally intense topic within the reach of all students (Vonder Embse, 1997), it is important that PSTs and their future students have a conceptual understanding of what it means to perform a least-square regression on data points and how to interpret the results.
We use the following activity to introduce students to linear regression in the context of examining a data set on smoking and lung cancer. We first give the PSTs the data set and ask them to do a qualitative analysis and conjecture a relationship between the two variables. Because this data is clustered by occupational groups, PSTs can discuss a variety of social and environmental factors that could affect either variable (e.g., professionals tend to have a low smoking rate, furnace and mill workers have the highest rate of death from lung cancer). One conjecture that PSTs make is that, as the rate for smoking increases, the rate for death from lung cancer also seems to increase. To help analyze this conjecture, the PSTs learn how to sort data in ascending order and create scatterplots in a spreadsheet. This allows them to see trends and examples of data that are contrary to the conjectured trend. The PSTs learn how to change the scale on the axes to represent the data in favorable and unfavorable ways for the tobacco industry and discuss the implications from each representation. (See Figure 10.)
Figure 10. Data displayed (a) favorable and (b) unfavorable for the tobacco industry.
The focus of the lesson then turns to comparing their analyses with each representation (table and graph) and mathematically describing a possible relationship between the variables. Questions about the "strength" of relationship lead into a discussion about the Pearson product-moment correlation coefficient ( r ), which gives a mathematical index of the strength of the relationship between two variables. Having secondary students understand the correlation coefficient poses "a dilemma, unknown a generation ago, for mathematics teachers" (Coes, 1995, p. 758). Instead of rotely memorizing the meaning of r , we have our PSTs use the Correlation Visualized spreadsheet (Neuwirth, 1995) shown in Figure 11 to construct their own interpretation of the values of r, and discuss how they could use this visualization to help their students understand the meaning of a correlation coefficient. (Click on the Figure 11 caption to view an animation of the changes in the scatterplot.)
Figure 11. The Correlation Visualized spreadsheet
The PSTs use the Correlation Visualized spreadsheet to estimate the value of r for the smoking and lung cancer data and then use the Excel function CORREL( y range, x range) to calculate the exact value of r . We discuss how to interpret the value ( r = 0.71) in terms of the data and using the square of the correlation coefficient ( r 2 = 0.49) to describe the proportion of variance in one variable (smoking) that is associated with the variance in the other variable (death from lung cancer). With a moderately strong value for r , it is reasonable to consider predictive questions such as "If an occupational group has a smoking index of 120, what would you predict their mortality index to be?" The question of prediction encourages the PSTs to use the numerical and graphical representations of data to make an estimate for the predicted mortality index and leads to a discussion of using a line to approximate the relationship and facilitate the prediction process. After using the drawing tools to place an estimated line of best fit on the scatterplot and using algebraic techniques to determine an equation for the line, the PSTs learn how to add a linear regression trendline to a scatterplot in Excel . (See Figure 12). The PSTs then compare their estimated line and equation with the ones calculated through Excel , and compare the predictions made from both techniques.
Figure 12. Scatterplot including estimated and calculated linear trendlines
The comparison of the PSTs' estimated line and the one calculated through Excel prompts an investigation into why the calculated line is the "best." We have the PSTs use a multimedia activity from the ExploreMath.com website to investigate how to minimize the sum of the squared error (numerically and geometrically) in fitting a line to a set of data points. The website includes linked algebraic, numeric, and graphic representations that can change dynamically through direct manipulation with all of the representations. The visualization of the squared error represented geometrically is a powerful tool in facilitating conceptual understanding of the least-squares regression model. (Click on the Figure 13 caption to see an animation of the activity.)
Figure 13. ExploreMath file illustrating a least-squares linear regression line
This investigation of the relationship between smoking and lung cancer also includes discussions about residuals and the effect of outliers on the linear regression equation. The PSTs learn many mathematics concepts and technology skills and engage in meaningful discourse about their own learning in this activity and the pedagogical issues surrounding teaching these topics and using the technology to effectively promote conceptual understanding, rather than mindless "speedy" calculations.
This "illustration-in-action" is rich in mathematics and pedagogy discussions and uses a variety of technology tools to facilitate the teaching and learning process. The PSTs learn how to use the technology tools in the context of the investigation, when the needed technology skill is appropriate (e.g., adding a trendline to a scatterplot). By focusing on conceptual understanding of both correlation coefficient and least-squares regression, we are modeling appropriate pedagogy and sound mathematical practices. We take advantage of technology as a vehicle for both procedural and conceptual understanding through using multiple representations, as well as facilitating classroom discussions. In addition, the mathematics is connected to a real-world controversial issue and also connects the study of linear functions in algebra to statistical techniques of linear regression.
Illustration in Action 2: Trigonometric Graphs
The study of trigonometry, which translates verbatim as "triangle measurement," began more than 2,000 years ago, partially as a means to solving land surveying problems. Traditionally, trigonometry is introduced as the ratios of the lengths of sides of a right triangle. This is commonly the first time the students hear the "SOH-CAH-TOA" acronym to remember the three common trigonometric ratios:
Because most students are not introduced to these ratios as functions of angle measurement until precalculus courses, the connections between the "right triangle" trigonometry and the graphs of functions of sine, cosine and tangent are not apparent. The following description is an illustration of an activity we use in our secondary methods class to connect right triangle and functional representations of trigonometry.
Using The Geometer's Sketchpad , PSTs construct a unit circle centered at the origin. We ask them to investigate a group of right triangles where: (a) the length of the hypotenuse is equal to the radius of the circle, (b) the vertex of the right angle is confined to the x-axis, and (c) a vertex is at the origin. Working in pairs, the PSTs construct a right triangle that satisfies these three conditions, comparing their sketch to the one in Figure 14.
Figure 14. Constructed right triangle with reference < DAB
Our PSTs drag point B around the circle and focus their attention on the lengths of sides a and b . They record qualitative descriptions of the lengths of sides a and b, as point B is dragged around the circle. As they discover, it is difficult to describe specifics about lengths a and b without being able to pinpoint locations around the unit circle. Introducing the measurement of angle DAB helps them describe where point B is located on the circle. One point of conflict arises when the PSTs observe that The Geometer's Sketchpad displays angle measurement as a number between -180° and 180°, rather than between 0° and 360°, to which they are accustomed. We discuss the possible conceptual and instructional problems that could arise in a high school classroom from this aspect of the technology. The PSTs then drag point B around the circle to find the maximum and minimum lengths of sides a and b .
In their sketches, the PSTs notice the lengths of the sides of the triangle changing as point B is dragged around the circle and devise a plan to graph the length of side a versus the measure of angle DAB geometrically . Once they finish the construction, we have the PSTs predict the shape of the graph on paper before activating their sketch (See Figure 15), and then reconcile any differences between their predicted graph and the graph sketched by the Sketchpad . (Click on the Figure 15 caption to activate the sketch.)
Figure 15. Sketchpad file of a geometric construction of the sine function
As an in-class exercise or for homework, we have the PSTs construct a graph of the length of side b versus the measure of angle DAB to illustrate the graph of the cosine function. This geometric construction is similar to the previous construction; however it is more challenging. Once the PSTs successfully construct the graph of side b versus the measure of angle DAB , we qualitatively and quantitatively compare and contrast the sine and cosine functions using a sketch similar to Figure 16. This investigation of the relationship between the right triangle and functional representations of trigonometry extend to activities challenging the PSTs to develop a geometric plan to construct the graphs of the other four trigonometric functions using the Sketchpad .
Figure 16. Sketchpad file of a geometric construction of the sine and cosine functions
PSTs follow these constructions with discussions of the various representations of trigonometric functions and how they relate to secondary students' conceptions and misconceptions. They then discuss how such constructions can be integrated into instruction. Subsequently, we continue our investigation of trigonometric functions with a graphing calculator activity that illustrates how technology facilitates use of mathematics to model real-world phenomena, in this case connecting trigonometry to physical geography. Together, these activities provide opportunities for the PSTs to view "mathematical ideas from multiple perspectives" (NCTM, 2000, p.24).
The PSTs examine the average monthly temperature data given in Table 1 and describe and interpret patterns in the data over one year.
Table 1Average Monthly Temperatures
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Jan
Feb
Mar
Apr
Wash. DC
54
64
73
77
75
68
57
46
37
34
36
45
54
Verhoyansk
5
32
54
57
48
36
5
-35
-53
-57
-48
-25
5
Buenos Aries
63
55
48
50
52
55
59
66
72
73
73
70
63
Next, they plot the data on their graphing calculators and compare the shape of their scatterplots with their descriptions and interpretations. We discuss mathematical models of the data and the general form of the sine function, y = A sin( B ( x + C ))+ D . We ask our PSTs to determine a "best fit" sine curve for the Washington data incrementally , by determining one coefficient at a time and graphing the resulting curves. Click on the Figure 17 caption to see an animation of an incremental curve fit that first adjusts the amplitude ( A ), then the vertical shift ( D ), then the period ( B ), and finally the horizontal shift ( C).
Figure 17. Incremental curve fitting
Our PSTs next discuss the fit of this curve and then compare it with the least squares sine regression equation determined by using the graphing calculator feature. (See Figure 18.)
Figure 18. Calculator screenshot of the calculation of a sine regression
PSTs are then able to estimate from the scatterplots the coefficients for the Verhoyansk and Buenos Aires data and later verify their estimates. We then discuss the relationships between the coefficients for the best-fit equations for the three cities, and the geographical locations of these cities (e.g., latitude, proximity to an ocean). We discuss advantages of the various forms of representation and how they should be connected in secondary mathematics instruction.
PSTs are often "rusty" with trigonometric equations, and from this activity they develop better understandings of the effect of each of the coefficients on the graph of the sine function. We discuss the teaching of this topic and how real-world phenomena can be incorporated into instruction. This is followed by investigations of periodicity in other contexts. For example, we use calculator-based probes to collect and display data from physical experiments (e.g., a light intensity probe to investigate cycles per second with florescent lights and an amplifier to investigate cycles per second of notes generated by tuning forks).
This "illustration-in-action" solidifies what we believe to be an appropriate use of technology in mathematics teaching. The PSTs learn how to use the technology tools (e.g., the trace , animation and movement features of Sketchpad) in the context of the investigation. Focusing on relationships between the right triangle and functional representations of trigonometric functions demonstrates worthwhile mathematics with appropriate pedagogy . Furthermore, this activity takes advantage of technology to connect trigonometric topics traditionally taught in geometry to those addressed in the pre-calculus curriculum, and connect trigonometry to another discipline, in each case using multiple representations of trigonometric relationships.
Conclusion
The preparation of secondary mathematics teachers who are able to use technology to enhance students' learning of mathematics is not a trivial matter. PSTs need to develop technology skills, enhance and extend their knowledge of mathematics with technological tools, and become critical developers and users of technology-enabled pedagogy. The creation of guidelines and materials represents a starting point in our efforts to prepare PSTs to use technology appropriately
We believe that our guidelines for the use of technology in mathematics teaching reflect the views of most mathematics educators and that our materials have the potential to provide mathematics educators with concrete starting places for integrating meaningful technology activities in their teacher education programs. Internal and external field-testing of our materials indicate that mathematics educators have found the activities useful in mathematics content courses, mathematics methods courses, and courses designed specifically on technology and mathematics education. However, we are now striving to create models for best using such materials in teacher education programs to help PSTs use technology to enhance their students' learning of mathematics.
We invite readers to access all of our graphing calculator, Excel , Geometer's Sketchpad , and MicroWorlds materials at the Curry Center for Technology and Teacher Education mathematics web site. We encourage you to use our activities in your courses for preservice and inservice mathematics teachers and send us your feedback and suggestions.
READINGS...
Learning to teach mathematics at the middle and secondary levels should include many opportunities for teachers to learn how to use technology to better understand mathematics themselves and promote students' learning of mathematical concepts with technology-enabled pedagogy. This article highlights work done in a variety of pre-service and in-service mathematics teacher education courses to help teachers use commonly available spreadsheets as an interactive exploratory learning tool. Several examples of teachers' subsequent use of spreadsheets in their own teaching are also discussed.
"Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students' learning" (National Council of Teachers of Mathematics ([NCTM], 2000, p. 11). This statement is one of six principles described in the NCTM's document, Principles and Standards for School Mathematics. The use of the word essential in the statement has many implications for school mathematics, as well as preservice and in-service mathematics teacher education. Not only are teachers charged with a vision of transforming their teaching and students' learning of mathematics, but teacher educators are challenged with the task of preparing teachers who can utilize technology as an essential tool in developing a deep understanding of mathematics, for themselves and for their students. Recent trends in teacher education have emphasized the importance of learning with technology rather than learning about technology. This implies that teachers should learn to use a computer as a cognitive tool to enhance student learning of content material (e.g., mathematics, social studies, or science) rather than acquiring isolated skills in basic computing applications (e.g., word processing, database, spreadsheets, or hypermedia) or merely learning a specific programming language (Abramovich & Drier, 1999). Thus, to promote the use of technology for students' conceptual development, mathematics teachers should learn how to use widely available software, such as spreadsheets, as a conceptual teaching and learning tool (Abramovich, et al., 1999). During the past decade, spreadsheets have been used in teacher education and K-12 classrooms to explore a variety of mathematical concepts and to help students use numerical and graphical methods to solve problems (Abramovich, 1995; Abramovich & Nabors, 1998; Clements & Samara, 1997; Dugdale, 1998; Neuwirth, 1995). These uses of spreadsheets allow students to explore alternative solution processes that go beyond symbolic manipulation and can provide students with a deeper understanding of concepts embedded in a problem. One unique use of spreadsheets is the ability to interactively model and simulate mathematical situations. In much the same way scientists use a laboratory to discover and test scientific laws, mathematics teachers can use spreadsheets to create dynamic experiential environments for discovering mathematical relationships. Such activities can facilitate students' engagement with mathematical concepts and their conceptualization of relationships among numerical, graphical, and algebraic representations. The flexibility and power of the Microsoft Excel (1997) spreadsheet software allows teachers to be engaged in meaningful mathematical activities that, in turn, empower them to use spreadsheets as interactive environments in their own teaching.
Teaching with the use of spreadsheets help them to explore more on the development of the skills of the skills especially on how they will deal on problems that uses spreadsheets. However, today’s students are engaging more on the use of computer.
On the other hand, calculators and other technological tools, such as computer algebra systems, interactive geometry software, applets, spreadsheets, and interactive presentation devices, are vital components of a high-quality mathematics education. With guidance from effective mathematics teachers, students at different levels can use these tools to support and extend mathematical reasoning and sense making, gain access to mathematical content and problem-solving contexts, and enhance computational fluency. In a well-articulated mathematics program, students can use these tools for computation, construction, and representation as they explore problems. The use of technology also contributes to mathematical reflection, problem identification, and decision making.
However, there are some arguments that the use of technology cannot replace conceptual understanding, computational fluency, or problem-solving skills. In a balanced mathematics program, the strategic use of technology enhances mathematics teaching and learning. Teachers must be knowledgeable decision makers in determining when and how their students can use technology most effectively. All schools and mathematics programs should provide students and teachers with access to instructional technology, including appropriate calculators, computers with mathematical software, Internet connectivity, handheld data-collection devices, and sensing probes. Curricula and courses of study should incorporate instructional technology in learning outcomes, lesson plans, and assessments of students’ progress.
"Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students' learning" (National Council of Teachers of Mathematics ([NCTM], 2000, p. 11). This statement is one of six principles described in the NCTM's document, Principles and Standards for School Mathematics. The use of the word essential in the statement has many implications for school mathematics, as well as preservice and in-service mathematics teacher education. Not only are teachers charged with a vision of transforming their teaching and students' learning of mathematics, but teacher educators are challenged with the task of preparing teachers who can utilize technology as an essential tool in developing a deep understanding of mathematics, for themselves and for their students. Recent trends in teacher education have emphasized the importance of learning with technology rather than learning about technology. This implies that teachers should learn to use a computer as a cognitive tool to enhance student learning of content material (e.g., mathematics, social studies, or science) rather than acquiring isolated skills in basic computing applications (e.g., word processing, database, spreadsheets, or hypermedia) or merely learning a specific programming language (Abramovich & Drier, 1999). Thus, to promote the use of technology for students' conceptual development, mathematics teachers should learn how to use widely available software, such as spreadsheets, as a conceptual teaching and learning tool (Abramovich, et al., 1999). During the past decade, spreadsheets have been used in teacher education and K-12 classrooms to explore a variety of mathematical concepts and to help students use numerical and graphical methods to solve problems (Abramovich, 1995; Abramovich & Nabors, 1998; Clements & Samara, 1997; Dugdale, 1998; Neuwirth, 1995). These uses of spreadsheets allow students to explore alternative solution processes that go beyond symbolic manipulation and can provide students with a deeper understanding of concepts embedded in a problem. One unique use of spreadsheets is the ability to interactively model and simulate mathematical situations. In much the same way scientists use a laboratory to discover and test scientific laws, mathematics teachers can use spreadsheets to create dynamic experiential environments for discovering mathematical relationships. Such activities can facilitate students' engagement with mathematical concepts and their conceptualization of relationships among numerical, graphical, and algebraic representations. The flexibility and power of the Microsoft Excel (1997) spreadsheet software allows teachers to be engaged in meaningful mathematical activities that, in turn, empower them to use spreadsheets as interactive environments in their own teaching.
Teaching with the use of spreadsheets help them to explore more on the development of the skills of the skills especially on how they will deal on problems that uses spreadsheets. However, today’s students are engaging more on the use of computer.
On the other hand, calculators and other technological tools, such as computer algebra systems, interactive geometry software, applets, spreadsheets, and interactive presentation devices, are vital components of a high-quality mathematics education. With guidance from effective mathematics teachers, students at different levels can use these tools to support and extend mathematical reasoning and sense making, gain access to mathematical content and problem-solving contexts, and enhance computational fluency. In a well-articulated mathematics program, students can use these tools for computation, construction, and representation as they explore problems. The use of technology also contributes to mathematical reflection, problem identification, and decision making.
However, there are some arguments that the use of technology cannot replace conceptual understanding, computational fluency, or problem-solving skills. In a balanced mathematics program, the strategic use of technology enhances mathematics teaching and learning. Teachers must be knowledgeable decision makers in determining when and how their students can use technology most effectively. All schools and mathematics programs should provide students and teachers with access to instructional technology, including appropriate calculators, computers with mathematical software, Internet connectivity, handheld data-collection devices, and sensing probes. Curricula and courses of study should incorporate instructional technology in learning outcomes, lesson plans, and assessments of students’ progress.
Reflection on the seminar…
EDUCATION for the FUTURE
Last August 28, 2008 we had a seminar with regard to the “Education for the Future”. It was composed of two parts. The first part was all about readings and its importance. The speaker was Dr. Sunga who is the Dean of the College of Education.
Reading? It is a well-known fact that when there were no televisions or computers, reading was a primary leisure activity. People would spend hours reading books and travel to lands far away-in their minds. The only tragedy is that, with time, people have lost their skill and passion to read. There are many other exciting and thrilling options available, aside from books. And that is a shame because reading offers a productive approach to improving vocabulary and word power. It is advisable to indulge in at least half an hour of reading a day to keep abreast of the various styles of writing and new vocabulary.
It is observed that children and teenagers who love reading have comparatively higher IQs. They are more creative and do better in school and college. It is recommended that parents to inculcate the importance of reading to their children in the early years. Reading is said to significantly help in developing vocabulary, and reading aloud helps to build a strong emotional bond between parents and children. The children who start reading from an early age are observed to have good language skills, and they grasp the variances in phonics much better.
Reading helps in mental development and is known to stimulate the muscles of the eyes. Reading is an activity that involves greater levels of concentration and adds to the conversational skills of the reader. It is an indulgence that enhances the knowledge acquired, consistently. The habit of reading also helps readers to decipher new words and phrases that they come across in everyday conversations. The habit can become a healthy addiction and adds to the information available on various topics. It helps us to stay in-touch with contemporary writers as well as those from the days of yore and makes us sensitive to global issues.
On the other hand, the second speaker was Dr. Macarandang. She discussed the uses of computer in teaching. However the uses of computer in teaching were underscored now.
In connection to that, here are the advantages of computers in education.
1. "Computers improve both teaching and student achievement."
2. "Computer literacy should be taught as early as possible; otherwise students will be left behind."
3. "Technology programs leverage support from the business community - badly needed today because schools are increasingly starved for funds."
4. "To make tomorrow's work force competitive in an increasingly high-tech world, learning computer skills must be a priority."
5. "Work with computers - particularly using the Internet - brings students valuable connections with teachers, other schools and students, and a wide network of professionals around the globe. Those connections spice the school day with a sense of real-world relevance, and broaden the educational community."
Computers made the work of teacher easy but it does not replace teachers. The learning of the students requires touch by the teachers not by the computers.
Last August 28, 2008 we had a seminar with regard to the “Education for the Future”. It was composed of two parts. The first part was all about readings and its importance. The speaker was Dr. Sunga who is the Dean of the College of Education.
Reading? It is a well-known fact that when there were no televisions or computers, reading was a primary leisure activity. People would spend hours reading books and travel to lands far away-in their minds. The only tragedy is that, with time, people have lost their skill and passion to read. There are many other exciting and thrilling options available, aside from books. And that is a shame because reading offers a productive approach to improving vocabulary and word power. It is advisable to indulge in at least half an hour of reading a day to keep abreast of the various styles of writing and new vocabulary.
It is observed that children and teenagers who love reading have comparatively higher IQs. They are more creative and do better in school and college. It is recommended that parents to inculcate the importance of reading to their children in the early years. Reading is said to significantly help in developing vocabulary, and reading aloud helps to build a strong emotional bond between parents and children. The children who start reading from an early age are observed to have good language skills, and they grasp the variances in phonics much better.
Reading helps in mental development and is known to stimulate the muscles of the eyes. Reading is an activity that involves greater levels of concentration and adds to the conversational skills of the reader. It is an indulgence that enhances the knowledge acquired, consistently. The habit of reading also helps readers to decipher new words and phrases that they come across in everyday conversations. The habit can become a healthy addiction and adds to the information available on various topics. It helps us to stay in-touch with contemporary writers as well as those from the days of yore and makes us sensitive to global issues.
On the other hand, the second speaker was Dr. Macarandang. She discussed the uses of computer in teaching. However the uses of computer in teaching were underscored now.
In connection to that, here are the advantages of computers in education.
1. "Computers improve both teaching and student achievement."
2. "Computer literacy should be taught as early as possible; otherwise students will be left behind."
3. "Technology programs leverage support from the business community - badly needed today because schools are increasingly starved for funds."
4. "To make tomorrow's work force competitive in an increasingly high-tech world, learning computer skills must be a priority."
5. "Work with computers - particularly using the Internet - brings students valuable connections with teachers, other schools and students, and a wide network of professionals around the globe. Those connections spice the school day with a sense of real-world relevance, and broaden the educational community."
Computers made the work of teacher easy but it does not replace teachers. The learning of the students requires touch by the teachers not by the computers.
Reflection on the Lecture…
Last August 27, 2008, we conducted a lecture about the use of MS Excel. It was part of our midterm grade to conduct a lecture as part of the requirement in Seminar of Technology in Mathematics. Our audience was the freshmen and sophomore math major who were free that day. We rendered the lecture in room 308 (speech lab).
So lets take a look the advantages and disadvantages of MS excel.
The following are the advantages of Ms Excel:
· First and foremost, if you have the Microsoft Office in your Pc, you don’t need anything else to be installed in your computer. You are good to go and can start wit your programming right away.
· Second, if you are a beginner, you can easily learn the Excel VBA Programming because you are half familiar with programming platform that you are going to use, which is excel.
· Third, you can start with small Excel database application and as the need grows especially on records that are getting huge, you can still use the same Excel PPLICATION and upgrade your database platform from Excel to other high-end databases like Access, SQL server, Oracle, MySOL, etc.
· Fourth, Excel is capable of connecting directly to OLAP databases and can integrate in Pivot Tables.
· Fifth, you don’t need to create your own financial modules. Excel is rich in financial functions liked Fixed Assets Depreciation, Amortization, etc.
· Sixth, Excel is portable. You can send it to someone through email.
The following are the disadvantages of Ms Excel:
· Not easily sharable (except for Tools, Share workbook)
· Non relational (except for lookup & reference commands like VLOOKUP)
· Not scalable (limited to 65536 rows in Excel 2003)
· Limited Data Validation (except for Data Validation)
· No Forms (except for VBA or Data form)
· Poor Reports (except for Pivot Tables)
So with that, I know now the advantages and disadvantages of using excel. On the other hand, I must consider its limitations. I must use not use Excel all the time. Excel makes less time in computations and accurate, but using manual method makes me sharper and helps me to develop my inner skills.
So lets take a look the advantages and disadvantages of MS excel.
The following are the advantages of Ms Excel:
· First and foremost, if you have the Microsoft Office in your Pc, you don’t need anything else to be installed in your computer. You are good to go and can start wit your programming right away.
· Second, if you are a beginner, you can easily learn the Excel VBA Programming because you are half familiar with programming platform that you are going to use, which is excel.
· Third, you can start with small Excel database application and as the need grows especially on records that are getting huge, you can still use the same Excel PPLICATION and upgrade your database platform from Excel to other high-end databases like Access, SQL server, Oracle, MySOL, etc.
· Fourth, Excel is capable of connecting directly to OLAP databases and can integrate in Pivot Tables.
· Fifth, you don’t need to create your own financial modules. Excel is rich in financial functions liked Fixed Assets Depreciation, Amortization, etc.
· Sixth, Excel is portable. You can send it to someone through email.
The following are the disadvantages of Ms Excel:
· Not easily sharable (except for Tools, Share workbook)
· Non relational (except for lookup & reference commands like VLOOKUP)
· Not scalable (limited to 65536 rows in Excel 2003)
· Limited Data Validation (except for Data Validation)
· No Forms (except for VBA or Data form)
· Poor Reports (except for Pivot Tables)
So with that, I know now the advantages and disadvantages of using excel. On the other hand, I must consider its limitations. I must use not use Excel all the time. Excel makes less time in computations and accurate, but using manual method makes me sharper and helps me to develop my inner skills.
thesis
CHAPTER 1
THE PROBLEM
Introduction
Technologies as link to new knowledge, resources and high order thinking skills have entered classrooms and schools worldwide. Personal computers, CD-ROMS, on line services, the World Wide Web and other innovative technologies have enriched curricula and have altered the types of teaching available in the classroom. Schools’ access to technology is increasing steadily everyday and most of these newer technologies are now used even in traditional classrooms (Bilbao et al., 2006).
The use of technology in the classroom has never been underscored but now. However, survey data like that of Smith (1996); Harper (2001); and Lesh and Doerr (2000), suggest that technology remains poorly integrated into schools, despite massive acquisition of hardware. Some recent observations such as of Norton (1996); Cole (2002); and Posey (2001) indicate that the most frequent use of computers is for drill-and-skill practice that supplement existing curricula and instructional practices.
Some thirty years or more ago, the dominant model of teaching was directed instruction or lecture in which students memorize facts. Because of its limitations educationists began exploring the use of technology that supports models of teaching that emphasize learning with understanding and more active involvement. Thus a decision to use technology that is beyond facts-based, memorization-oriented curricula to curricula in which learning with understanding are emphasized was embraced. When to use technology, what technology to use, and for what purpose cannot be isolated from theories of teaching and learning that support learning with understanding (Bilbao et al., 2006).
On the other hand, (Goldman cited in Bilbao et.al., 2006) states that there are some roles of technology in achieving the goal of learning with understanding such as: (a) technology provides support to the solution of meaningful problems; (b) technology acts as cognitive support; and (c) technology promotes collaboration as well as independent learning.
Calculators and other technological tools, such as computer algebra systems, interactive geometry software, applets, spreadsheets, and interactive presentation devices, are vital components of a high-quality mathematics education. With guidance from effective mathematics teachers, students at different levels can use these tools to support and extend mathematical reasoning and sense making, gain access to mathematical content and problem-solving contexts, and enhance computational fluency. In a well-articulated mathematics program, students can use these tools for computation, construction, and representation as they explore problems. The use of technology also contributes to mathematical reflection, problem identification, and decision making (Cole et al., 2002).
However, there are some arguments that the use of technology cannot replace conceptual understanding, computational fluency, or problem-solving skills. In a balanced mathematics program, the strategic use of technology enhances mathematics teaching and learning. Teachers must be knowledgeable decision makers in determining when and how their students can use technology most effectively. All schools and mathematics programs should provide students and teachers with access to instructional technology, including appropriate calculators, computers with mathematical software, Internet connectivity, handheld data-collection devices, and sensing probes. Curricula and courses of study should incorporate instructional technology in learning outcomes, lesson plans, and assessments of students’ progress.
As a student of Mathematics, the researcher observed that the use of modern technology gadgets like computer and scientific calculator was employed and widely used among students. However, these observations made the researcher work on the effectiveness of the utilization of the said gadgets in learning mathematics. The traditional chalk and board are the technology used by the students and teachers in public schools and yet their level of understanding and learning Mathematics is not far behind from the level of the private school students who use modern technology gadgets in learning mathematics.
Which then is more useful: the modern – technology – aided Mathematics instructions or the traditional chalk and board method? What teaching strategies work for effective Mathematics learning? Does the use of computers and scientific calculators really help students?
These questions were examined in this study. In this research, the perceived effects of the utilization of modern technology gadgets in learning mathematics was analyzed for the attempt of providing concrete explanations of the effects of utilization of modern technology gadgets in learning mathematics.
Conceptual Framework
It is commonly observed that teaching strategies defined as the organized, orderly, and logical procedure in imparting knowledge and information among pupils or students (Zulueta, 2006), affect the learning of students. This somehow is true based on the researchers’ experience where he learned more with teachers who utilize varied teaching strategies that suit his needs in learning mathematics. Nowadays, the utilization of modern technology gadgets such as computer and scientific calculator seems to have a great effect on students’ learning.
Scientific calculator is an electronic calculator that has provisions for handling exponential, trigonometric, and sometimes other special functions in addition to performing arithmetic operations. On the other hand, computer is also an electronic devise design to manipulate data so that useful information can be generated (Cole et al., 2002).
To find out if these gadgets affect students’ learning, this study tried to trace the frequency of the use in varied teaching strategies. The assumption of the researcher was that, the frequency of using computer and scientific calculator in teaching, does not affect the learning of students.
This study started with determining the teaching strategy that used modern technology gadgets such as computer and scientific calculator. After determining these teaching strategies, the frequency of their use was identified. When the frequency of its utilization was identified. Their effects in learning mathematics were examined. This was done by relating it to its frequency.
The design of this conceptual model was an original work of the researcher and is not found in any book. This best suited the manner of analyzing the statement of the problem.
The concepts were made simple in the conceptual model on next page.
Teaching Strategies
Using Modern
Technology
Gadgets
Computer
Scientific Calculator
Frequency of
Modern Technology
Utilization
Effects of Using Modern
Technology Gadgets
In Learning
Mathematics
Figure 1. Conceptual Model of the Study
Statement of the Problem
The primary purpose of this study was to determine the perceived effects of the utilization of modern technology gadgets such as computers and scientific calculators in learning mathematics. It involved fourth year high school students of First Asia Institute of Technology and Humanities enrolled in Academic Year 2008 – 2009.
Specifically, this study sought answers to the following questions:
In what teaching strategies does the teacher utilize modern technology gadgets?
How often are these modern technology gadgets used?
How does the use of modern technology gadgets affect the respondents learning?
What are the perceived effects of using modern technology gadgets in learning mathematics?
Is there a significant relationship between the frequency of the utilization of modern technology gadgets and learning of mathematics?
Hypothesis
The study was guided by the null hypothesis:
There is no significant relationship between the frequency of the utilization of modern technology gadgets in the effective learning of mathematics
Significance of the Study
This research study would be useful to the following:
To the teachers, it would help them identify the teaching strategies suited to the needs of their students. The findings of this research would offer them understanding on how to utilize modern technology gadgets to appropriately address students learning style in mathematics.
To the parents, the result of this study would help them encourage their children to be keen on the varied styles of learning.
To the students, it would help them determine their preferred teaching strategies as well as the use of modern technology gadgets in learning.
To the Mathematics Heads, this study would help them identify the needs of the students with regard to the utilization of modern technology gadgets in the learning Mathematics.
Finally, this research would serve as reference material to other students and future researchers who will be interested to work on similar topic.
Scope and Delimitation of the Study
This study aimed to determine the perceived effects of the utilization of modern technology gadgets in learning mathematics. It involved one hundred two (102) fourth year high school students of First Asia Institute of Technology and Humanities enrolled in the Academic Year 2008 – 2009. They were chosen as respondents because it is the year level when Trigonometry is taken up which require the use of calculator in solving trigonometric functions.
This research aimed to point out the use of modern technology gadgets such as computer and scientific calculator. Computers are observed to be usually used in presenting the lesson, in research and video presentations. Similarly, scientific calculators are utilized in solving large numbers, factoring and games.
The teaching strategies made by the teacher were also analyzed because the researcher assumed that the teachers’ teaching strategies play important roles in students’ learning. This study further analyzed the frequency of using computers and calculators in teaching were believed to affect the respondents learning. This study also attempted to find the significant relationship between the frequency of the utilization of modern technology gadgets in the effective learning of mathematics.
This study only examined the use of two modern technology gadgets – computer and scientific calculator in learning mathematics. It did not cover any other types of gadgets like graphing calculator and projector.
The researcher used a validated, self-constructed questionnaire as an instrument in measuring responses of the respondents based on their perception about the effects of modern technology gadgets in their learning in mathematics.
Definition of Terms
The following terms were conceptually and operationally defined so as to give clearer understanding of some concepts used in the present study.
Analogy. It is a strategy that can be utilized in promoting comprehension and reasoning among students (Gagne, 1985). In this study, it is the strategy wherein the student will find out the pattern of the particular problem or how he/she will reason it out.
Computers. It is an electronic devise designed to manipulate data so that useful information can be generated (Cole et al., 2002). In this study, computer is used as one of the modern technology teaching gadgets used by the teacher in teaching mathematics.
Cooperative Learning. It is an instructional paradigm in which teams of students work on structured tasks (e.g., homework assignments, laboratory experiments, or design projects) under conditions that meet five criteria: positive interdependence, individual accountability, face-to-face interaction, appropriate use of collaborative skills, and regular self-assessment of team functioning (Johnson, Johnson, and Smith cited in Zulueta , 2006). In this study, it deals with the kind of group work wherein the students seek involvement and cooperation with one another.
Discovery Learning. It consists of instruction in which the learners draw their own conclusions from information they were able to glean by themselves as provided by their teachers or others (Clark and Starr cited in Zulueta , 2006). In this study, it is used as the students will discover for themselves the particular topic presented to them.
Discussion. This method allows for interaction between the teacher and the students as well as among themselves (Salandanan et al., 1996). In this study, it is one of the strategies where the students will have interaction between their teachers or among themselves.
Graphic Organizers. It is a collective group of strategies that provide visual representations as a means of organizing and presenting information. They make visible the thinking of the students. They help students represent abstract concepts and ideas in concrete forms. They display the relationships between pieces of information, connect new learning to prior learning and generally organize information into a more useful form (Salandanan et al., 1996). In this study, it refers to the visual materials used by the mathematics teacher.
Guided Discovery. It is the process that teachers used to introduce new materials, explore centers or areas of the classroom, and prepare students for various aspects of the curriculum (Salandanan et al., 1996). In this study, it is the teacher who is acting as a facilitator of learning and guiding his students in learning mathematical problem.
Lecture. It is a teaching procedure of classifying or explaining a major idea cast in the form of question or a problem (Bossing cited in Zulueta , 2006). In this study, it is the teacher’s teaching strategy when he is providing the students all the needed information without interaction at all.
Mathematics. It is the study of quantities and relations through the use of numbers and symbols (Norton et al., 1996). In this study, it is the subject or course that the researchers worked on.
Model-Making. It is a strategy that teachers can use in teaching application (Eby and Kujawa cited in Zulueta , 2006). In this study, it refers to the strategy wherein the students make a model for them to visualize the given worded problem.
Power point. A computer based media presentation using slideshow (Cole et al., 2002). In this study, it is one of the computer software that is used by the teacher in presenting the lesson.
Reasoning It is the cognitive process of looking for reasons for beliefs, conclusions, actions or feelings (Kirwin cited in de Leon, 2000). In this study, it refers to how the students reason out their feelings for something.
Research. It is defined as human activity based on intellectual application in the investigation of matter. The primary aim for applied research is discovering, interpreting, and the development of methods and systems for the advancement of human knowledge on a wide variety of scientific matters of our world and the universe (Trochim, 2006). In this study, it refers to the teaching strategy that helps the students develop their students’ thinking, reasoning and research skills.
School. It is an educational institution, private and public, undertaking educational operation with specific age-group of pupils or students pursuing defined studies at defined levels, receiving instruction from teachers, usually located in a building or a group of buildings in a particular physical or cyber site (Oreta, 2001). In this study, it refers to First Asia Institute of Technology and Humanities Unified School where the respondents are currently enrolled.
Scientific Calculators. It is a type of electronic calculator, usually but not always handheld, designed to calculate problems in science (especially physics), engineering, and mathematics. They have almost completely replaced slide rules in almost all traditional applications, and are widely used in both education and professional settings (Cole et al., 2002). In this study, it is one of the teaching gadgets used by the teacher and also the students for them to be guided for finding the accurate answer.
Students. It refers to those who enrolled in and who regularly attend an educational institution of secondary or higher level or a person engaged in formal study (Salandanan et al., 1996). In this study, it refers to the Fourth Year students of First Asia Institute of Technology and Humanities Unified School enrolled in Academic Year 2008 – 2009.
Teachers. They are the key-learning person who is responsible for supervising/ facilitating the learning processes and activities of the students (Oreta, 2001). In this study, it refers to all concerned mathematics teacher of First Asia Institute of Technology and Humanities.
Teaching Strategies. It refers to the organized, orderly, and logical procedure in imparting knowledge and information among pupils or students (Zulueta, 2006). In this study, it refers to the strategies used by the teacher in presenting or delivering the topic.
Technology. It refers to how we apply science slideshow (Cole et al., 2002). In this study, it refers to computer and scientific calculator as means of teaching and learning lessons in Mathematics quickly.
Trigonometry. It is a branch of mathematics that combines arithmetic, algebra and geometry (Frere cited in de Leon, 2000). In this study, it refers to the subject that requires the use of modern technology gadgets.
CHAPTER II
REVIEW OF LITERATURE
This chapter presents a review of existing and available literature as well as related studies on the perceived effects of utilization of modern technology gadgets in learning mathematics of fourth year high school students. Only those studies that showed certain aspects of relationship with the present study were selected and their findings were discussed. Its last part is a synthesis of all the reviewed literature.
Conceptual Literature
Definition and Use of Technology in Mathematics. The definition and use of technology in mathematics education is constantly evolving. Technology may refer to the use of graphing calculators, student response systems, online laboratories, simulations and visualizations, mathematical software, spreadsheets, multimedia, computers or the Internet, and other innovations yet to be discovered (Cole, 2002).
Technology can be used to learn mathematics, to do mathematics, and to communicate mathematical information and ideas. The Internet hosts a wealth of mathematical materials that are easily accessible through the use of search engines, creating additional avenues to enhance teaching and facilitate learning. Outside of class, students and faculty can pose problems and offer solutions through e-mail, chat rooms, or websites. Technology provides opportunities for educators to develop and nurture learning communities, embrace collaboration, provide community-based learning, and address diverse learning styles of students and teaching styles of teachers (NCTM, 2000).
The integration of appropriately used technology can enhance student understanding of mathematics through pattern recognition, connections, and dynamic visualizations. Electronic teaching activities can attract attention to the mathematics to be learned and promote the use of multiple methods. Learning can be enhanced with electronic questioning that engages students with technology in small groups and facilitates skills development through guided-discovery exercise sets. Using electronic devices for communication, all students can answer mathematics questions posed in class and instructors can have an instantaneous record of the answers given by each student. This immediate understanding of what students know, and don’t know, can direct the action of the instructor in the teaching session (NCTM, 2000).
According to Pea (cited in NCTM, 2000), cognitive technologies serve two transcendent functions. First, technologies have purpose functions. They serve to engage students in the activity of mathematical and scientific inquiry. This provides meaning for engagement, ownership of the mathematics and science being learned, and empowerment through the generation of personal agency. Technologies engage students in more powerful mathematical activity in a way that could not be approached without them.
But technologies are not by nature engaging. To achieve this quality, they must be both functional (teachers and students must be able to do with them something that they could not do without them), and they must increase communication and facilitate collaboration. Second, technologies have process functions. Some of the tools available for students should free up their working memory so that they are able to concentrate on problem formulation and modeling (Cole, 2002).
If a high school student is bogged down with computing or graphing, the big picture of number systems, functions, families of curves, etc., is lost. Other tools must provide opportunities for exploration and discovery. In a mediated learning environment, some agent (teacher, peer, and tool) must bridge the informal knowledge of the student and the formalism of mathematical and scientific structure. Still other tools must provide ways of representing mathematical and scientific models and linking representations to make the underlying commonalties transparent (Lesh cited in Cole, 2000).
A single technology rarely has all these process functions. However, a careful selection of tools and software as described in this article can help achieve the necessary complementarities. Two other features of cognitive technologies are necessary for the development of coherent mathematical and scientific structures. The first is what Roschelle (cited in Cole, 2000) called epistemic fidelity. This refers to the requirement that any teaching tool must reflect and develop understandings that are true to the field of study. Students’ mathematical and scientific activity should develop the kinds of understandings that experts in the field would recognize. Two caveats are in order. The road from novice to expert goes through several transformational periods and may not be immediately recognizable as important without an understanding of students’ cognitive development.
Second, the sophisticated knowledge of handed to students. The path taken is as much a part of expert understanding as the final product. The other necessary feature of cognitive technologies should focus the students’ attention on the mathematical structure of the experiences and provide them with a means of communicating their thinking about this structure to others. This is, in its basic form, the engagement of students in mathematical and scientific modeling. The vision that guides the integration of technology, science, and mathematics is the engagement of students in activity that elicits the development of mathematical and scientific models with a coherent epistemological framework. The movement from informal discovery to more formal models marks an authentic transition between the exploratory knowledge of the student, and the theoretical knowledge of the expert (Kozulin & Presseisen cited in Cole, 2000).
Teaching with Technology. In recent years integration of instructional technology (IT) into the university classroom has become a significant part of education. As such resources for educating and assisting faculty in this new arena have become crucial. Workshops on how to use software are not always enough. Teachers need to understand the ways in which these new tools can make a significant difference in student teaching (Cole, 2000).
Developments and advances in technology–hardware and software–have had a tremendous impact on our lives. The infusion of technology into education presents interesting opportunities for teaching and learning, especially in mathematics. Technology changes not only how mathematics is taught, but also when and what mathematics is taught (Posey, 2001).
In one review by Smith (1996), as can be inferred from the six principles, the kind of software and the way it is used are also crucial elements. Common features of the software used in this program are that it can be used by high school students; it is user friendly; it is designed for the kind of computers available in schools; and most important, students are in control, telling the computer what to do rather than the computer telling students what to do. The kinds of software used range from general purpose tools to specialized programs for science and mathematics learning. The particular software used can change from year to year. Typically, four or five kinds of technology are used in depth, including computer-based software and graphing calculators. Although prospective teachers become quite expert in the use of the technology, the main goal is that their future students use technology to explore concepts and solve problems in science and mathematics. Prospective teachers use it in conjunction with hands-on materials, such as geoboards and polyhedra, and activities such as paper folding. Use of natural objects and outdoor activities are also an important part of integration (Harper cited in Stiles, 2002).
However, in the study of Middleton and Goepfert (cited in Papert, 1980), there are six principles that guide the design of choosing the equipment and software, such as: (1) technologies are only tools. Technologies neither supplant the thought processes of students, nor do they make learning fun or easy. Technologies are instruments that should be used judiciously at the proper time in the proper place. (2) Technologies should enable students do what they could not do without them. When used appropriately, technologies help students expand their zone of proximal development. This can serve to make learning more intentional, powerful, and connected. In addition, computer technologies can represent situations unfeasible with other types of tools. (3) Technologies must be on hand all the time. The context, social setting, and tools that students use to construct their mathematical and scientific knowledge are inseparable from the knowledge itself. For technologies to be authentically integrated into students’ learning activity, they must be available when the question arises.
The fourth (4) principle is that tools should facilitate the creation of sharable, modifiable, transportable models of mathematical and scientific concepts (Lesh & Doerr, cited in Papert, 1980). Technologies facilitate the development of public records of thought. These records should be shared as students develop, refine, and test models of mathematical and scientific phenomena. It is crucial that students can modify them, as most models students construct in the beginning are either incomplete, or contain misconceptions. Through discourse, the shared model can be pared down into a workable model that can serve the class as a whole. (5) Sharing of data/resources should be simple. Technological systems should be user friendly. The mechanism of communication should not be more complex than the learning process itself. (6) The setup of the workstations should facilitate collaboration between students. As collaborative tools, technologies are imbedded within the geography, culture, and psychology of the classroom. The setup should facilitate collaborative inquiry, but also engage students in independent exploration.
The National Council of Teachers in Mathematics in USA (2000) stressed that technology can be used to learn mathematics, to do mathematics, and to communicate mathematical information and ideas. The Internet hosts a wealth of mathematical materials that are easily accessible through the use of search engines, creating additional avenues to enhance teaching and facilitate learning. Outside of class, students and faculty can pose problems and offer solutions through e-mail, chat rooms, or websites. Technology provides opportunities for educators to develop and nurture learning communities, embrace collaboration, provide community-based learning, and address diverse learning styles of students and teaching styles of teachers. The integration of appropriately used technology can enhance student understanding of mathematics through pattern recognition, connections, and dynamic visualizations. Electronic teaching activities can attract attention to the mathematics to be learned and promote the use of multiple methods. Learning can be enhanced with electronic questioning that engages students with technology in small groups and facilitates skills development through guided-discovery exercise sets. Using electronic devices for communication, all students can answer mathematics questions posed in class and instructors can have an instantaneous record of the answers given by each student. This immediate understanding of what students know, and don’t know, can direct the action of the instructor in the teaching session.
According to Cole (2002), technology helps students document the validity of their mathematical/critical thinking process, facilitating and enriching the learning processes and the development of problem-solving skills. The use of technology should be guided by consideration of what mathematics is to be learned, the ways students might learn it, the research related to successful practices, and the standards and recommendations recommended by professional organizations in education. Technology can be used by mathematics educators to enhance conceptual understanding through a comparison of verbal, numerical, symbolic, and graphical representations of the same problem.
Teaching with PowerPoint. Both courses have been regularly taught with PowerPoint to supplement lecture and discussion. The choice of PowerPoint as the technological solution was driven substantially by two factors: the suitability of a visual tool to studying politics in foreign countries, and the ease with which PowerPoint can be learned and integrated with existing course material. Comparative politics, the systematic study of domestic political conditions and tendencies in various countries, is ideally suited to a highly visual approach, particularly in imparting to American students a broad overview of the historical, social, and economic conditions which underlie the politics of countries such as Great Britain, France, Germany, the European Union, Japan, Russia, China, India, Nigeria and Mexico. Since the students in question in many cases have not been out of the United States, each unit is begun with a "virtual tour" of the country in PowerPoint, featuring photographs of people and places in the country in question, accompanied by a loop of national music. Clipart from Microsoft and other vendors is extensively used to illustrate the flags, crests, maps, currencies, and landmarks of the country and region. Pie charts are used to express ethnic diversity and budgetary allocations. Line and bar charts are used for economic trends and comparative public policy performance, such as budget deficits, health care expenditures, inflation and unemployment rates. In most cases these can be constructed within PowerPoint itself using the Microsoft Graph 5.0, but in a few instances of advanced mixed chart formats, the charts are created in Microsoft Excel and pasted into the PowerPoint slide. Photographs of political leaders from Microsoft Bookshelf, the World Wide Web and other sources are used to illustrate slides. Short video clips from CNN and other sources are inserted to add variety and a dynamic quality to the presentations. Finally, complex processes can be illustrated easily by use of the Auto shapes. It is also possible to build complex scenario presentations in which students must discuss and choose between difficult alternatives facing poor developing countries, in this case Nigeria. The choices made then branch off to other short presentations which explore the advantages and drawbacks of the choices made by the students (Sammons, 1995).
In the same text, a further advantage of teaching comparative politics with PowerPoint is that it is now possible to go beyond mere text as the focus of testing and broaden the scope of what teachers can expect students to learn and retain. Quizzes and tests can be presented as a PowerPoint presentation, and ask essay, fill-in or multiple-choice questions, reducing photocopying costs for departments in an era of diminishing resources and increased expectations. Furthermore, a PowerPoint quiz can test students' recognition of leaders, flags, and maps; such a quiz may involve an essay reacting to a chart, graph, or photograph, moving students beyond the goal of grasping secondary knowledge and toward reacting to and interpreting primary data.
One additional advantage of using PowerPoint is the ability to easily produce handout sheets with the bullet points clearly printed out. These were produced in the three-slides-per-page format, allowing students plenty of room to write additional information from class lecture and discussion. These sheets are then photocopied as a course pack by a local vendor and available to students at the beginning of the semester (Sammons, 1995).
In the same text student reactions to the use of PowerPoint have been overwhelmingly positive. Surveys distributed to students during the semester asked students their reactions to a variety of statements concerning the use of PowerPoint in the classroom, using a five-point feeling thermometer (ranging from "strongly agree" to "strongly disagree"). The instrument was based partly on the instrument used in the Wright State University Pilot Project which used presentational software (though not PowerPoint) in general education courses. Although the reaction in both Wright State and IUP to the use of presentational software was quite positive, reactions at IUP were in most cases higher than at Wright State. That having been said, the much smaller sample size of the IUP experiment prevents any broad comparative conclusions at this point. The results from two semesters of comparative politics (total of forty-seven students responding) are reproduced below.
PowerPoint is one of many alternatives in teaching with technology. PowerPoint works very well in comparative politics, but is far less suited to a subject such as political theory, where visual elements are limited. Political theory, on the other hand, is ideal for instruction via the World Wide Web, since many of the great books of Western political thought are available in full-text online. What works for one subfield does not necessarily work well in another. In examining the use of PowerPoint as an alternative in technological approaches to teaching, a number of factors should be borne in mind. First, the suitability of the subject matter to a highly visual approach is key. Although PowerPoint can be used effectively without photos, clipart, or charts of any kind, the real attraction of the software is the seamless integration of text and visual elements. Subjects such as art, history, nutrition, area studies, safety sciences, geography, anatomy, zoology, physical education, computer science, archaeology and military sciences all have a wealth of visual elements which are easily inserted in PowerPoint. Other subjects, such as philosophy, linguistics, English composition and law would need much more imaginative application of PowerPoint, since many of these disciplines are highly textual in nature. PowerPoint is ideal for teaching with the case study method, beginning with the "facts of the case" and then turning to the questions and discussion. For other disciplines, such as economics, the challenge is to go beyond the charts and bring additional visual elements that enliven and illustrate abstract principles with concrete examples (Sammons, 1995).
In the same text, faculty willingness to learn new technologies and apply them to their teaching is a second challenge in technological selection. PowerPoint has the advantage that it is - for a remarkably powerful presentation package - surprisingly easy to learn and use. That ideal combination of power and ease of use is rarely seen in educational software. Faculty can be easily persuaded that learning PowerPoint does not represent the same kind of learning curve that mastering HTML might. Furthermore, teaching with PowerPoint does not necessarily involve radical changes to teaching approaches, though it can if the instructor so desires. Even as a tool to create better-designed black and white or color transparencies, PowerPoint enforces simple but important rules of highly effective media design in the point sizes of text, bullets, framing, and layout. Once faculty begins to use PowerPoint in this simple way, it is a short step to using PowerPoint as an electronic slide show, where the marginal cost of a new slide is virtually nil. As a simple supplement to traditional lecture and discussion, most instructors see PowerPoint as a simple yet highly effective step forward. Naturally, the technology can be taken much further, and should be, but bringing technology into education is not simply a matter of establishing the cutting edge. As often as not, advancing technology means taking the small steps that introduce new approaches to a broader audience.
The number one challenge to the effective use of PowerPoint in the classroom is clear: the need for effective, cost-efficient, flexible projection systems. This shows up in surveys more than any other complaint. A number of alternatives exist, but there are serious trade-offs in the choices. Linking a laptop computer with PowerPoint to a frame capture or scan capture system which converts monitor images to TV images is fairly cheap, but visibility is limited, making this suitable for smaller classes only. A computer linked to an LCD panel on an overhead projector is a slightly more expensive system, but not only does it require a special highly reflective screen, it also needs a higher-power overhead, both raising the cost of the system. In any case, the image is washed out in all but the most darkened rooms. Ceiling-mounted RGB projectors can show a much better image, but are somewhat expensive and are not portable. These are well-suited to larger auditorium, but instructors who wish to teach their courses with both PowerPoint and cooperative learning techniques which break classes into small groups will find that venue difficult at best. The optimal solution are the LCD projectors on the market, which when combined with a high-performance laptop computer loaded with PowerPoint can give a crisp, visible image for small or large groups, and are portable. This optimal solution, however, is not cheap, and better quality LCD projectors run several thousand dollars. Fortunately, the prices on these devices are dropping, and their flexibility favors sharing them with different instructors or different departments (Sammons, 1995).
Using Calculators versus Paper and Pencil in Mathematics. According to Ray (2000), when she conducts workshops, she does an activity where she asks the participants to get into groups of 4. She has 12 math problems from a 5th grade textbook on an overhead transparency and she asks each group to solve the problems one at a time. One participant has to use a calculator, one uses paper and pencil, and one does it in their head. The fourth decided who gets the problem correct first. Each time that I've done this, mental math wins out about 60% of the time with the calculator coming in about 35% and paper and pencil 5%. What does this say? These are the rules that she uses when it comes to appropriate use of a calculator: (1) Use a calculator as a tool in problem solving... sort of like a "fast pencil." (2) Use a calculator for complex computations but NOT for basic facts! (3) Use a calculator to develop number concepts and skills. (4) Use a calculator in testing situations when not assessing computational proficiency. In general, she uses calculators to help teach mathematics matter, not to replace the teaching of mathematics.
However, according to Ediger (1997), the use of scientific calculator is more enjoyable, excellent to notice fewer errors made in the computation, students have more fun and checking one’s work with a calculator could be done just as easily and accurately that’s why teacher preferred the students to use it to enhance their learning.
On the other hand, when using paper and pencil, pupils have more time to deliberate in problem solving as compared with the use of calculator. Calculators provide the correct results if the user touches the proper keys. Pupils get frustrated if they see long answers and lastly, they preferred to use paper and pencil method when it comes to solve less complex problems (Ediger, 1997).
Guiding Principles and Effective Practices of Graphic Organizers. Visual displays and representations of information, commonly called graphic organizers, have become standard practice in most educational settings. But simply using a graphic organizer does not guarantee enhanced student understanding or achievement. Research and best practices have shown that, for graphic organizers to be effective instructional tools, several factors must be addressed. First, the graphic organizers need to be very straightforward and coherent. Next, students must be taught how to use the graphic organizer. Finally, teachers should consistently use graphic organizers during all aspects of instruction so that students begin to internalize the organizational skills of the graphic display (Boyle, 1997).
Graphic organizers are visual displays of key content information designed to benefit learners who have difficulty organizing information (Fisher & Schumaker cited in Boyle, 1997). Sometimes referred to as concept maps, cognitive maps, or content webs, no matter what name is used, the purpose is the same: Graphic organizers are meant to help students clearly visualize how ideas are organized within a text or surrounding a concept. Through use of graphic organizers, students have a structure for abstract ideas. Graphic organizers can be categorized in many ways according to the way they arrange information: hierarchical, conceptual, sequential, or cyclical (Bromley, Irwin-DeVitis, & Modlo cited in Boyle, 1997). Some graphic organizers focus on one particular content area. For example, a vast number of graphic organizers have been created solely around reading and prereading strategies (Merkley & Jeffries cited in Boyle, 1997).
In presenting a graphic organizer, there are things to be considered, like: (1) keep them simple. For graphic organizers to be effective instructional tools, they must be clear and straightforward (Boyle & Yeager, 1997; Egan cited in Boyle, 1997). The connections and relationships between the ideas depicted in the organizer should be obvious, otherwise the academic benefits will be limited. If an organizer is poorly constructed, includes too much information, or contains distractions, students can easily become confused and even more disorganized than before in their understanding of the target concepts (Robinson cited in Boyle, 1997). Therefore, teachers must keep graphic organizers simple. Suggestions for following this principle include: (a) limit the number of ideas covered in each organizer. Focus on essential concepts that students need to understand and remember; (b) include clear labels and arrows to identify the relationships between concepts; and (c) be careful of graphic organizers that accompany teacher resource materials. They often contain many pictures or background visuals that are distracting to students.
(2) Teach to and with the organizer. As with all instructional tools, students need to be taught how to use graphic organizers effectively and efficiently. Students enter the classroom with varied experiences using graphic organizers. Therefore, teachers must give explicit instructions about how to organize information and when a particular organizer is beneficial. With such guidance and scaffolding, students gain greater independence with graphic organizers.
Once students understand how to use an organizer, teachers need to implement it in creative and engaging ways to enhance effectiveness (Bromley cited in Boyle, 1997). As organizers have become more common, simply using an organizer is no longer enough to maintain students’ attention and focus. The following ideas will help ensure that students are engaged with organizers. (a) Allow students to add illustrations. As long as the pictures add to a student’s understanding of the concepts displayed and do not distract, illustrations can be very engaging. (b) Implement organizers with cooperative groups or pairs of students. Organizers can be excellent tools for discussion and student engagement with each other. (c) Allow students to make their own organizers and share them with the class. As students become more comfortable using organizers, they can teach the strategies they use to organize information for the whole group.
(3) Use graphic organizers often. Many students benefit from routine and structure, so using graphic organizers consistently in the classroom will help them internalize the organizing techniques that are being taught (Griffin & Tulbert cited in Boyle, 1997). The more students are exposed to organizers, the more familiar and comfortable they will become using them. Here are some things to consider when trying being consistent: (a) establish a routine for using organizers during instruction. For example, always use a web when starting a new unit, no matter what the subject area is. Use the same sequence chart when ordering events or steps in math, reading, writing, science, or social studies. (b) Incorporate organizers into all phases of instruction. When students see them used as a warm-up, a guided practice, or a homework assignment, they better understand the purpose and the benefits of the organizer. (c) If students have difficulty using a particular organizer, don’t give up. Students will often struggle with new approaches. Stay consistent and keep providing them guidance and practice. When students see the teacher using an organizer consistently, they are more likely to understand it themselves.
Development of Students’ Skills through the Use of Technology. Technology helps students document the validity of their mathematical/critical thinking process, facilitating and enriching the learning processes and the development of problem-solving skills. The use of technology should be guided by consideration of what mathematics is to be learned, the ways students might learn it, the research related to successful practices, and the standards and recommendations recommended by professional organizations in education. Technology can be used by mathematics educators to enhance conceptual understanding through a comparison of verbal, numerical, symbolic, and graphical representations of the same problem (Cole, 2002).
Students can use technology to search for patterns in data, while allowing the technology to perform routine and repeated calculations. The use of technology should not be used as a substitute for an understanding of and mastery of basic mathematical skills. Technology should be used to enhance conceptual understanding, while simultaneously improving performance in basic skills (Posey, 2001).
Educational Technologies in Learning. Educational technologies are not single technologies but complex combinations of hardware and software. These technologies may employ some combination of audio channels, computer code, data, graphics, video, or text. Although technology applications are frequently characterized in terms of their most obvious hardware feature (e.g., a VCR or a computer), from the standpoint of education, it is the nature of the instruction delivered that is important rather than the equipment delivering it. In this chapter, we review the history and current status of educational technologies, categorized into four basic uses: tutorial, exploratory, application, and communication. Our categories are designed to highlight differences in the instructional purposes of various technology applications, but we recognize that purposes are not always distinct, and a particular application may in fact be used in several of these ways (Conaty, 1993).
In the same text, tutorial uses are those in which the technology does the teaching, typically in a lecture-like or workbook-like format in which the system controls what material will be presented to the student. In our classification scheme, tutorial uses include (1) expository learning, in which the system provides information; (2) demonstration, in which the system displays a phenomenon; and (3) practice, in which the system requires the student to solve problems, answer questions, or engage in some other procedure.
Exploratory uses of technology are those in which the student is free to roam around the information displayed or presented in the medium. Exploratory applications may promote discovery or guided discovery approaches to helping students learn information, knowledge, facts, concepts, or procedures. We also include reference applications, such as CD-ROM encyclopedias, in this category. In contrast to tutorial uses in which the technology acts on the student, in exploratory uses the student controls the learning (as in exploring microworlds or hypermedia stacks) (Conaty, 1993)..
In the same text. application uses, such as word processors and spreadsheets, help students in the educational process by providing them with tools to facilitate writing tasks, analysis of data, and other uses. In addition to word processors and spreadsheets, applications include database management programs, graphing software, desktop publishing systems, and videotape recording and editing equipment.
Communication uses are those that allow students and teachers to send and receive messages and information to one another through networks or other technologies. Interactive distance learning via satellite, computer and modem, cable links, or other technologies constitutes another example of communication uses (Conaty, 1993).
Historically, the dominant teaching-learning model has been one of transmission: teachers transmitting information to students. Not surprisingly, the first uses of educational technology supported this mode. Although other ways of using technology to support learning are now available, tutorial uses continue to be the most widespread, especially with disadvantaged students (Becker cited in Conaty, 1993).
Computer-Assisted Instruction--Some of the first computer-assisted instruction (CAI), developed by Patrick Suppes at Stanford University during the 1960s, set standards for subsequent instructional software. After systematically analyzing courses in arithmetic and other subjects, Suppes designed highly structured computer systems featuring learner feedback, lesson branching, and student record keeping (Coburn cited in Teaching with Computer in Education, 2008).
During the 1970s, a particularly widespread and influential source of computer-assisted instruction was the University of Illinois PLATO system. This system included hundreds of tutorial and drill-and-practice programs. Like other systems of the time, PLATO's resources were available through timesharing on a mainframe computer (Coburn cited in Teaching with Computer in Education, 2008).
Today, microcomputers are powerful enough to act as file servers, and CAI can be delivered either through an integrated learning system or as stand-alone software. Typical CAI software provides text and multiple-choice questions or problems to students, offers immediate feedback, notes incorrect responses, summarizes students' performance, and generates exercises for worksheets and tests. CAI typically presents tasks for which there is one (and only one) correct answer; it can evaluate simple numeric or very simple alphabetic responses, but it cannot evaluate complex student responses.
Integrated learning systems (ILSs) are networked CAI systems that manage individualized instruction in core curriculum areas (mathematics, science, language arts, reading, writing). ILSs differ from most stand-alone CAI in their use of a network (i.e., computer terminals are connected to a central computer) and in their more extensive student record-keeping capabilities. The systems are sold as packages, incorporating both the hardware and software for setting up a computer lab.
ILSs are typically sold in sets of 30 workstations, with an average cost of about $125,000. Major producers include Josten's Learning Corporation, WICAT Systems, and CCC (founded by Patrick Suppes). About 10,000 ILSs are in use in U.S. schools, most of them purchased with funds from the ESEA Chapter 1 program for at-risk students (Mageau cited in Teaching with Computer in Education, 2008).
The instructional software within ILSs is typically conventional CAI: instruction is organized into discrete content areas (mathematics, reading, etc.) and requires simple responses from students. ILS developers have also made a point of developing systems that tie into the major basal textbooks. Mageau (cited in Teaching with Computer in Education, 2008) notes that the systems "can correlate almost objective by objective to a district's K-8.... language arts, reading, math, and even science curricula". Users of ILSs enjoy the advantage of having one coordinated system, making it easy for students to use a large selection of software.
A new trend in integrated learning systems is represented by ClassWorks, developed by Computer Networking Specialists. ClassWorks offers the school access to whatever variety of third-party software the teachers select, along with all the instructional management features associated with an ILS (Mageau cited in Teaching with Computer in Education, 2008).
CAI in general, and integrated learning systems in particular, have found a niche in America's schools by fitting into existing school structures (Newman cited in Teaching with Computer in Education, 2008). Cohen (cited in Teaching with Computer in Education, 2008) describes these structures as follows: (a) most instruction occurs in groups of 25 to 35 students in small segments from 45 to 50 minutes long; (b) instruction is usually either whole-class or completely individual; (c) instruction is teacher dominated, with teachers doing most of the talking and student talk confined largely to brief answers to teacher questions; (d) when students work on their own, they complete handouts devised or selected by the teacher. Students have little responsibility for selecting goals or deadlines and little chance to explore issues in depth. Most responses are brief; (d) knowledge is represented as mastery of isolated bits of information and discrete skills.
Many features of tutorial CAI are consistent with the traditional classroom described by Cohen. Tutorial CAI provides a one-way (computer to student) transmission of knowledge; it presents information and the student is expected to learn the information presented. Much CAI software presents information in a single curriculum area (e.g., arithmetic or vocabulary) and uses brief exercises that can easily is accommodated within the typical 50-minute academic period. CAI is designed for use by a single student and can be accommodated into a regular class schedule if computers are placed in a laboratory into which various whole classes are scheduled.
Basic skills (such as the ability to add or spell) lend themselves to drill- and-practice activities, and CAI, with its ability to generate exercises (e.g., mathematics problems or vocabulary words) is well suited to providing extensive drill and practice in basic skills. Students at risk of academic failure often seen as lacking in basic skills and therefore unable to acquire advanced thinking skills become logical candidates for CAI drill-and-practice instruction. Recent research and thinking on the needs of disadvantaged students stress a different need. Disadvantaged students need the opportunity to acquire advanced thinking skills and can acquire basic skills within the context of complex, meaningful problems. This latter approach to instruction, which is stressed in education reform, has not been well served by traditional CAI.
Intelligent Computer-Assisted Instruction-- Intelligent computer-assisted instruction (ICAI, also known as intelligent tutoring systems or ITSs) grew out of generative computer-assisted instruction. Programs that generated problems and tasks in arithmetic and vocabulary learning eventually were designed to select problems at a difficulty level appropriate for individual students (Suppes cited in Teaching with Computer in Education, 2008).
These adaptive systems (i.e., adapting problems to the student's learning level) were based on summaries of a student's performance on earlier tasks, however, rather than on representations of the student's knowledge of the subject matter (Sleeman & Brown cited in Teaching with Computer in Education, 2008). The truly intelligent systems that followed were able to present problems based on models of the student's knowledge, to solve problems themselves, and to diagnose and explain student capabilities.
Historically, ICAI systems have been developed in more mathematically oriented domains--arithmetic, algebra, programming--and have been more experimental in nature than has conventional CAI. Although ICAI is an area of active research projects, ICAI programs in the schools are not widespread. ICAI tends to call for more meaningful interactions than traditional CAI and tends to deal with more complex subject matter.
ICAI's focus on modeling student knowledge lends itself to applications in teaching advanced thinking skills. ICAI has not been used extensively with disadvantaged students (traditional targets for basic skills instruction).
One intelligent tutoring system, Geometry Tutor, provides students with instruction in planning and problem solving to prove theorems in geometry (Office of Technology Assessment cited in Teaching with Computer in Education, 2008). Geometry Tutor comprises an expert system containing knowledge of how to construct geometry proofs, a tutor to teach students strategies and to identify their errors, and an interface to let students communicate with the computer. Geometry Tutor monitors students as they try to prove theorems, instructing and guiding them throughout the problem-solving process (Anderson et al. 1985). Schofield, Evans-Rhodes, and Huber (cited in Teaching with Computer in Education, 2008) studied the implementation of Geometry Tutor in a public high school and found changes in the behavior of teachers and students using this system: teachers spent more time with students having problems, collaborated more with students, and based more of a student's grade on effort; students increased their level of effort and were more involved in the academic tasks. Thus, ICAI can be implemented in ways that support the kind of learning that education reformers advocate. Although most of these applications control instructional content, they can be used within a broader instructional framework that stresses joint work with the automated tutor.
Technologies for tutorial learning typically use a transmission rather than constructivist model of instruction. For this reason, although they have found their place in education and have the greatest rate of adoption within schools thus far, they are unlikely to serve as a catalyst for restructuring education. The focus of drill-and-practice CAI on basic skills allows little room for the presentation of complex tasks, multistep problems, or collaborative learning. ICAI, on the other hand, has the potential to deal with complex domains, to provide models of higher- order thinking, and to probe students understanding, but has seldom been well integrated into a school's mainstream curriculum. One-way video technologies can be very motivating but are nearly always viewed as enrichment and have not instigated fundamental changes within schools (Conaty, 1993).
Research Literature
The following studies discussed some of the basic tenets pertinent to the present study. These studies were included in the view of clarifying concepts related to this research.
In a study by Setzer entitled “Computers in Education”, administered in University of Sao Paulo, Brazil in 2000, he introduced some arguments in using computers in education, at home and at school. They are the following: (1). Computers improve both teaching and student achievement. (2). Computer literacy should be taught as early as possible; otherwise students will be left behind. (3). Technology programs leverage support from the business community - badly needed today because schools are increasingly starved for funds. (4). To make tomorrow's work force competitive in an increasingly high-tech world, learning computer skills must be a priority. (5). Work with computers - particularly using the Internet - brings students valuable connections with teachers, other schools and students, and a wide network of professionals around the globe. Those connections spice the school day with a sense of real-world relevance, and broaden the educational community (Oppenheimer, 2000).
According to Papert (cited in Setzer, 2000) in examining those arguments, the following patterns emerge. These patterns are the following: (a) computers should be learned and used as soon as possible because they will be essential for the individual in the professional working place; (b) students who do not master computers will not keep pace with their classmates; (c) computers are good tools for learning; (d) computers improve students' achievements; (e) computers accelerate children's development, mainly intellectual; (f) computers may provide a free environment for learning; (g) computers may promote social (and family) cohesion; (h) computers provide a fascinating learning environment, one that attracts children and young people; (i) computers provide for a challenge of traditional educational methods and values; (j) computers induce a certain vision of the world; (k) computers make it possible to learn without tensions and pressures; (l) computers (through the Internet) make students get interested in foreign cultures and people; (m) computers develop self-control; (n) computers may provide for a more humanistic teaching; (o) computers may enhance imagination and creativity; (p) computers may be used to make children conscious of their own thinking process; (q) computers provide for an individual way and pace of learning. (r) Children have to learn computers otherwise they will be afraid of them at adult ages; (s) children who don't use a computer at home may develop psychological and social problems (e.g. a sense of inferiority); and (t) through the Internet, computers make it possible for students to access all sorts of information not available through other means.
In addition, the study of Fulton and Wenglinsky about the “Technology and Students Learning” administered on September 1997, states some of the questions they want to study like: “does it work?" and "is it effective?" are legitimate questions about educational technology. When educators ask these questions, they are really asking if technology helps students learn. But technology is only a tool and the question cannot just be "Does the presence of technology improve learning?" It is clear that when the researchers try to evaluate the educational uses of technology, what they are really evaluating are the broader pedagogical practices being used. The question, then, becomes: What kinds of technology are being used, under what context, and in what ways that help promote student learning?
Not all the research paints a rosy picture of technology in schools. Some show no academic improvement; no pay off for costly investments (Mathews, 2000). Other authors believe technology takes funding away from other resources and programs that may be more beneficial to students (Healy, 1999; Oppenheimer, 1997); that technology sits idle and is underused (Cuban, 2001); and that an over-reliance on technology can rob from children opportunities to express creativity, build human relationships, and experience hands-on learning (Alliance for Childhood, 2000).
Furthermore this study found out that under the right conditions, technology: (a). accelerates, enriches, and deepens basic skills; (b) motivates and engages students in learning; (c) helps relate academics to the practices of today's workforce.; (d) increases economic viability of tomorrow's workers; (e) strengthens teaching.; (f) contributes to change in schools; and lastly, (g) connects schools to the world.
Other studies with negative results indicate that the initiatives themselves focused on hardware and software, or teachers taught about the technology instead of using the technology to enhance learning experiences. Bracewell, Breuleux, Laferriere, Beniot, and Abdous (cited in Fulton and Wellingski, 1997) asserted that the integration of educational technology into the classroom, in conjunction with supportive pedagogy, typically leads to increased student interest and motivation in learning, more student-centered classroom environments, and increased real-life or authentic learning opportunities. Davis (cited in Fulton and Wellingski, 1997) agreed that technology integration led to student-centered classrooms, which increased student self-esteem. Schacter (cited in Fulton and Wellingski, 1997) concluded that technology initiatives have to focus on teaching and learning, not the technology, to be successful: "One of the enduring difficulties about technology and education is that a lot of people think about the technology first and the education later". Educators are starting to recognize it is more important to use technology for learning than it is to learn how to use the technology.
Becker (2000) examined data from the 1998 national survey of teachers, Teaching, Learning, and Computing (TLC), and concluded: under the right conditions—where teachers are personally comfortable and at least moderately skilled in using computers themselves, where the school's daily class schedule permits allocating time for students to use computers as part of class assignments, where enough equipment is available and convenient to permit computer activities to flow seamlessly alongside other learning tasks, and where teachers' personal philosophies support a student-centered, constructivist pedagogy that incorporates collaborative projects defined partly by student interest—computers are clearly becoming a valuable and well-functioning instructional tool.
Educational visionaries are often frustrated that technology has primarily been used only to automate traditional education. They see the various ways technology will be used to revolutionize education through ‘learning by doing' and in the kinds of collaborative communities young people are creating with technology (Richardson, 2006; Tapscott, 1998). Further, "Computer based technology has been called an essential ingredient in restructuring because it can provide the diversity in instructional methods necessary to reach all school children," according to Polin (1991). Papert (1996) described the important role technology can play in learning (Cherniavsky & Soloway, 2002; Papert, 1989, 1996; Schank, 2001; Schank & Cleary, 1995).
Synthesis
Upon discussing the related literatures, several realizations helped the researcher in the development of this study. Also, mathematics educators cannot separate the vision of how they should prepare high school teachers in mathematics from the vision of what and how students should learn mathematics in the high school. Prospective teachers should have the same kind of experiences integrating mathematics, and technology as their future students. One of the goals of the high school concept is the integration of mathematics with other areas. Teachers should experience how technology can be integrated in an authentic way, so that the integrity of mathematics is preserved. Different high schools incorporate to different degrees the ideal of the high school concept. Prospective teachers can also take part of the approach presented here to implement change and support the necessary reform in mathematics teaching over time, regardless of the degree of implementation of the high school concept in their placement school.
Studies and researches mentioned above were considered very significant because they help the researcher acquire a lot of knowledge and understanding regarding the concept of what modern technology gadgets (computer and scientific calculator) really are and how it affects the learning of the students in the present time. From this study, the researcher was given full view of modern technology gadgets, which will be the starting point in delving more with this research.
Fourth year high school students were the specific respondents of the present study since this was the time when they will be taking up Trigonometry and the use of scientific calculator is a must for them.
This study shares the same view on the aforementioned studies and researches conducted – that the use of modern technology gadgets and the frequency of use greatly affect the learning in mathematics of the students today.
CHAPTER III
RESEARCH METHOD AND PROCEDURE
This chapter deals with the research design, the subjects of the study, the data gathering procedure and the statistical treatment of the gathered data.
Research Design
This study used the descriptive type of research. Descriptive method was used in this study because it best describes the frequency of use of technology-aided teaching strategy, the effects of utilizing modern technology gadgets and the its effect of using in students learning. It is defined by Travers (1978) as the design which aims “to describe the nature of a situation as it exists at the time of the study and to explore the cases of particular phenomena”.
Respondents of the Study
The total population of the fourth year high school students of First Asia Institute of Technology and Humanities (FAITH) is one hundred thirty-nine (139) for the Academic Year 2008-2009 but one hundred thirty-six (136) answered the survey because three of them were absent. The Slovin’s Formula was used to get the number of respondents. From one hundred thirty-six (136), the researcher got one hundred two (102) students, which were composed of fifty-one (51) males and fifty-one (51) females. They were randomly selected using stratified sampling. Their ages ranges from fifteen (15) to seventeen (17) years old.
The researcher employed them as the respondents of the study since this was the time when they are taking up Trigonometry and the use of scientific calculator is a must for them.
Data Gathering Instrument
The researcher, in order to attain this objective, used a self-constructed, validated questionnaire as the main instrument in gathering data. The researcher looked for the unpublished and published materials available in the library and electronic sources for topics that are related to the present study. Items in the questionnaire were constructed based on the statement of the problem that was presented in the study such as the technology-aided teaching strategy that utilize modern technology gadgets and its frequency of use; the effects of utilizing modern technology gadgets in learning mathematics; and lastly the effects of using modern technology gadgets in students learning.
After constructing the questionnaire, the researcher had it validated, first with the thesis adviser by asking how to correct on the items and its format, then to two Mathematics teacher by checking the mathematical words used and finally to an English teacher by correcting the grammar. In validating the questionnaire, instead of using formula as a standard process of validation, a focused group was involved which composed of ten (10) college students of First Asia Institute of Technology and Humanities in the Tertiary Level. Since all items in the questionnaire were answered correctly by the focused group, it was assumed that they are valid – the items can answer the posted statement of the problems in this study. After the validation, it was made ready for actual administration to the respondents. It was constructed to gather responses from the fourth year high school students of First Asia Institute of Technology and Humanities.
The questionnaire consisted of three main parts. The first part contained the technology-aided teaching strategies and the frequency of their use wherein they were given scalar values and description such as: 4- Always; 3- Often; 2- Sometimes and 1- Never. There are twelve (12) items in this part.
The second part of the questionnaire contains fourteen (14) items about the effects of the utilization of modern technology gadgets in learning mathematics. The respondents were asked to use the scalar values and description such as: 4- Very Much; 3- Much; 2- Not so much; 1- Not at all, in answering each item.
Finally, the last part of the questionnaire is consisted of fourteen (14) items on the effects of using modern technology gadgets in students learning. Like in the first and second part, scalar values and its description were given such as: 4- Strongly agree; 3- Agree; 2- Disagree and 1- Strongly disagree, as the respondents guide in for answering the questions.
To further interpret the weighted mean, this scale was used.
Scale
Part I
Part II
Part III
3.49 – 4
Always
Very Much
Strongly Agree
2.49 – 3.48
Often
Much
Agree
1.49 – 2.48
Sometimes
Not Much
Disagree
1 – 1.48
Never
Not at all
Strongly Disagree
Data Gathering Procedure
When the researcher came up with the topic he wanted to study, he presented his chosen topic to the thesis adviser, and defended it. After some critical analysis and examination, the proposed topic was accepted. Thereupon, the researcher started to construct the questionnaire. With the help of the thesis adviser, grammarian and Mathematics teachers; the questionnaire was then approved and advised for a dry-run.
The researcher then prepared the questionnaire ready for dry-run administered to ten (10) college students, four first year Industrial Engineering students, one second year Secondary Education major in Mathematics, two third year Elementary Education major in Special Education and three fourth year Elementary Education major in Special Education. Since all items in the questionnaire were answered correctly by the focused group, it was assumed that they are valid – the items can answer the posted statement of the problems in this study. After tallying the results, the answers yielded were manifested that the items were clear because they were answered correctly by those students.
The researcher then ready for the actual administration of the questionnaire for the target respondents, made a formal letter to the high school principal of First Asia Institute of Technology and Humanities. The letter was duly signed by the adviser and the College Dean and was approved by the high school principal. He advised the researcher to administer the questionnaire to the whole population of the fourth year high school students. This is to keep records of all students to the perceived effects of the utilization of modern technology gadget in learning mathematics. After the approval of the letter, the data gathering instrument was reproduced and distributed to the respondents on August 15, 2008 from 1:30pm to 2:30 pm at room 30, 31, 40 and 41. The questionnaire was personally administered by the researcher with the assistance of the concerned high school teacher.
In tallying the answers the researcher used the number of respondents obtained through Slovin’s Formula. From one hundred thirty-six (136), only one hundred two (102) were tallied. The one hundred two (102) students were randomly selected to properly represent each section. This was personally analyzed by the researcher.
A one hundred percent (100%) retrieval of questionnaire was tallied and appropriate statistical tools were applied.
Statistical Treatment of Data
After the questionnaire was collected, the researcher made the appropriate tabulation of the responses and statistical tools were applied to the study which was follows:
Slovin’s Formula was used to determine the sample size of the respondents.
Formula:
Where:
N is the population size
e is the margin of error
Weighted Mean was used to determine the technology-aided teaching strategy that utilize modern technology gadgets and its frequency of use, the effects of utilizing modern technology gadgets in learning mathematics and lastly the effects of using modern technology gadgets in students learning.
Formula:
=
where:
w is the weights
is the mean.
is the sum of all score.
n is the total number of respondents.
Composite mean was used to determine the sum of the entire weighted mean in each part.
Formula:
Where:
is the weighted mean of the first item.
is the weighted mean of the second item.
is the weighted mean of the nth item.
n is the number of items.
c is the composite mean.
Pearson product moment correlation coefficient was used to determine the correlation between the frequency of the utilization of modern technology gadgets and learning of mathematics.
Formula:
)
Where:
x is the value of independent variable. is the mean of the values of x.
y is the value of dependent variable.
is the mean of the value y.
T-test was used to determine the significant relationship between the frequency of the utilization of modern technology gadgets and learning of mathematics.
Formula:
Where:
r is the value of the correlation (Pearson R)
N is the sample size
CHAPTER IV
PRESENTATION, ANALYSIS AND INTERPRETATION OF DATA
This chapter presents the analysis and interpretations of the data gathered on the perceived effects of the utilization of modern technology gadgets in learning mathematics of fourth year high school students of First Asia Institute of Technology and Humanities.
1. Teaching Strategy Utilizing Modern Technology Gadgets
Varied teaching strategy that utilizes modern technology gadgets were presented to the respondents as follows: (1) lecture method with power point presentation; (2) reporting with power point presentation; (3) independent research through the use of internet; (4) model-making through the use of related software; (5) computer hands-on activities; (6) problem solving with the use of scientific calculator; (7) cooperative learning with power point presentation; (8) use of graphic organizers in solving problems; (9) discussion with the use of power point presentation; (10) use of analogy (reasoning) with power point presentation; (11) discovery learning through the use of scientific calculators; and lastly, (12) guided discovery through the use of related software.
The abovementioned teaching strategies were assumed by the researcher to be utilized by the teacher in teaching mathematics as they are commonly used by the teachers in Mathematics across schools.
2. Frequency of Using Technology-Aided Teaching Strategy that Utilizes Modern Technology Gadgets
Table 1 shows the frequency of using the teaching strategy that utilizes modern technology gadgets. It was analyze by the researcher because it seems that the frequency of utilization has its effect in students learning.
Table 1. Frequency of Using Technology-Aided Teaching Strategy that Utilizes Modern Technology Gadgets
Technology-Aided Teaching Strategy
that Utilizes Modern Technology Gadgets
WEIGHTED
MEAN
VERBAL INTERPRETATION
1. Lecture method with power point presentation
3.12
Often
2. Reporting with power point presentation
3.15
Often
3. Independent research through the use of internet
3.00
Often
4. Model-making through the use of related software
2.88
Often
5. Computer hands-on activities
3.06
Often
6. Problem solving with the use of scientific calculator
3.07
Often
7. Cooperative learning with power point presentation
3.14
Often
8. Use of graphic organizers in solving problems
2.79
Often
9. Discussion with the use of power point presentation
3.21
Often
10. Use of analogy (reasoning) with power point presentation
2.85
Often
11. Discovery learning through the use of scientific calculators
2.69
Often
12. Guided discovery through the use of related software
2.73
Often
COMPOSITE MEAN
2.97
Often
It can be gleaned in the table that the three most commonly used teaching strategies used by the teachers in teaching Mathematics are: cooperative learning with power point presentation with a weighted mean of 3.14; second is reporting with power point presentation with a weighted mean of 3.15; and third is discussion with the use of power point presentation with a weighted mean of 3.21. All of them have a verbal interpretation, “often”.
It can be inferred from the results that the use of power point presentation is commonly used in different teaching strategy such as cooperative learning, reporting and discussion. This probably is made possible because in First Asia Institute of Technology and Humanities (FAITH), technology use is one of the focuses of teaching and learning. Students and teachers are encouraged to deliver their lessons through the use of computers and one effective means is through the use of power point presentation. As the school promotes technology, it is but natural that the lessons must be presented through the use of technology.
This is further explained by Sammons (1995) when he said that power point works best for things that are presented visually, not verbally. It helps when the pupil need to draw a picture. Communication delivered over multiple channels is more efficient than communication over a single channel. Multiple channels make it more likely that the whole message will be received. An appropriate picture adds another channel. A picture aids in memory by making a visual connection to an abstract idea like memory rests on connections and a vivid picture forms a solid connection. Lastly, power point makes it easy to create visuals, and, by using a template, make it easy to be consistent.
On the other hand, the three least used technology – aided teaching strategies are the following: (a) discovery learning through the use of scientific calculators with a weighted mean of 2.69; (b) guided discovery through the use of related software with a weighted mean of 2.73; and (c) use of graphic organizers in solving problems with a weighted mean of 2.79. All of them have a verbal interpretation of often.
These three seem to have lower score because calculator, graphic organizer and related software are not regularly used in the classroom. The teacher seems to use the computer through the power point presentation than the use calculator in solving, graphic organizer in the lesson and the related software because here in First Asia Institute of Technology and Humanities (FAITH), computer is the instrument in teaching and/or presenting the lessons while scientific calculator, graphic organizer and related software are already installed or programmed in the computer.
This is similar to Ray’s findings (2000), from her workshop where she concluded that there are rules when it comes to appropriate use of a calculator: (1) use a calculator as a tool in problem solving... sort of like a "fast pencil"; (2) use a calculator for complex computations but NOT for basic facts; (3) use a calculator to develop number concepts and skills; and (4) use a calculator in testing situations when not assessing computational proficiency. In general, she uses calculators to help teach mathematics matter, not to replace the teaching of mathematics.
Likewise Mitchel, (cited in Boyle, 1997), said that the use of graphic organizer can develop students’ visual skills. He added however that using graphic organizer must be based on the suitability of the lessons presented are therefore cannot be used regularly.
3. Effects of Utilizing Modern Technology Gadgets in Learning Mathematics
Table 2 shows the effects of utilizing computer in learning mathematics. It was analyzed in the research because it seems that the utilization of computer in teaching and/or presenting the lesson will enhance the learning of the students in mathematics.
Table 2. Effects of Utilizing Computer in Learning Mathematics
Computer
My learning is enhanced when…
WEIGHTED
MEAN
VERBAL INTERPRETATION
1. the teacher uses power point presentation in
presenting the lesson.
3.22
Much Enhanced
2. the teacher shows animation in the monitor
which is related to the topic.
2.93
Much Enhanced
3. the teacher gives an independent research
work using the internet.
2.81
Much Enhanced
4. we work in groups on a particular problem
when using the internet .
2.53
Much Enhanced
5. the lectures of my teacher were sent to my e-
mail.
2.22
Not so Much Enhanced
6. the teacher shows some presentation related
to our topic.
2.77
Much Enhanced
7. we always have computer hands-on
activities.
2.85
Much Enhanced
COMPOSITE MEAN
2.76
Much Enhanced
It is shown in Table 2 that the teachers up three ways of presenting and/or teaching the lesson with the use of computer wherein the students can enhance their learning are the following: (a) having computer hands-on activities with a weighted mean of 2.85; (b) the teacher showing animation in the monitor which is related to the topic with a weighted mean of 2.93; and (c) the teacher uses power point presentation in presenting the lesson with a weighted mean of 3.22. All of them have a verbal interpretation of “much enhanced”.
The three most commonly used ways of teacher in presenting and/or teaching the lesson with the use of computer that enhances students’ learning with the use of power point presentation is believed to be useful probably because through power point presentation, students can visibly see concepts in the simplest way – with bulleted information, readable and with interesting slide design that captures the attention of the students. It is user-friendly and its purpose is to really present concepts in a more interesting manner.
This is similar to Sammons (1995) statements saying that, that the teachers, in using power point presentation to enhance their students learning, must observed the following: (1) use it judiciously for a few key graphics or illustrations; (2) avoid text slides; (3) use text occasionally as a reference point for big ideas, (e.g. the three main objectives of a lesson); (4) remember other kinds of visuals; (5) handouts may be a more appropriate alternative. (6) Don’t be seduced by textbook publishers that offer canned presentations that go with a textbook; (7) the teacher should not be the publisher nor the textbook; (8) the teacher should make careful choices of what to use and what to avoid; (9) avoid using power point for discussion or coaching sessions.
He added that power point does not facilitate spontaneous discussion or discovery. The use of whatever media in the classroom must work to help students make connections. Making connections is the foundation of memory and ingenuity. Students learn as they make connections. An efficient use of visuals in the classroom can help students make connections between parts and the whole, between cause and effect, between problem and solution, between principle and practice.
On the other hand, the three least used ways in presenting and/or teaching the lesson with the use of computer wherein students can enhance their learning are the following: (a) the lectures sent to e-mail with a weighted mean of 2.22 which has the lowest score and verbal interpretation of “not so much enhance”; (b)working in groups on a particular problem when using computer with a weighted mean of 2.53; and (c) the teachers’ showing of some presentation related to the topic with a weighted mean of 2.77. The remaining two has a verbal interpretation of “much enhanced”.
The three least used in presenting and/or teaching the lesson with the use of computer wherein the students will enhanced their learning, are not preferable for the students. The sending of lectures to students’ email may not be preferred probably because of minimal contacts between the students and teachers. For example, students may still need more explanations and instructions that are lacking when the lecture is already sent through e-mail. Also, students may find it so taxing sending, downloading, and printing those lectures instead of understanding it with the help of the teacher. Similarly, not all students have their own e-mail and computer or internet at home.
Meanwhile, working in groups may not be preferred because most students want to discover for himself how a problem may be solved through computer hands-on activities. Computer hands-on may be one of the most interesting activities for students but since they will be working in group, it becomes minimal, for they only have a one-at-a time chance in using the calculator. Also in grouping some students become dependent to their leader.
TV showing of some presentations related to the topic may not be so attractive to students probably because they still have to develop the so-called integrative learning skills. Or it can also inferred that the teachers way lack the skills of integrative teaching, thus making students not interested to the presentations which really are not their lessons.
According to (Suber, 2008) note taking is the practice of writing pieces of information, often in an informal or unstructured manner. In taking down notes, the students can: (1) organized notes that help in identifying the core of important ideas in the lecture; (2) It serves as permanent records that help in learning and remembering lecture; (3) the lecture may contain information not available anywhere else, this will be the only chance to students to learn it; (4) lecture is where the students learn what the instructor thinks is important, and makes up the exams; (5) class assignments are usually given in the lecture; and (6) the underlying organization and purpose of the lecture will become clear through note taking.
Another gadget used in presenting Mathematics lesson is the scientific calculator. Table 3 shows the effects of utilizing scientific calculator in learning mathematics. It was analyzed by the researcher because it seems that the utilization of scientific calculator in teaching and/or presenting the lesson will enhance the learning of the students in mathematics.
It is shown in Table 3 that the up three ways of presenting and/or teaching the lesson with the use of scientific calculator wherein the students will enhanced their learning are the following: (a) use scientific calculator in finding the factors of a given large number with a weighted mean of 3.11; (b) use scientific calculator in solving problems involving trigonometric functions with a weighted mean of 3.12; and (c) evaluate, press the keys and see the results on the display screen of the scientific calculator with a weighted mean of 3.12. All of them have a verbal interpretation of “much enhanced”.
Table 3. Effects of Utilizing Scientific Calculator in Learning Mathematics
SCIENTIFIC CALCULATOR
My learning is enhanced when…
WEIGHTED
MEAN
VERBAL INTERPRETATION
1. I use scientific calculator in solving problems
involving trigonometric functions.
3.12
Much Enhanced
2. I evaluate, press the keys and see the results
on the display screen of the scientific
calculator.
3.24
Much Enhanced
3. the teacher encourages me to discover by
myself certain facts about numbers using
scientific calculator.
2.67
Much Enhanced
4. I challenge my classmate to a calculator
game like in being the first to solve the
problem and get the right answer.
2.24
Not so Much Enhanced
5. I learn how to minimize the number of
keystrokes in solving problems.
2.82
Much Enhanced
6. I use scientific calculator in finding the
factors of a given large number.
3.11
Much Enhanced
7. I use scientific calculator in doing my
homework.
3.03
Much Enhanced
COMPOSITE MEAN
2.89
Much Enhanced
It is observed by the researcher that the learning of the students are enhanced when they use scientific calculator in finding the factors, solving problems involving trigonometric functions and evaluating large numbers because they seems to have fun and enjoyment by means of pressing the keys of the scientific calculator and at the same time they are learning what their doing.
This also observed by Ediger (1997), who said that the use of scientific calculator is more enjoyable, excellent to notice fewer errors made in the computation, students have more fun and checking one’s work with a calculator could be done just as easily and accurately that’s why teacher preferred the students to use it to enhance their learning.
On the other hand, it is shown in Table 3 that the three least used ways in presenting and/or teaching the lesson with the use of scientific calculator with less effects on influence are the following (a) challenging classmate to a calculator game like in being the first to solve the problem and get the right answer with a weighted mean of 2.24 and has a verbal interpretation of “not so much enhanced”, then (b) the teacher encourages students to discover certain facts about numbers by themselves using scientific calculator with a weighted mean of 2.67 and lastly (c) learning how to minimize the number of keystrokes in solving problems with a weighted mean of 2.82. The last two have a verbal interpretation of “much enhanced”.
It is observed by the researcher that the students do not pay much attention to the other uses, functions and importance of scientific calculator. They only consider the use of calculators in solving problems that is why they do not use it for a calculator game; they do not notice the number of keystrokes; and also the facts about numbers.
According to Ediger (1997), the calculator programs allow individual students to work at their own speed through a set sequence of stages, problems or challenges. Where a student is consistently successful, the calculator program can progress to the next level of difficulty. This "responsiveness" of the exercise is something that cannot easily be emulated with a textbook. The programs typically generate problems with randomly selected numbers. As a result of this students can't copy answers from their neighbor, but rather they spend time discussing how to do questions and identifying methods that work. The motivational aspect of use of IT in the teaching and learning process cannot be undervalued - especially with the weaker, or younger, mathematician. Students typically enjoy lessons that are based around calculator programs - the desire to obtain a faster time, or a higher score, often adds an additional competitive element to the chosen exercise.
4. Effects of Using Modern Technology Gadgets in the Development of Students Skills
Table 4 shows the effects of using computer in the development of students’ skills. It was analyzed by the researcher because it seems that the using of computers can develop the skills of the students.
Table 4. Effects of Using Computer in the Development of Students Skills
ITEMS
By using computers…
WEIGHTED
MEAN
VERBAL INTERPRETATION
1. I become attentive to learn new concepts.
3.40
Agree
2. I understand the lesson easier.
3.28
Agree
3. I can visualize the underlying principles of
th topic.
3.21
Agree
4. I become more participative in class
discussion.
3.05
Agree
5. I learn while having fun.
3.37
Agree
6. I become interactive in group activities.
3.23
Agree
7. I am more receptive in the discussion.
3.03
Agree
COMPOSITE MEAN
3.22
Agree
It is shown in Table 4 that the up three behavior/skills mostly develop in learning mathematics with the use computer are as follows: (a) understanding the lesson easier with a verbal interpretation of 3.28; (b) learning while having fun with a weighted mean of 3.37; and (c) becoming attentive to learn new concepts with a weighted mean of 3.40. All of them have a verbal interpretation of “agree”.
It can be said that when using computer, the students will understand the lesson easier because it helps them to have a one click a way information in surfing the net, in terms of having fun, because of its graphics and because of that the (graphics and animation) it helps them to become attentive listener and they will pay more attention.
Similarly, the National Council of Teachers in Mathematics in USA (2000) stressed that technology can be used to learn mathematics, to do mathematics, and to communicate mathematical information and ideas. The Internet hosts a wealth of mathematical materials that are easily accessible through the use of search engines, creating additional avenues to enhance teaching and facilitate learning. Outside of class, students and faculty can pose problems and offer solutions through e-mail, chat rooms, or websites.
Technology provides opportunities for educators to develop and nurture learning communities, embrace collaboration, provide community-based learning, and address diverse learning styles of students and teaching styles of teachers. The integration of appropriately used technology can enhance student understanding of mathematics through pattern recognition, connections, and dynamic visualizations. Electronic teaching activities can attract attention to the mathematics to be learned and promote the use of multiple methods.
Learning can be enhanced with electronic questioning that engages students with technology in small groups and facilitates skills development through guided-discovery exercise sets. Using electronic devices for communication, all students can answer mathematics questions posed in class and instructors can have an instantaneous record of the answers given by each student. This immediate understanding of what students know, and don’t know, can direct the action of the instructor in the teaching session.
However, the three behavior/skills least developed in learning mathematics with the use computer are the following: (a) becoming more receptive in the discussion with a weighted mean of 3.03; (b) becoming more participative in class discussion with a weighted mean of 3.05; and (c) visualizing the underlying principles of the topic with weighted mean of 3.21. All of them have a verbal interpretation of “agree”.
The results can be associated to the observations that computer poses less interaction with the students and is useful to the visual learners only. In that case the other kind of learners which is not visual type cannot learn that much with computer.
According to Cole (2002), technology helps students document the validity of their mathematical/critical thinking process, facilitating and enriching the learning processes and the development of problem-solving skills. The use of technology should be guided by consideration of what mathematics is to be learned, the ways students might learn it, the research related to successful practices, and the standards and recommendations recommended by professional organizations in education. Technology can be used by mathematics educators to enhance conceptual understanding through a comparison of verbal, numerical, symbolic, and graphical representations of the same problem.
Consequently, Table 5 shows the effects of using scientific calculator in the development of students’ skills. It was analyzed by the researcher because it seems that the using of scientific calculator can develop the skills of the students.
It is shown in Table 5 that the up three behavior/skills mostly develop in learning mathematics with the use scientific calculator are the following: (a) skipping the step-by-step process on how to solve problems with a weighted mean of 3.32; (b) finding it easy to solve difficult problems with a weighted mean of 3.35; and (c) spending less time in solving large numbers with a weighted mean of 3.43. All of them have a verbal interpretation “agree”.
Table 5. Effects of Using Scientific Calculator in the Development of Students Skills
ITEMS
By using scientific calculator…
WEIGHTED
MEAN
VERBAL INTERPRETATION
1. I develop my skills in solving problems.
3.12
Agree
2. I reinforce my skills in computation.
3.04
Agree
3. I improve my reasoning to a higher-level of
thinking.
3.07
Agree
4. I enhance my analytical thinking skills.
3.07
Agree
5. I spend less time in solving large numbers.
3.43
Agree
6. I can skip the step-by-step process on how to
solve problems.
3.32
Agree
7. I find it easy to solve difficult problems.
3.35
Agree
COMPOSITE MEAN
3.20
Agree
It is then believe by the researcher that the using of calculators helped the students skip the step-by-step process, spend less time in solving large numbers, and find it easy to solve problems. Because of that they will have the time to review their answers, compare the results to others, helps to improve their solutions and they can track which is incorrect in their solutions.
According to Ediger (1997), the use of scientific calculator is more enjoyable, excellent to notice fewer errors made in the computation, students have more fun and checking one’s work with a calculator could be done just as easily and accurately that’s why students preferred to use it to develop their skills.
On the hand, the three behavior/skills least developed in learning mathematics with the use scientific calculator are the following: (a) reinforcing skills in computation with a weighted mean of 3.04, and then both (b) improving reasoning to a higher-level of thinking and (c) enhancing analytical thinking skills have a weighted mean of 3.07. All of them have a verbal interpretation of “agree”.
It is then believed by the researcher that the students don’t know the importance of scientific calculator, they are just compute a problem using scientific calculator without knowing that that their skills are to be developed because of that they become more dependent in using it. If they rely on the calculator, even the simple 1+1 will be solve using calculator.
According to Ediger (1997), when using paper and pencil, pupils have more time to deliberate in problem solving as compared with the use of calculator. Calculators provide the correct results if the user touches the proper keys. Pupils get frustrated if they see long answers and lastly, they preferred to use paper and pencil method when it comes to solve less complex problems.
5. Relationship between the Frequency of the Utilization of Modern Technology Gadgets in the Effective Learning of Mathematics
Table 6 shows the relationship between the frequency of the utilization of modern technology gadgets in the effective learning of mathematics. In this table it is also shows the variable, value, the decision and its verbal interpretation.
Table 6. Significant Relationship between the Frequency of the Utilization of Modern Technology Gadgets in the Effective Learning of Mathematics
Variables
Value
Decision
Verbal
Interpretation
Technology-Aided Teaching Strategies and the Frequency of their Uses And Effects of Utilizing Modern Technology Gadgets (computer and scientific calculator) in Learning Mathematics
r= 0.3824
Reject the null hypothesis
The frequency of using technology-aided teaching strategies relates to the effective learning of Mathematics.
The computed value of 0.3824 at 0.05 level of significance has a verbal interpretation of “moderately correlated” and therefore the hypothesis: there is no significant relationship between the frequency of the utilization of modern technology gadgets in the effective learning of mathematics, was rejected. Therefore, it could be said that the frequency of using technology-aided teaching strategies relates to the effective learning of Mathematics.
According to The National Council of Teachers in Mathematics in USA (2000), it says that technology can be used to learn mathematics, to do mathematics, and to communicate mathematical information and ideas. The Internet hosts a wealth of mathematical materials that are easily accessible through the use of search engines, creating additional avenues to enhance teaching and facilitate learning. Outside of class, students and faculty can pose problems and offer solutions through e-mail, chat rooms, or websites. Technology provides opportunities for educators to develop and nurture learning communities, embrace collaboration, provide community-based learning, and address diverse learning styles of students and teaching styles of teachers. The integration of appropriately used technology can enhance student understanding of mathematics through pattern recognition, connections, and dynamic visualizations. Electronic teaching activities can attract attention to the mathematics to be learned and promote the use of multiple methods. Learning can be enhanced with electronic questioning that engages students with technology in small groups and facilitates skills development through guided-discovery exercise sets. Using electronic devices for communication, all students can answer mathematics questions posed in class and instructors can have an instantaneous record of the answers given by each student. This immediate understanding of what students know, and don’t know, can direct the action of the instructor in the teaching session.
However, according to Cole (2002), technology helps students document the validity of their mathematical/critical thinking process, facilitating and enriching the learning processes and the development of problem-solving skills. The use of technology should be guided by consideration of what mathematics is to be learned, the ways students might learn it, the research related to successful practices, and the standards and recommendations recommended by professional organizations in education. Technology can be used by mathematics educators to enhance conceptual understanding through a comparison of verbal, numerical, symbolic, and graphical representations of the same problem.
CHAPTER V
SUMMARY, CONCLUSION AND RECOMMENDATION
This chapter presents the summary the findings, the conclusions and the recommendations of the study.
SUMMARY
The researcher conducted this study to find out the perceived effects of utilizing modern technology gadgets in learning mathematics of fourth year high school students of First Asia Institute of Technology and Humanities.
Specifically, this study sought answers to the following questions:
1. In what teaching strategies does the teacher utilize modern technology gadgets?
2. How often are these modern technology gadgets used?
3. How does the use of modern technology gadgets affect the respondents learning?
4. What are the perceived effects of using modern technology gadgets in learning mathematics?
5. Is there a significant relationship between the frequency of the utilization of modern technology gadgets in the effective learning of mathematics?
The study tested the null hypothesis of no relationship between the frequency of the utilization of modern technology gadgets in the effective learning of mathematics.
This study is limited to the responses of the one hundred two (102) fourth year high school students of First Asia Institute of Technology and Humanities (FAITH) enrolled in Academic year 2008-2009. The population was identified through the use of proportional stratified random sampling. This study includes only two kinds of modern technology gadgets – computer and scientific calculator.
The descriptive method of research was used and a self-constructed, validated questionnaire served as the basic instrument in gathering data.
Slovin’s formula, weighted mean, composite mean and Pearson product moment correlation coefficient were the statistical treatments used to qualify the data.
Findings
After the analysis and interpretation of the data gathered, the researcher was able to come up with the following findings:
1. Teaching Strategy Utilizing Modern Technology Gadgets
Varied teaching strategy that utilizes modern technology gadgets were presented to the respondents as follows: (1) lecture method with power point presentation; (2) reporting with power point presentation; (3) independent research through the use of internet; (4) model-making through the use of related software; (5) computer hands-on activities; (6) problem solving with the use of scientific calculator; (7) cooperative learning with power point presentation; (8) use of graphic organizers in solving problems; (9) discussion with the use of power point presentation; (10) use of analogy (reasoning) with power point presentation; (11) discovery learning through the use of scientific calculators; and lastly, (12) guided discovery through the use of related software.
The abovementioned teaching strategies were assumed by the researcher to be utilized by the teacher in teaching mathematics as they are commonly used by the teachers in Mathematics across schools.
2. Frequency of Using Technology-Aided Teaching Strategy that Utilizes Modern Technology Gadgets
The three most commonly used teaching strategies used by the teachers in teaching Mathematics are: cooperative learning with power point presentation with a weighted mean of 3.14; second is reporting with power point presentation with a weighted mean of 3.15; and third is discussion with the use of power point presentation with a weighted mean of 3.21. All of them have a verbal interpretation, “often”.
On the other hand, the three least used technology – aided teaching strategies are the following: (a) discovery learning through the use of scientific calculators with a weighted mean of 2.69; (b) guided discovery through the use of related software with a weighted mean of 2.73; and (c) use of graphic organizers in solving problems with a weighted mean of 2.79. All of them have a verbal interpretation of “often”.
3. Effects of Utilizing Modern Technology Gadgets in Learning Mathematics
a. Effects of Utilizing Computer in Learning Mathematics
The teachers up three ways of presenting and/or teaching the lesson with the use of computer wherein the students can enhance their learning are the following: (a) having computer hands-on activities with a weighted mean of 2.85; (b) the teacher showing animation in the monitor which is related to the topic with a weighted mean of 2.93; and (c) the teacher uses power point presentation in presenting the lesson with a weighted mean of 3.22. All of them have a verbal interpretation of “much enhanced”.
On the other hand, the three least used in presenting and/or teaching the lesson with the use of computer wherein students can enhance their learning are the following: (a) the lectures sent to e-mail with a weighted mean of 2.22 which has the lowest score and verbal interpretation of “not so much enhance”; (b)working in groups on a particular problem when using computer with a weighted mean of 2.53; and (c) the teachers’ showing of some presentation related to the topic with a weighted mean of 2.77. The remaining two has a verbal interpretation of “much enhanced”.
b. Effects of Utilizing Scientific Calculator in Learning Mathematics
The up three ways of presenting and/or teaching the lesson with the use of scientific calculator wherein the students will enhance their learning are the following: (a) use scientific calculator in finding the factors of a given large number with a weighted mean of 3.11; (b) use scientific calculator in solving problems involving trigonometric functions with a weighted mean of 3.12; and (c) evaluate, press the keys and see the results on the display screen of the scientific calculator with a weighted mean of 3.12. All of them have a verbal interpretation of “much enhanced”.
On the other hand, the three least used in presenting and/or teaching the lesson with the use of scientific calculator with less effects on influence are the following (a) challenging classmate to a calculator game like in being the first to solve the problem and get the right answer with a weighted mean of 2.24 and has a verbal interpretation of “not so much enhanced”, then (b) the teacher encourages students to discover certain facts about numbers by themselves using scientific calculator with a weighted mean of 2.67 and lastly (c) learning how to minimize the number of keystrokes in solving problems with a weighted mean of 2.82. The last two have a verbal interpretation of “much enhanced”.
4. Effects of Using Modern Technology Gadgets in the Development of Students Skills
a. Effects of Using Computer in the Development of Students Skills
The up three behavior/skills mostly develop in learning mathematics with the use computer are as follows: (a) understanding the lesson easier with a verbal interpretation of 3.28; (b) enjoying learning while having fun with a weighted mean of 3.37; and (c) becoming attentive to learn new concepts with a weighted mean of 3.40. All of them have a verbal interpretation of “agree”.
However, the three behavior/skills least developed in learning mathematics with the use computer are the following: (a) becoming more receptive in the discussion with a weighted mean of 3.03; (b) becoming more participative in class discussion with a weighted mean of 3.05; and (c) visualizing the underlying principles of the topic with weighted mean of 3.21. All of them have a verbal interpretation of “agree”.
b. Effects of Using Scientific Calculator in the Development of Students Skills
The first three skills to be developed in learning mathematics with the use scientific calculator are the following: (a) skipping the step-by-step process on how to solve problems with a weighted mean of 3.32; (b) finding it easy to solve difficult problems with a weighted mean of 3.35; and (c) spending less time in solving large numbers with a weighted mean of 3.43. All of them have a verbal interpretation “agree”.
On the hand, the last three skills to be developed in learning mathematics with the use scientific calculator are the following: (a) reinforcing skills in computation with a weighted mean of 3.04, and then both (b) improving reasoning to a higher-level of thinking and (c) enhancing analytical thinking skills have a weighted mean of 3.07. All of them have a verbal interpretation of “agree”.
5. Significant Relationship Between the Frequency of the Utilization of Modern Technology Gadgets in the Effective Learning of Mathematics
The computed value of 0.3824 at 0.05 level of significance has a verbal interpretation of “moderately correlated” and therefore the hypothesis: there is no significant relationship between the frequency of the utilization of modern technology gadgets in the effective learning of mathematics, was rejected. Therefore, it could be said that the frequency of using technology-aided teaching strategies relates to the effective learning of Mathematics.
Conclusions
With the major findings of the study as bases, the researcher had drawn the following conclusions:
1. Mathematics teachers are using varied technology-aided teaching strategies.
2. Power point presentation is the most commonly used strategy in presenting lessons in Mathematics while the use of scientific calculators, graphic organizers and related software are the least utilized.
3. The use of power point presentation, computer hands-on activities and animations were found to enhance students learning in Mathematics while sending lectures to e-mail, working in groups on a particular problem and showing of some presentation related to the topic were found not that much to enhance students learning. On the other hand, when it comes to utilizing scientific calculators, finding the factors of a given large number, solving problems involving trigonometric functions and evaluating, pressing the keys and seeing the results on the display screen were found to enhance students learning in Mathematics while challenging classmate to a calculator game, encouraging students to discover certain facts about numbers by themselves, and learning how to minimize the number of keystrokes in solving problems, were found not that much to enhance students learning.
4. The use of computers help the students to understand the lesson easier, to learn with fun, and to become attentive to learn new concepts while being receptive and participative in class discussion, and helping to visualize the underlying principles of the topic with the use of computer do not help that much in the development of students’ skills. However, when it comes to the use of scientific calculator, it helps the students to skip the step-by-step process in solving, to find it easy to solve difficult problems, to spend less time in solving large numbers while reinforcing skills in computation, improving reasoning to a higher-level of thinking, and enhancing analytical thinking, when using scientific calculator do not help that much in the development of students’ skills.
5. The frequent use of modern technology gadgets enhances the students’ learning in Mathematics.
Recommendations:
Based on the findings and conclusions, the following recommendations are offered:
1. Since it was found out that the students learn a lot in using the modern technology gadgets, the school should focus more on how to improve the learning of the students in terms of the utilization of modern technology gadgets (computer and scientific calculator) in learning Mathematics by providing many activities involving the use of modern technology gadgets.
2. As concluded, the frequency of utilizing varied technology-aided teaching strategies has an effect to the learning of the students, the Mathematics teacher should use different kinds of technology-aided teaching strategy which can lead the students to fully develop their skills and knowledge by being flexible enough in dealing with the students.
3. Since the study confirmed that there is a significant relationship between the frequency of utilizing modern technology gadgets and the learning of Mathematics, frequency of using technology-aided teaching strategy that utilizes modern technology gadgets should be based on the needs of the students which capture their interest towards the development of their knowledge and skills.
BIBLIOGRAPHY
A. Books
Bilbao, Purita, et.al., The Teaching Profession. Cubao, Quezon City: Lorimar Publishing Co., Inc. 2006
De Leon, Cecille. Integrated Mathematics IV. Sta. Mesa, Quezon City: Vibal Publishing House, Inc. 2000
Ediger, Marlow. Teaching Mathematics in the Elementary School, Kirksville, Missouri: Simpson Publishing. 1997
Ediger, Marlow. Teaching Mathematics Successfully. Kirsksville, Missouri, U.S.A: Simpson Publishing Company. 1997
Fabra, Norberto. Elementary Statistics for Secondary Schools. Las Pinas, Metor Manila: Fabra, N. F. Educational Books. 1979
Gagne, Robert (1985), The Conditioning of Learning and Theory of Instruction, New York: Holt, Rinehart and Winstonbrain teaser gamesmarble popper gamespc game downloadsmanagement gamescard gameshidden object gamessimulation gamesaction gamesmatch 3 games
Goldstein, I. (1947). Training: Program Development and Evaluation. Monterey, California: Brooks/Cole.
National Council of Teachers of Mathematics. Principles and standards for school mathematics. Reston, VA: Author. 2000.
Norton, Mary. Math Power I. Chicago, Illinois, U. S. A: World Book Inc. 1995
Papert, S.(1980). Children, Computers and Powerful Ideas. Basic Books, New York
Salandanan, Gloria, et.al., The Teaching of Science and Health, Mathematics, and Home Economics and Practical Arts. Quezon City: Katha Publishing Co., Inc. 1996
Zulueta, Francisco. Principles and Methods of Teaching. Mandaluyong City: National Bookstore. 2006
B. Unpublished Thesis
Acuna, Shirley, et.al., Perceived Influence of Rock Music to the Behavior of Third Year High School Students of San Pascual National High School. Batangas State University, Batangas City. 2003
C. Journals
Sammons, Martha C. (1995). Students Assess Computer-Aided Classroom Presentations, T.H.E. Journal 22 (May 1995): 66-69.
D. Electronic Data
Developing Mathematical skills (Loughborough University)
(Website: http://mlsc.lboro.ac.uk) Date Retrieved: July 16, 2008
Glossary of Mathematics Teaching Strategies
(Website: http://cehd.umn.edu/NCEO/Presentations/NCTMLEPIEPStrategies
MathGlossaryHandout.pdf) Date Retrieved: June 22, 2008
Implementation Standard: Instruction with Technology
(Website: http://www.beyondcrossroads.com/doc/CH7.html)
Date Retrieved: August 2, 2008
PowerPoint Presentations: The Good, the Bad and the Ugly
(Website: http://technologysource.org/article/use_of_powerpoint_in_teaching_ comparative_politics/) Date Retrieved: September 24, 2008
Ray, Susan (2000). The Use of Calculators Gets at the Heart of Good Teaching. Louisville, KY (Website: http://www.middleweb.com/CurrMathCalc.html)
Date Retrieved: September 24, 2008
Setzer, Valdemar W. (2000). Computers in Education: A Review of Arguments for the Use of Computers in Elementary Education
(Website: http://www.southerncrossreview.org/4/review.html)
Date Retrieved: August 3, 2008
Stiles, Mark (2002). Computers In Teaching And Learning.
(Website: http://www.staffs.ac.uk/cital/) Date Retrieved: October 2, 2008
Suber (2008). Importance of Notetaking.
(Website: http://www.edchange.org/multicultural/papers/math.html)
Date Retrieved: September 26, 2008
Teaching with Computers in Education
(Website: http://712educators.about.com/cs/technology/a/integratetech.htm)
Date Retrieved: October 1, 2008
Appendices
Appendix A
Letter of Request
First Asia Institute
Of Technology and Humanities
August 12, 2008
MR. ARNOLD CATAPANG
Principal
First Asia Institute of Technology and Humanities Unified School
Sir:
Greetings!
The undersigned BS Secondary Education major in Mathematics student from College of Education is currently undergoing a research entitled, “Perceived Effects of the Utilization of Modern Technology Gadgets in Learning Mathematics of Fourth Year High School students of First Asia Institute of Technology and Humanities.
In view of this, may I request for your approval to allow me distribute my questionnaire to the fourth year students of your institution.
Rest assured that the data and information that will be generated from this study will be utilized solely for academic purposes.
I’m hoping for your usual kind consideration.
Thank you!
Respectfully yours,
(SGD.) MARVELINO M. NIEM
(SGD.) ROSANNI DEL MUNDO, M.A.
Thesis Adviser
(SGD.) EVELIA S. ORBETA, Ed.D
Dean, College of Education
Approved by:
(SGD.) MR. ARNOLD CATAPANG
Appendix B
Cover Letter
First Asia Institute
Of Technology and Humanities
August 12, 2008
Dear Respondents,
Greetings!
The undersigned BS Secondary Education major in Mathematics student from College of Education is presently conducting a research entitled, “Perceived Effects of the Utilization of Modern Technology Gadgets in Learning Mathematics of Fourth Year High School students of First Asia Institute of Technology and Humanities.
In connection with this, may I request for your honest response to the attached questionnaire? This intends to gather information that would greatly enhance the quality of my study. Please do not leave any items unanswered. Rest assured that any information you will give will be treated with utmost confidentiality.
Thank you very much for your cooperation.
Respectfully yours,
(SGD.) MARVELINO M. NIEM
Noted by:
(SGD.) ROSANNI DEL MUNDO, M.A.
Thesis Adviser
(SGD.) EVELIA S. ORBETA, Ed.D
Dean, College of Education
Appendix C
Questionnaire
Name: _(optional)________________ Year &Section: __________
General Directions: Please accomplish the questionnaire very carefully and honestly. Rest assured that any information you will give will be treated with utmost confidentiality.
PART I: Technology-Aided Teaching Strategies and the Frequency of their Uses
Directions: Encircle the number that corresponds to the frequency of the use of such teaching strategy.
Always Often Sometimes Never
1. Lecture method with power 4 3 2 1
point presentation
2. Reporting with power 4 3 2 1
point presentation
3. Independent research through 4 3 2 1
the use of internet
4. Model-making through the 4 3 2 1
use of related software
5. Computer hands-on activities 4 3 2 1
6. Problem solving with scientific 4 3 2 1
calculator
7. Cooperative learning with power 4 3 2 1
point presentation
8. Use of graphic organizers in 4 3 2 1
solving problems
9. Discussion with the use of 4 3 2 1
power point presentation
10. Use of analogy (reasoning) with 4 3 2 1
the use of power point presentation
11. Discovery learning through the 4 3 2 1
use of scientific calculator
12. Guided discovery through the 4 3 2 1
use of related software
PART II: Effects of Utilizing Modern Technology in Learning Mathematics
Directions: Below are the possible effects in learning aided by modern technology. Using the scale, encircle the number that corresponds to your answer:
4- Very much 2- Not so much
3-Much 1- Not at all
Computer
My learning is enhanced when…
the teacher uses power point 4 3 2 1
presentation in presenting the
lesson.
2. the teacher shows animation 4 3 2 1
related to the topic in the monitor.
3. the teacher gives an independent 4 3 2 1
research work using the internet.
4. we work in groups on a particular 4 3 2 1
problem when using computer.
5. the lectures of my teacher were sent 4 3 2 1
to my e-mail.
6. the teacher shows some video 4 3 2 1
presentation related to our topic.
7. we always have computer hands-on 4 3 2 1
activities.
Scientific Calculator
My learning is enhanced when…
1. I use scientific calculator in solving 4 3 2 1
problems involving trigonometric
functions.
2. I evaluate, press the keys and see 4 3 2 1
the results on the display screen of the
scientific calculator.
3. the teacher encourages me to discover 4 3 2 1
by myself certain facts about numbers
using scientific calculator.
4. I challenge my classmate to a calculator 4 3 2 1
game like in being the first to solve
the problem and get the right answer.
5. I learn how to minimize the number 4 3 2 1
of keystrokes in solving problems.
6. I use scientific calculator in finding 4 3 2 1
the factors of a given large number.
7. I use scientific calculator in doing my 4 3 2 1
homework.
PART III: Effects of Using Modern Technology Gadgets in the Development of Students Skills
Directions: Below are the possible effects in learning which is aided by modern technology. Using the scale, encircle the number that corresponds to your answer:
4- Strongly Agree 2- Disagree
3-Agree 4- Strongly Disagree
Computer
By using computers…
1. I become attentive to learn 4 3 2 1
new concepts.
2. I understand the lesson easier. 4 3 2 1
3. I can visualize the underlying 4 3 2 1
principles of the topic.
4. I become more participative in 4 3 2 1
class discussion.
5. I learn while having fun. 4 3 2 1
6. I become interactive in group 4 3 2 1
activities.
7. I am more receptive in the discussion. 4 3 2 1
Scientific Calculator
By using scientific calculator…
1. I develop my skills in solving 4 3 2 1
problems.
2. I reinforce my skills in computation. 4 3 2 1
3. I improve my reasoning to a 4 3 2 1
higher-level of thinking.
4. I enhance my analytical thinking skills. 4 3 2 1
5. I spend less time in solving large 4 3 2 1
numbers.
6. I can skip the step-by-step process 4 3 2 1
on how to solve problems.
7. I find it easy to solve difficult 4 3 2 1
problems.
Appendix D
Tallied Data
PART I: Technology-Aided Teaching Strategies and the Frequency of their Uses.
Item Number
Always
Often
Sometimes
Never
1
39
47
16
0
2
35
41
24
2
3
27
48
25
2
4
17
58
25
2
5
37
37
22
6
6
32
39
22
6
7
24
63
15
0
8
12
55
33
2
9
34
51
15
2
10
14
54
30
4
11
17
36
46
3
12
13
57
30
2
PART II: Effects of Utilizing Modern Technology in Learning Mathematics.
a. Effects of Utilizing Computer in Learning Mathematics
Item Number
Very Much
Much
Satisfactorily
Not Satisfactorily
1
38
58
4
2
2
24
43
34
1
3
16
47
38
1
4
8
45
41
8
5
11
26
30
35
6
20
35
31
16
7
24
41
32
5
b. Effects of Utilizing Scientific Calculator in Learning Mathematics
Item Number
Very Much
Much
Satisfactorily
Not Satisfactorily
1
38
36
25
3
2
33
49
19
1
3
18
46
35
3
4
13
32
35
22
5
18
51
31
2
6
43
33
24
2
7
42
39
18
3
PART III: Effects of Using Modern Technology Gadgets in the Development of Students Skills.
a. Effects of Using Computer Gadgets in the Development of Students Skills.
Item Number
Strongly Agree
Agree
Disagree
Strongly Disagree
1
41
56
5
0
2
25
66
9
2
3
27
54
18
3
4
17
69
16
0
5
43
53
6
0
6
23
57
19
3
7
20
66
16
0
b. Effects of Using Scientific Calculator Gadgets in the Development of Students Skills.
Item Number
Strongly Agree
Agree
Disagree
Strongly Disagree
1
31
52
16
3
2
24
56
21
1
3
26
60
15
1
4
22
61
18
1
5
54
34
14
0
6
45
44
12
1
7
49
42
9
2
CURRICULUM VITAE
MARVELINO MANIPOL NIEM
H
151 San Felix, Sto. Tomas, Batangas
(
09195347768
*
vel_capcom@yahoo.com
PERSONAL INFORMATION
Nickname : Marvel
Birthday : April 20, 1988
Age : 20
Civil Status : Single
Nationality : Filipino
Religion : Roman Catholic
Height : 5’ 5”
Weight : 110 lbs.
Father : Mario Caponpon Niem
Mother : Maria Manipol Niem
Address : 151 San Felix, Sto. Tomas, Batangas
EDUCATIONAL BACKGROUND
2005 –2009
Bachelor of Science in Secondary Education
Major in Mathematics
First Asia Institute of Technology and Humanities
2 Pres. Laurel Hi-way, Darasa, Tanauan City, Batangas
2001-2005
St. Thomas Academy
Poblacion 3, Sto. Tomas, Batangas
Achiever
1995-2001
1994-1995
Sto. Tomas Central School
Poblacion 4, Sto. Tomas, Batangas
Achiever
San Felix Elementary School
San Felix, Sto. Tomas, Batangas
THE PROBLEM
Introduction
Technologies as link to new knowledge, resources and high order thinking skills have entered classrooms and schools worldwide. Personal computers, CD-ROMS, on line services, the World Wide Web and other innovative technologies have enriched curricula and have altered the types of teaching available in the classroom. Schools’ access to technology is increasing steadily everyday and most of these newer technologies are now used even in traditional classrooms (Bilbao et al., 2006).
The use of technology in the classroom has never been underscored but now. However, survey data like that of Smith (1996); Harper (2001); and Lesh and Doerr (2000), suggest that technology remains poorly integrated into schools, despite massive acquisition of hardware. Some recent observations such as of Norton (1996); Cole (2002); and Posey (2001) indicate that the most frequent use of computers is for drill-and-skill practice that supplement existing curricula and instructional practices.
Some thirty years or more ago, the dominant model of teaching was directed instruction or lecture in which students memorize facts. Because of its limitations educationists began exploring the use of technology that supports models of teaching that emphasize learning with understanding and more active involvement. Thus a decision to use technology that is beyond facts-based, memorization-oriented curricula to curricula in which learning with understanding are emphasized was embraced. When to use technology, what technology to use, and for what purpose cannot be isolated from theories of teaching and learning that support learning with understanding (Bilbao et al., 2006).
On the other hand, (Goldman cited in Bilbao et.al., 2006) states that there are some roles of technology in achieving the goal of learning with understanding such as: (a) technology provides support to the solution of meaningful problems; (b) technology acts as cognitive support; and (c) technology promotes collaboration as well as independent learning.
Calculators and other technological tools, such as computer algebra systems, interactive geometry software, applets, spreadsheets, and interactive presentation devices, are vital components of a high-quality mathematics education. With guidance from effective mathematics teachers, students at different levels can use these tools to support and extend mathematical reasoning and sense making, gain access to mathematical content and problem-solving contexts, and enhance computational fluency. In a well-articulated mathematics program, students can use these tools for computation, construction, and representation as they explore problems. The use of technology also contributes to mathematical reflection, problem identification, and decision making (Cole et al., 2002).
However, there are some arguments that the use of technology cannot replace conceptual understanding, computational fluency, or problem-solving skills. In a balanced mathematics program, the strategic use of technology enhances mathematics teaching and learning. Teachers must be knowledgeable decision makers in determining when and how their students can use technology most effectively. All schools and mathematics programs should provide students and teachers with access to instructional technology, including appropriate calculators, computers with mathematical software, Internet connectivity, handheld data-collection devices, and sensing probes. Curricula and courses of study should incorporate instructional technology in learning outcomes, lesson plans, and assessments of students’ progress.
As a student of Mathematics, the researcher observed that the use of modern technology gadgets like computer and scientific calculator was employed and widely used among students. However, these observations made the researcher work on the effectiveness of the utilization of the said gadgets in learning mathematics. The traditional chalk and board are the technology used by the students and teachers in public schools and yet their level of understanding and learning Mathematics is not far behind from the level of the private school students who use modern technology gadgets in learning mathematics.
Which then is more useful: the modern – technology – aided Mathematics instructions or the traditional chalk and board method? What teaching strategies work for effective Mathematics learning? Does the use of computers and scientific calculators really help students?
These questions were examined in this study. In this research, the perceived effects of the utilization of modern technology gadgets in learning mathematics was analyzed for the attempt of providing concrete explanations of the effects of utilization of modern technology gadgets in learning mathematics.
Conceptual Framework
It is commonly observed that teaching strategies defined as the organized, orderly, and logical procedure in imparting knowledge and information among pupils or students (Zulueta, 2006), affect the learning of students. This somehow is true based on the researchers’ experience where he learned more with teachers who utilize varied teaching strategies that suit his needs in learning mathematics. Nowadays, the utilization of modern technology gadgets such as computer and scientific calculator seems to have a great effect on students’ learning.
Scientific calculator is an electronic calculator that has provisions for handling exponential, trigonometric, and sometimes other special functions in addition to performing arithmetic operations. On the other hand, computer is also an electronic devise design to manipulate data so that useful information can be generated (Cole et al., 2002).
To find out if these gadgets affect students’ learning, this study tried to trace the frequency of the use in varied teaching strategies. The assumption of the researcher was that, the frequency of using computer and scientific calculator in teaching, does not affect the learning of students.
This study started with determining the teaching strategy that used modern technology gadgets such as computer and scientific calculator. After determining these teaching strategies, the frequency of their use was identified. When the frequency of its utilization was identified. Their effects in learning mathematics were examined. This was done by relating it to its frequency.
The design of this conceptual model was an original work of the researcher and is not found in any book. This best suited the manner of analyzing the statement of the problem.
The concepts were made simple in the conceptual model on next page.
Teaching Strategies
Using Modern
Technology
Gadgets
Computer
Scientific Calculator
Frequency of
Modern Technology
Utilization
Effects of Using Modern
Technology Gadgets
In Learning
Mathematics
Figure 1. Conceptual Model of the Study
Statement of the Problem
The primary purpose of this study was to determine the perceived effects of the utilization of modern technology gadgets such as computers and scientific calculators in learning mathematics. It involved fourth year high school students of First Asia Institute of Technology and Humanities enrolled in Academic Year 2008 – 2009.
Specifically, this study sought answers to the following questions:
In what teaching strategies does the teacher utilize modern technology gadgets?
How often are these modern technology gadgets used?
How does the use of modern technology gadgets affect the respondents learning?
What are the perceived effects of using modern technology gadgets in learning mathematics?
Is there a significant relationship between the frequency of the utilization of modern technology gadgets and learning of mathematics?
Hypothesis
The study was guided by the null hypothesis:
There is no significant relationship between the frequency of the utilization of modern technology gadgets in the effective learning of mathematics
Significance of the Study
This research study would be useful to the following:
To the teachers, it would help them identify the teaching strategies suited to the needs of their students. The findings of this research would offer them understanding on how to utilize modern technology gadgets to appropriately address students learning style in mathematics.
To the parents, the result of this study would help them encourage their children to be keen on the varied styles of learning.
To the students, it would help them determine their preferred teaching strategies as well as the use of modern technology gadgets in learning.
To the Mathematics Heads, this study would help them identify the needs of the students with regard to the utilization of modern technology gadgets in the learning Mathematics.
Finally, this research would serve as reference material to other students and future researchers who will be interested to work on similar topic.
Scope and Delimitation of the Study
This study aimed to determine the perceived effects of the utilization of modern technology gadgets in learning mathematics. It involved one hundred two (102) fourth year high school students of First Asia Institute of Technology and Humanities enrolled in the Academic Year 2008 – 2009. They were chosen as respondents because it is the year level when Trigonometry is taken up which require the use of calculator in solving trigonometric functions.
This research aimed to point out the use of modern technology gadgets such as computer and scientific calculator. Computers are observed to be usually used in presenting the lesson, in research and video presentations. Similarly, scientific calculators are utilized in solving large numbers, factoring and games.
The teaching strategies made by the teacher were also analyzed because the researcher assumed that the teachers’ teaching strategies play important roles in students’ learning. This study further analyzed the frequency of using computers and calculators in teaching were believed to affect the respondents learning. This study also attempted to find the significant relationship between the frequency of the utilization of modern technology gadgets in the effective learning of mathematics.
This study only examined the use of two modern technology gadgets – computer and scientific calculator in learning mathematics. It did not cover any other types of gadgets like graphing calculator and projector.
The researcher used a validated, self-constructed questionnaire as an instrument in measuring responses of the respondents based on their perception about the effects of modern technology gadgets in their learning in mathematics.
Definition of Terms
The following terms were conceptually and operationally defined so as to give clearer understanding of some concepts used in the present study.
Analogy. It is a strategy that can be utilized in promoting comprehension and reasoning among students (Gagne, 1985). In this study, it is the strategy wherein the student will find out the pattern of the particular problem or how he/she will reason it out.
Computers. It is an electronic devise designed to manipulate data so that useful information can be generated (Cole et al., 2002). In this study, computer is used as one of the modern technology teaching gadgets used by the teacher in teaching mathematics.
Cooperative Learning. It is an instructional paradigm in which teams of students work on structured tasks (e.g., homework assignments, laboratory experiments, or design projects) under conditions that meet five criteria: positive interdependence, individual accountability, face-to-face interaction, appropriate use of collaborative skills, and regular self-assessment of team functioning (Johnson, Johnson, and Smith cited in Zulueta , 2006). In this study, it deals with the kind of group work wherein the students seek involvement and cooperation with one another.
Discovery Learning. It consists of instruction in which the learners draw their own conclusions from information they were able to glean by themselves as provided by their teachers or others (Clark and Starr cited in Zulueta , 2006). In this study, it is used as the students will discover for themselves the particular topic presented to them.
Discussion. This method allows for interaction between the teacher and the students as well as among themselves (Salandanan et al., 1996). In this study, it is one of the strategies where the students will have interaction between their teachers or among themselves.
Graphic Organizers. It is a collective group of strategies that provide visual representations as a means of organizing and presenting information. They make visible the thinking of the students. They help students represent abstract concepts and ideas in concrete forms. They display the relationships between pieces of information, connect new learning to prior learning and generally organize information into a more useful form (Salandanan et al., 1996). In this study, it refers to the visual materials used by the mathematics teacher.
Guided Discovery. It is the process that teachers used to introduce new materials, explore centers or areas of the classroom, and prepare students for various aspects of the curriculum (Salandanan et al., 1996). In this study, it is the teacher who is acting as a facilitator of learning and guiding his students in learning mathematical problem.
Lecture. It is a teaching procedure of classifying or explaining a major idea cast in the form of question or a problem (Bossing cited in Zulueta , 2006). In this study, it is the teacher’s teaching strategy when he is providing the students all the needed information without interaction at all.
Mathematics. It is the study of quantities and relations through the use of numbers and symbols (Norton et al., 1996). In this study, it is the subject or course that the researchers worked on.
Model-Making. It is a strategy that teachers can use in teaching application (Eby and Kujawa cited in Zulueta , 2006). In this study, it refers to the strategy wherein the students make a model for them to visualize the given worded problem.
Power point. A computer based media presentation using slideshow (Cole et al., 2002). In this study, it is one of the computer software that is used by the teacher in presenting the lesson.
Reasoning It is the cognitive process of looking for reasons for beliefs, conclusions, actions or feelings (Kirwin cited in de Leon, 2000). In this study, it refers to how the students reason out their feelings for something.
Research. It is defined as human activity based on intellectual application in the investigation of matter. The primary aim for applied research is discovering, interpreting, and the development of methods and systems for the advancement of human knowledge on a wide variety of scientific matters of our world and the universe (Trochim, 2006). In this study, it refers to the teaching strategy that helps the students develop their students’ thinking, reasoning and research skills.
School. It is an educational institution, private and public, undertaking educational operation with specific age-group of pupils or students pursuing defined studies at defined levels, receiving instruction from teachers, usually located in a building or a group of buildings in a particular physical or cyber site (Oreta, 2001). In this study, it refers to First Asia Institute of Technology and Humanities Unified School where the respondents are currently enrolled.
Scientific Calculators. It is a type of electronic calculator, usually but not always handheld, designed to calculate problems in science (especially physics), engineering, and mathematics. They have almost completely replaced slide rules in almost all traditional applications, and are widely used in both education and professional settings (Cole et al., 2002). In this study, it is one of the teaching gadgets used by the teacher and also the students for them to be guided for finding the accurate answer.
Students. It refers to those who enrolled in and who regularly attend an educational institution of secondary or higher level or a person engaged in formal study (Salandanan et al., 1996). In this study, it refers to the Fourth Year students of First Asia Institute of Technology and Humanities Unified School enrolled in Academic Year 2008 – 2009.
Teachers. They are the key-learning person who is responsible for supervising/ facilitating the learning processes and activities of the students (Oreta, 2001). In this study, it refers to all concerned mathematics teacher of First Asia Institute of Technology and Humanities.
Teaching Strategies. It refers to the organized, orderly, and logical procedure in imparting knowledge and information among pupils or students (Zulueta, 2006). In this study, it refers to the strategies used by the teacher in presenting or delivering the topic.
Technology. It refers to how we apply science slideshow (Cole et al., 2002). In this study, it refers to computer and scientific calculator as means of teaching and learning lessons in Mathematics quickly.
Trigonometry. It is a branch of mathematics that combines arithmetic, algebra and geometry (Frere cited in de Leon, 2000). In this study, it refers to the subject that requires the use of modern technology gadgets.
CHAPTER II
REVIEW OF LITERATURE
This chapter presents a review of existing and available literature as well as related studies on the perceived effects of utilization of modern technology gadgets in learning mathematics of fourth year high school students. Only those studies that showed certain aspects of relationship with the present study were selected and their findings were discussed. Its last part is a synthesis of all the reviewed literature.
Conceptual Literature
Definition and Use of Technology in Mathematics. The definition and use of technology in mathematics education is constantly evolving. Technology may refer to the use of graphing calculators, student response systems, online laboratories, simulations and visualizations, mathematical software, spreadsheets, multimedia, computers or the Internet, and other innovations yet to be discovered (Cole, 2002).
Technology can be used to learn mathematics, to do mathematics, and to communicate mathematical information and ideas. The Internet hosts a wealth of mathematical materials that are easily accessible through the use of search engines, creating additional avenues to enhance teaching and facilitate learning. Outside of class, students and faculty can pose problems and offer solutions through e-mail, chat rooms, or websites. Technology provides opportunities for educators to develop and nurture learning communities, embrace collaboration, provide community-based learning, and address diverse learning styles of students and teaching styles of teachers (NCTM, 2000).
The integration of appropriately used technology can enhance student understanding of mathematics through pattern recognition, connections, and dynamic visualizations. Electronic teaching activities can attract attention to the mathematics to be learned and promote the use of multiple methods. Learning can be enhanced with electronic questioning that engages students with technology in small groups and facilitates skills development through guided-discovery exercise sets. Using electronic devices for communication, all students can answer mathematics questions posed in class and instructors can have an instantaneous record of the answers given by each student. This immediate understanding of what students know, and don’t know, can direct the action of the instructor in the teaching session (NCTM, 2000).
According to Pea (cited in NCTM, 2000), cognitive technologies serve two transcendent functions. First, technologies have purpose functions. They serve to engage students in the activity of mathematical and scientific inquiry. This provides meaning for engagement, ownership of the mathematics and science being learned, and empowerment through the generation of personal agency. Technologies engage students in more powerful mathematical activity in a way that could not be approached without them.
But technologies are not by nature engaging. To achieve this quality, they must be both functional (teachers and students must be able to do with them something that they could not do without them), and they must increase communication and facilitate collaboration. Second, technologies have process functions. Some of the tools available for students should free up their working memory so that they are able to concentrate on problem formulation and modeling (Cole, 2002).
If a high school student is bogged down with computing or graphing, the big picture of number systems, functions, families of curves, etc., is lost. Other tools must provide opportunities for exploration and discovery. In a mediated learning environment, some agent (teacher, peer, and tool) must bridge the informal knowledge of the student and the formalism of mathematical and scientific structure. Still other tools must provide ways of representing mathematical and scientific models and linking representations to make the underlying commonalties transparent (Lesh cited in Cole, 2000).
A single technology rarely has all these process functions. However, a careful selection of tools and software as described in this article can help achieve the necessary complementarities. Two other features of cognitive technologies are necessary for the development of coherent mathematical and scientific structures. The first is what Roschelle (cited in Cole, 2000) called epistemic fidelity. This refers to the requirement that any teaching tool must reflect and develop understandings that are true to the field of study. Students’ mathematical and scientific activity should develop the kinds of understandings that experts in the field would recognize. Two caveats are in order. The road from novice to expert goes through several transformational periods and may not be immediately recognizable as important without an understanding of students’ cognitive development.
Second, the sophisticated knowledge of handed to students. The path taken is as much a part of expert understanding as the final product. The other necessary feature of cognitive technologies should focus the students’ attention on the mathematical structure of the experiences and provide them with a means of communicating their thinking about this structure to others. This is, in its basic form, the engagement of students in mathematical and scientific modeling. The vision that guides the integration of technology, science, and mathematics is the engagement of students in activity that elicits the development of mathematical and scientific models with a coherent epistemological framework. The movement from informal discovery to more formal models marks an authentic transition between the exploratory knowledge of the student, and the theoretical knowledge of the expert (Kozulin & Presseisen cited in Cole, 2000).
Teaching with Technology. In recent years integration of instructional technology (IT) into the university classroom has become a significant part of education. As such resources for educating and assisting faculty in this new arena have become crucial. Workshops on how to use software are not always enough. Teachers need to understand the ways in which these new tools can make a significant difference in student teaching (Cole, 2000).
Developments and advances in technology–hardware and software–have had a tremendous impact on our lives. The infusion of technology into education presents interesting opportunities for teaching and learning, especially in mathematics. Technology changes not only how mathematics is taught, but also when and what mathematics is taught (Posey, 2001).
In one review by Smith (1996), as can be inferred from the six principles, the kind of software and the way it is used are also crucial elements. Common features of the software used in this program are that it can be used by high school students; it is user friendly; it is designed for the kind of computers available in schools; and most important, students are in control, telling the computer what to do rather than the computer telling students what to do. The kinds of software used range from general purpose tools to specialized programs for science and mathematics learning. The particular software used can change from year to year. Typically, four or five kinds of technology are used in depth, including computer-based software and graphing calculators. Although prospective teachers become quite expert in the use of the technology, the main goal is that their future students use technology to explore concepts and solve problems in science and mathematics. Prospective teachers use it in conjunction with hands-on materials, such as geoboards and polyhedra, and activities such as paper folding. Use of natural objects and outdoor activities are also an important part of integration (Harper cited in Stiles, 2002).
However, in the study of Middleton and Goepfert (cited in Papert, 1980), there are six principles that guide the design of choosing the equipment and software, such as: (1) technologies are only tools. Technologies neither supplant the thought processes of students, nor do they make learning fun or easy. Technologies are instruments that should be used judiciously at the proper time in the proper place. (2) Technologies should enable students do what they could not do without them. When used appropriately, technologies help students expand their zone of proximal development. This can serve to make learning more intentional, powerful, and connected. In addition, computer technologies can represent situations unfeasible with other types of tools. (3) Technologies must be on hand all the time. The context, social setting, and tools that students use to construct their mathematical and scientific knowledge are inseparable from the knowledge itself. For technologies to be authentically integrated into students’ learning activity, they must be available when the question arises.
The fourth (4) principle is that tools should facilitate the creation of sharable, modifiable, transportable models of mathematical and scientific concepts (Lesh & Doerr, cited in Papert, 1980). Technologies facilitate the development of public records of thought. These records should be shared as students develop, refine, and test models of mathematical and scientific phenomena. It is crucial that students can modify them, as most models students construct in the beginning are either incomplete, or contain misconceptions. Through discourse, the shared model can be pared down into a workable model that can serve the class as a whole. (5) Sharing of data/resources should be simple. Technological systems should be user friendly. The mechanism of communication should not be more complex than the learning process itself. (6) The setup of the workstations should facilitate collaboration between students. As collaborative tools, technologies are imbedded within the geography, culture, and psychology of the classroom. The setup should facilitate collaborative inquiry, but also engage students in independent exploration.
The National Council of Teachers in Mathematics in USA (2000) stressed that technology can be used to learn mathematics, to do mathematics, and to communicate mathematical information and ideas. The Internet hosts a wealth of mathematical materials that are easily accessible through the use of search engines, creating additional avenues to enhance teaching and facilitate learning. Outside of class, students and faculty can pose problems and offer solutions through e-mail, chat rooms, or websites. Technology provides opportunities for educators to develop and nurture learning communities, embrace collaboration, provide community-based learning, and address diverse learning styles of students and teaching styles of teachers. The integration of appropriately used technology can enhance student understanding of mathematics through pattern recognition, connections, and dynamic visualizations. Electronic teaching activities can attract attention to the mathematics to be learned and promote the use of multiple methods. Learning can be enhanced with electronic questioning that engages students with technology in small groups and facilitates skills development through guided-discovery exercise sets. Using electronic devices for communication, all students can answer mathematics questions posed in class and instructors can have an instantaneous record of the answers given by each student. This immediate understanding of what students know, and don’t know, can direct the action of the instructor in the teaching session.
According to Cole (2002), technology helps students document the validity of their mathematical/critical thinking process, facilitating and enriching the learning processes and the development of problem-solving skills. The use of technology should be guided by consideration of what mathematics is to be learned, the ways students might learn it, the research related to successful practices, and the standards and recommendations recommended by professional organizations in education. Technology can be used by mathematics educators to enhance conceptual understanding through a comparison of verbal, numerical, symbolic, and graphical representations of the same problem.
Teaching with PowerPoint. Both courses have been regularly taught with PowerPoint to supplement lecture and discussion. The choice of PowerPoint as the technological solution was driven substantially by two factors: the suitability of a visual tool to studying politics in foreign countries, and the ease with which PowerPoint can be learned and integrated with existing course material. Comparative politics, the systematic study of domestic political conditions and tendencies in various countries, is ideally suited to a highly visual approach, particularly in imparting to American students a broad overview of the historical, social, and economic conditions which underlie the politics of countries such as Great Britain, France, Germany, the European Union, Japan, Russia, China, India, Nigeria and Mexico. Since the students in question in many cases have not been out of the United States, each unit is begun with a "virtual tour" of the country in PowerPoint, featuring photographs of people and places in the country in question, accompanied by a loop of national music. Clipart from Microsoft and other vendors is extensively used to illustrate the flags, crests, maps, currencies, and landmarks of the country and region. Pie charts are used to express ethnic diversity and budgetary allocations. Line and bar charts are used for economic trends and comparative public policy performance, such as budget deficits, health care expenditures, inflation and unemployment rates. In most cases these can be constructed within PowerPoint itself using the Microsoft Graph 5.0, but in a few instances of advanced mixed chart formats, the charts are created in Microsoft Excel and pasted into the PowerPoint slide. Photographs of political leaders from Microsoft Bookshelf, the World Wide Web and other sources are used to illustrate slides. Short video clips from CNN and other sources are inserted to add variety and a dynamic quality to the presentations. Finally, complex processes can be illustrated easily by use of the Auto shapes. It is also possible to build complex scenario presentations in which students must discuss and choose between difficult alternatives facing poor developing countries, in this case Nigeria. The choices made then branch off to other short presentations which explore the advantages and drawbacks of the choices made by the students (Sammons, 1995).
In the same text, a further advantage of teaching comparative politics with PowerPoint is that it is now possible to go beyond mere text as the focus of testing and broaden the scope of what teachers can expect students to learn and retain. Quizzes and tests can be presented as a PowerPoint presentation, and ask essay, fill-in or multiple-choice questions, reducing photocopying costs for departments in an era of diminishing resources and increased expectations. Furthermore, a PowerPoint quiz can test students' recognition of leaders, flags, and maps; such a quiz may involve an essay reacting to a chart, graph, or photograph, moving students beyond the goal of grasping secondary knowledge and toward reacting to and interpreting primary data.
One additional advantage of using PowerPoint is the ability to easily produce handout sheets with the bullet points clearly printed out. These were produced in the three-slides-per-page format, allowing students plenty of room to write additional information from class lecture and discussion. These sheets are then photocopied as a course pack by a local vendor and available to students at the beginning of the semester (Sammons, 1995).
In the same text student reactions to the use of PowerPoint have been overwhelmingly positive. Surveys distributed to students during the semester asked students their reactions to a variety of statements concerning the use of PowerPoint in the classroom, using a five-point feeling thermometer (ranging from "strongly agree" to "strongly disagree"). The instrument was based partly on the instrument used in the Wright State University Pilot Project which used presentational software (though not PowerPoint) in general education courses. Although the reaction in both Wright State and IUP to the use of presentational software was quite positive, reactions at IUP were in most cases higher than at Wright State. That having been said, the much smaller sample size of the IUP experiment prevents any broad comparative conclusions at this point. The results from two semesters of comparative politics (total of forty-seven students responding) are reproduced below.
PowerPoint is one of many alternatives in teaching with technology. PowerPoint works very well in comparative politics, but is far less suited to a subject such as political theory, where visual elements are limited. Political theory, on the other hand, is ideal for instruction via the World Wide Web, since many of the great books of Western political thought are available in full-text online. What works for one subfield does not necessarily work well in another. In examining the use of PowerPoint as an alternative in technological approaches to teaching, a number of factors should be borne in mind. First, the suitability of the subject matter to a highly visual approach is key. Although PowerPoint can be used effectively without photos, clipart, or charts of any kind, the real attraction of the software is the seamless integration of text and visual elements. Subjects such as art, history, nutrition, area studies, safety sciences, geography, anatomy, zoology, physical education, computer science, archaeology and military sciences all have a wealth of visual elements which are easily inserted in PowerPoint. Other subjects, such as philosophy, linguistics, English composition and law would need much more imaginative application of PowerPoint, since many of these disciplines are highly textual in nature. PowerPoint is ideal for teaching with the case study method, beginning with the "facts of the case" and then turning to the questions and discussion. For other disciplines, such as economics, the challenge is to go beyond the charts and bring additional visual elements that enliven and illustrate abstract principles with concrete examples (Sammons, 1995).
In the same text, faculty willingness to learn new technologies and apply them to their teaching is a second challenge in technological selection. PowerPoint has the advantage that it is - for a remarkably powerful presentation package - surprisingly easy to learn and use. That ideal combination of power and ease of use is rarely seen in educational software. Faculty can be easily persuaded that learning PowerPoint does not represent the same kind of learning curve that mastering HTML might. Furthermore, teaching with PowerPoint does not necessarily involve radical changes to teaching approaches, though it can if the instructor so desires. Even as a tool to create better-designed black and white or color transparencies, PowerPoint enforces simple but important rules of highly effective media design in the point sizes of text, bullets, framing, and layout. Once faculty begins to use PowerPoint in this simple way, it is a short step to using PowerPoint as an electronic slide show, where the marginal cost of a new slide is virtually nil. As a simple supplement to traditional lecture and discussion, most instructors see PowerPoint as a simple yet highly effective step forward. Naturally, the technology can be taken much further, and should be, but bringing technology into education is not simply a matter of establishing the cutting edge. As often as not, advancing technology means taking the small steps that introduce new approaches to a broader audience.
The number one challenge to the effective use of PowerPoint in the classroom is clear: the need for effective, cost-efficient, flexible projection systems. This shows up in surveys more than any other complaint. A number of alternatives exist, but there are serious trade-offs in the choices. Linking a laptop computer with PowerPoint to a frame capture or scan capture system which converts monitor images to TV images is fairly cheap, but visibility is limited, making this suitable for smaller classes only. A computer linked to an LCD panel on an overhead projector is a slightly more expensive system, but not only does it require a special highly reflective screen, it also needs a higher-power overhead, both raising the cost of the system. In any case, the image is washed out in all but the most darkened rooms. Ceiling-mounted RGB projectors can show a much better image, but are somewhat expensive and are not portable. These are well-suited to larger auditorium, but instructors who wish to teach their courses with both PowerPoint and cooperative learning techniques which break classes into small groups will find that venue difficult at best. The optimal solution are the LCD projectors on the market, which when combined with a high-performance laptop computer loaded with PowerPoint can give a crisp, visible image for small or large groups, and are portable. This optimal solution, however, is not cheap, and better quality LCD projectors run several thousand dollars. Fortunately, the prices on these devices are dropping, and their flexibility favors sharing them with different instructors or different departments (Sammons, 1995).
Using Calculators versus Paper and Pencil in Mathematics. According to Ray (2000), when she conducts workshops, she does an activity where she asks the participants to get into groups of 4. She has 12 math problems from a 5th grade textbook on an overhead transparency and she asks each group to solve the problems one at a time. One participant has to use a calculator, one uses paper and pencil, and one does it in their head. The fourth decided who gets the problem correct first. Each time that I've done this, mental math wins out about 60% of the time with the calculator coming in about 35% and paper and pencil 5%. What does this say? These are the rules that she uses when it comes to appropriate use of a calculator: (1) Use a calculator as a tool in problem solving... sort of like a "fast pencil." (2) Use a calculator for complex computations but NOT for basic facts! (3) Use a calculator to develop number concepts and skills. (4) Use a calculator in testing situations when not assessing computational proficiency. In general, she uses calculators to help teach mathematics matter, not to replace the teaching of mathematics.
However, according to Ediger (1997), the use of scientific calculator is more enjoyable, excellent to notice fewer errors made in the computation, students have more fun and checking one’s work with a calculator could be done just as easily and accurately that’s why teacher preferred the students to use it to enhance their learning.
On the other hand, when using paper and pencil, pupils have more time to deliberate in problem solving as compared with the use of calculator. Calculators provide the correct results if the user touches the proper keys. Pupils get frustrated if they see long answers and lastly, they preferred to use paper and pencil method when it comes to solve less complex problems (Ediger, 1997).
Guiding Principles and Effective Practices of Graphic Organizers. Visual displays and representations of information, commonly called graphic organizers, have become standard practice in most educational settings. But simply using a graphic organizer does not guarantee enhanced student understanding or achievement. Research and best practices have shown that, for graphic organizers to be effective instructional tools, several factors must be addressed. First, the graphic organizers need to be very straightforward and coherent. Next, students must be taught how to use the graphic organizer. Finally, teachers should consistently use graphic organizers during all aspects of instruction so that students begin to internalize the organizational skills of the graphic display (Boyle, 1997).
Graphic organizers are visual displays of key content information designed to benefit learners who have difficulty organizing information (Fisher & Schumaker cited in Boyle, 1997). Sometimes referred to as concept maps, cognitive maps, or content webs, no matter what name is used, the purpose is the same: Graphic organizers are meant to help students clearly visualize how ideas are organized within a text or surrounding a concept. Through use of graphic organizers, students have a structure for abstract ideas. Graphic organizers can be categorized in many ways according to the way they arrange information: hierarchical, conceptual, sequential, or cyclical (Bromley, Irwin-DeVitis, & Modlo cited in Boyle, 1997). Some graphic organizers focus on one particular content area. For example, a vast number of graphic organizers have been created solely around reading and prereading strategies (Merkley & Jeffries cited in Boyle, 1997).
In presenting a graphic organizer, there are things to be considered, like: (1) keep them simple. For graphic organizers to be effective instructional tools, they must be clear and straightforward (Boyle & Yeager, 1997; Egan cited in Boyle, 1997). The connections and relationships between the ideas depicted in the organizer should be obvious, otherwise the academic benefits will be limited. If an organizer is poorly constructed, includes too much information, or contains distractions, students can easily become confused and even more disorganized than before in their understanding of the target concepts (Robinson cited in Boyle, 1997). Therefore, teachers must keep graphic organizers simple. Suggestions for following this principle include: (a) limit the number of ideas covered in each organizer. Focus on essential concepts that students need to understand and remember; (b) include clear labels and arrows to identify the relationships between concepts; and (c) be careful of graphic organizers that accompany teacher resource materials. They often contain many pictures or background visuals that are distracting to students.
(2) Teach to and with the organizer. As with all instructional tools, students need to be taught how to use graphic organizers effectively and efficiently. Students enter the classroom with varied experiences using graphic organizers. Therefore, teachers must give explicit instructions about how to organize information and when a particular organizer is beneficial. With such guidance and scaffolding, students gain greater independence with graphic organizers.
Once students understand how to use an organizer, teachers need to implement it in creative and engaging ways to enhance effectiveness (Bromley cited in Boyle, 1997). As organizers have become more common, simply using an organizer is no longer enough to maintain students’ attention and focus. The following ideas will help ensure that students are engaged with organizers. (a) Allow students to add illustrations. As long as the pictures add to a student’s understanding of the concepts displayed and do not distract, illustrations can be very engaging. (b) Implement organizers with cooperative groups or pairs of students. Organizers can be excellent tools for discussion and student engagement with each other. (c) Allow students to make their own organizers and share them with the class. As students become more comfortable using organizers, they can teach the strategies they use to organize information for the whole group.
(3) Use graphic organizers often. Many students benefit from routine and structure, so using graphic organizers consistently in the classroom will help them internalize the organizing techniques that are being taught (Griffin & Tulbert cited in Boyle, 1997). The more students are exposed to organizers, the more familiar and comfortable they will become using them. Here are some things to consider when trying being consistent: (a) establish a routine for using organizers during instruction. For example, always use a web when starting a new unit, no matter what the subject area is. Use the same sequence chart when ordering events or steps in math, reading, writing, science, or social studies. (b) Incorporate organizers into all phases of instruction. When students see them used as a warm-up, a guided practice, or a homework assignment, they better understand the purpose and the benefits of the organizer. (c) If students have difficulty using a particular organizer, don’t give up. Students will often struggle with new approaches. Stay consistent and keep providing them guidance and practice. When students see the teacher using an organizer consistently, they are more likely to understand it themselves.
Development of Students’ Skills through the Use of Technology. Technology helps students document the validity of their mathematical/critical thinking process, facilitating and enriching the learning processes and the development of problem-solving skills. The use of technology should be guided by consideration of what mathematics is to be learned, the ways students might learn it, the research related to successful practices, and the standards and recommendations recommended by professional organizations in education. Technology can be used by mathematics educators to enhance conceptual understanding through a comparison of verbal, numerical, symbolic, and graphical representations of the same problem (Cole, 2002).
Students can use technology to search for patterns in data, while allowing the technology to perform routine and repeated calculations. The use of technology should not be used as a substitute for an understanding of and mastery of basic mathematical skills. Technology should be used to enhance conceptual understanding, while simultaneously improving performance in basic skills (Posey, 2001).
Educational Technologies in Learning. Educational technologies are not single technologies but complex combinations of hardware and software. These technologies may employ some combination of audio channels, computer code, data, graphics, video, or text. Although technology applications are frequently characterized in terms of their most obvious hardware feature (e.g., a VCR or a computer), from the standpoint of education, it is the nature of the instruction delivered that is important rather than the equipment delivering it. In this chapter, we review the history and current status of educational technologies, categorized into four basic uses: tutorial, exploratory, application, and communication. Our categories are designed to highlight differences in the instructional purposes of various technology applications, but we recognize that purposes are not always distinct, and a particular application may in fact be used in several of these ways (Conaty, 1993).
In the same text, tutorial uses are those in which the technology does the teaching, typically in a lecture-like or workbook-like format in which the system controls what material will be presented to the student. In our classification scheme, tutorial uses include (1) expository learning, in which the system provides information; (2) demonstration, in which the system displays a phenomenon; and (3) practice, in which the system requires the student to solve problems, answer questions, or engage in some other procedure.
Exploratory uses of technology are those in which the student is free to roam around the information displayed or presented in the medium. Exploratory applications may promote discovery or guided discovery approaches to helping students learn information, knowledge, facts, concepts, or procedures. We also include reference applications, such as CD-ROM encyclopedias, in this category. In contrast to tutorial uses in which the technology acts on the student, in exploratory uses the student controls the learning (as in exploring microworlds or hypermedia stacks) (Conaty, 1993)..
In the same text. application uses, such as word processors and spreadsheets, help students in the educational process by providing them with tools to facilitate writing tasks, analysis of data, and other uses. In addition to word processors and spreadsheets, applications include database management programs, graphing software, desktop publishing systems, and videotape recording and editing equipment.
Communication uses are those that allow students and teachers to send and receive messages and information to one another through networks or other technologies. Interactive distance learning via satellite, computer and modem, cable links, or other technologies constitutes another example of communication uses (Conaty, 1993).
Historically, the dominant teaching-learning model has been one of transmission: teachers transmitting information to students. Not surprisingly, the first uses of educational technology supported this mode. Although other ways of using technology to support learning are now available, tutorial uses continue to be the most widespread, especially with disadvantaged students (Becker cited in Conaty, 1993).
Computer-Assisted Instruction--Some of the first computer-assisted instruction (CAI), developed by Patrick Suppes at Stanford University during the 1960s, set standards for subsequent instructional software. After systematically analyzing courses in arithmetic and other subjects, Suppes designed highly structured computer systems featuring learner feedback, lesson branching, and student record keeping (Coburn cited in Teaching with Computer in Education, 2008).
During the 1970s, a particularly widespread and influential source of computer-assisted instruction was the University of Illinois PLATO system. This system included hundreds of tutorial and drill-and-practice programs. Like other systems of the time, PLATO's resources were available through timesharing on a mainframe computer (Coburn cited in Teaching with Computer in Education, 2008).
Today, microcomputers are powerful enough to act as file servers, and CAI can be delivered either through an integrated learning system or as stand-alone software. Typical CAI software provides text and multiple-choice questions or problems to students, offers immediate feedback, notes incorrect responses, summarizes students' performance, and generates exercises for worksheets and tests. CAI typically presents tasks for which there is one (and only one) correct answer; it can evaluate simple numeric or very simple alphabetic responses, but it cannot evaluate complex student responses.
Integrated learning systems (ILSs) are networked CAI systems that manage individualized instruction in core curriculum areas (mathematics, science, language arts, reading, writing). ILSs differ from most stand-alone CAI in their use of a network (i.e., computer terminals are connected to a central computer) and in their more extensive student record-keeping capabilities. The systems are sold as packages, incorporating both the hardware and software for setting up a computer lab.
ILSs are typically sold in sets of 30 workstations, with an average cost of about $125,000. Major producers include Josten's Learning Corporation, WICAT Systems, and CCC (founded by Patrick Suppes). About 10,000 ILSs are in use in U.S. schools, most of them purchased with funds from the ESEA Chapter 1 program for at-risk students (Mageau cited in Teaching with Computer in Education, 2008).
The instructional software within ILSs is typically conventional CAI: instruction is organized into discrete content areas (mathematics, reading, etc.) and requires simple responses from students. ILS developers have also made a point of developing systems that tie into the major basal textbooks. Mageau (cited in Teaching with Computer in Education, 2008) notes that the systems "can correlate almost objective by objective to a district's K-8.... language arts, reading, math, and even science curricula". Users of ILSs enjoy the advantage of having one coordinated system, making it easy for students to use a large selection of software.
A new trend in integrated learning systems is represented by ClassWorks, developed by Computer Networking Specialists. ClassWorks offers the school access to whatever variety of third-party software the teachers select, along with all the instructional management features associated with an ILS (Mageau cited in Teaching with Computer in Education, 2008).
CAI in general, and integrated learning systems in particular, have found a niche in America's schools by fitting into existing school structures (Newman cited in Teaching with Computer in Education, 2008). Cohen (cited in Teaching with Computer in Education, 2008) describes these structures as follows: (a) most instruction occurs in groups of 25 to 35 students in small segments from 45 to 50 minutes long; (b) instruction is usually either whole-class or completely individual; (c) instruction is teacher dominated, with teachers doing most of the talking and student talk confined largely to brief answers to teacher questions; (d) when students work on their own, they complete handouts devised or selected by the teacher. Students have little responsibility for selecting goals or deadlines and little chance to explore issues in depth. Most responses are brief; (d) knowledge is represented as mastery of isolated bits of information and discrete skills.
Many features of tutorial CAI are consistent with the traditional classroom described by Cohen. Tutorial CAI provides a one-way (computer to student) transmission of knowledge; it presents information and the student is expected to learn the information presented. Much CAI software presents information in a single curriculum area (e.g., arithmetic or vocabulary) and uses brief exercises that can easily is accommodated within the typical 50-minute academic period. CAI is designed for use by a single student and can be accommodated into a regular class schedule if computers are placed in a laboratory into which various whole classes are scheduled.
Basic skills (such as the ability to add or spell) lend themselves to drill- and-practice activities, and CAI, with its ability to generate exercises (e.g., mathematics problems or vocabulary words) is well suited to providing extensive drill and practice in basic skills. Students at risk of academic failure often seen as lacking in basic skills and therefore unable to acquire advanced thinking skills become logical candidates for CAI drill-and-practice instruction. Recent research and thinking on the needs of disadvantaged students stress a different need. Disadvantaged students need the opportunity to acquire advanced thinking skills and can acquire basic skills within the context of complex, meaningful problems. This latter approach to instruction, which is stressed in education reform, has not been well served by traditional CAI.
Intelligent Computer-Assisted Instruction-- Intelligent computer-assisted instruction (ICAI, also known as intelligent tutoring systems or ITSs) grew out of generative computer-assisted instruction. Programs that generated problems and tasks in arithmetic and vocabulary learning eventually were designed to select problems at a difficulty level appropriate for individual students (Suppes cited in Teaching with Computer in Education, 2008).
These adaptive systems (i.e., adapting problems to the student's learning level) were based on summaries of a student's performance on earlier tasks, however, rather than on representations of the student's knowledge of the subject matter (Sleeman & Brown cited in Teaching with Computer in Education, 2008). The truly intelligent systems that followed were able to present problems based on models of the student's knowledge, to solve problems themselves, and to diagnose and explain student capabilities.
Historically, ICAI systems have been developed in more mathematically oriented domains--arithmetic, algebra, programming--and have been more experimental in nature than has conventional CAI. Although ICAI is an area of active research projects, ICAI programs in the schools are not widespread. ICAI tends to call for more meaningful interactions than traditional CAI and tends to deal with more complex subject matter.
ICAI's focus on modeling student knowledge lends itself to applications in teaching advanced thinking skills. ICAI has not been used extensively with disadvantaged students (traditional targets for basic skills instruction).
One intelligent tutoring system, Geometry Tutor, provides students with instruction in planning and problem solving to prove theorems in geometry (Office of Technology Assessment cited in Teaching with Computer in Education, 2008). Geometry Tutor comprises an expert system containing knowledge of how to construct geometry proofs, a tutor to teach students strategies and to identify their errors, and an interface to let students communicate with the computer. Geometry Tutor monitors students as they try to prove theorems, instructing and guiding them throughout the problem-solving process (Anderson et al. 1985). Schofield, Evans-Rhodes, and Huber (cited in Teaching with Computer in Education, 2008) studied the implementation of Geometry Tutor in a public high school and found changes in the behavior of teachers and students using this system: teachers spent more time with students having problems, collaborated more with students, and based more of a student's grade on effort; students increased their level of effort and were more involved in the academic tasks. Thus, ICAI can be implemented in ways that support the kind of learning that education reformers advocate. Although most of these applications control instructional content, they can be used within a broader instructional framework that stresses joint work with the automated tutor.
Technologies for tutorial learning typically use a transmission rather than constructivist model of instruction. For this reason, although they have found their place in education and have the greatest rate of adoption within schools thus far, they are unlikely to serve as a catalyst for restructuring education. The focus of drill-and-practice CAI on basic skills allows little room for the presentation of complex tasks, multistep problems, or collaborative learning. ICAI, on the other hand, has the potential to deal with complex domains, to provide models of higher- order thinking, and to probe students understanding, but has seldom been well integrated into a school's mainstream curriculum. One-way video technologies can be very motivating but are nearly always viewed as enrichment and have not instigated fundamental changes within schools (Conaty, 1993).
Research Literature
The following studies discussed some of the basic tenets pertinent to the present study. These studies were included in the view of clarifying concepts related to this research.
In a study by Setzer entitled “Computers in Education”, administered in University of Sao Paulo, Brazil in 2000, he introduced some arguments in using computers in education, at home and at school. They are the following: (1). Computers improve both teaching and student achievement. (2). Computer literacy should be taught as early as possible; otherwise students will be left behind. (3). Technology programs leverage support from the business community - badly needed today because schools are increasingly starved for funds. (4). To make tomorrow's work force competitive in an increasingly high-tech world, learning computer skills must be a priority. (5). Work with computers - particularly using the Internet - brings students valuable connections with teachers, other schools and students, and a wide network of professionals around the globe. Those connections spice the school day with a sense of real-world relevance, and broaden the educational community (Oppenheimer, 2000).
According to Papert (cited in Setzer, 2000) in examining those arguments, the following patterns emerge. These patterns are the following: (a) computers should be learned and used as soon as possible because they will be essential for the individual in the professional working place; (b) students who do not master computers will not keep pace with their classmates; (c) computers are good tools for learning; (d) computers improve students' achievements; (e) computers accelerate children's development, mainly intellectual; (f) computers may provide a free environment for learning; (g) computers may promote social (and family) cohesion; (h) computers provide a fascinating learning environment, one that attracts children and young people; (i) computers provide for a challenge of traditional educational methods and values; (j) computers induce a certain vision of the world; (k) computers make it possible to learn without tensions and pressures; (l) computers (through the Internet) make students get interested in foreign cultures and people; (m) computers develop self-control; (n) computers may provide for a more humanistic teaching; (o) computers may enhance imagination and creativity; (p) computers may be used to make children conscious of their own thinking process; (q) computers provide for an individual way and pace of learning. (r) Children have to learn computers otherwise they will be afraid of them at adult ages; (s) children who don't use a computer at home may develop psychological and social problems (e.g. a sense of inferiority); and (t) through the Internet, computers make it possible for students to access all sorts of information not available through other means.
In addition, the study of Fulton and Wenglinsky about the “Technology and Students Learning” administered on September 1997, states some of the questions they want to study like: “does it work?" and "is it effective?" are legitimate questions about educational technology. When educators ask these questions, they are really asking if technology helps students learn. But technology is only a tool and the question cannot just be "Does the presence of technology improve learning?" It is clear that when the researchers try to evaluate the educational uses of technology, what they are really evaluating are the broader pedagogical practices being used. The question, then, becomes: What kinds of technology are being used, under what context, and in what ways that help promote student learning?
Not all the research paints a rosy picture of technology in schools. Some show no academic improvement; no pay off for costly investments (Mathews, 2000). Other authors believe technology takes funding away from other resources and programs that may be more beneficial to students (Healy, 1999; Oppenheimer, 1997); that technology sits idle and is underused (Cuban, 2001); and that an over-reliance on technology can rob from children opportunities to express creativity, build human relationships, and experience hands-on learning (Alliance for Childhood, 2000).
Furthermore this study found out that under the right conditions, technology: (a). accelerates, enriches, and deepens basic skills; (b) motivates and engages students in learning; (c) helps relate academics to the practices of today's workforce.; (d) increases economic viability of tomorrow's workers; (e) strengthens teaching.; (f) contributes to change in schools; and lastly, (g) connects schools to the world.
Other studies with negative results indicate that the initiatives themselves focused on hardware and software, or teachers taught about the technology instead of using the technology to enhance learning experiences. Bracewell, Breuleux, Laferriere, Beniot, and Abdous (cited in Fulton and Wellingski, 1997) asserted that the integration of educational technology into the classroom, in conjunction with supportive pedagogy, typically leads to increased student interest and motivation in learning, more student-centered classroom environments, and increased real-life or authentic learning opportunities. Davis (cited in Fulton and Wellingski, 1997) agreed that technology integration led to student-centered classrooms, which increased student self-esteem. Schacter (cited in Fulton and Wellingski, 1997) concluded that technology initiatives have to focus on teaching and learning, not the technology, to be successful: "One of the enduring difficulties about technology and education is that a lot of people think about the technology first and the education later". Educators are starting to recognize it is more important to use technology for learning than it is to learn how to use the technology.
Becker (2000) examined data from the 1998 national survey of teachers, Teaching, Learning, and Computing (TLC), and concluded: under the right conditions—where teachers are personally comfortable and at least moderately skilled in using computers themselves, where the school's daily class schedule permits allocating time for students to use computers as part of class assignments, where enough equipment is available and convenient to permit computer activities to flow seamlessly alongside other learning tasks, and where teachers' personal philosophies support a student-centered, constructivist pedagogy that incorporates collaborative projects defined partly by student interest—computers are clearly becoming a valuable and well-functioning instructional tool.
Educational visionaries are often frustrated that technology has primarily been used only to automate traditional education. They see the various ways technology will be used to revolutionize education through ‘learning by doing' and in the kinds of collaborative communities young people are creating with technology (Richardson, 2006; Tapscott, 1998). Further, "Computer based technology has been called an essential ingredient in restructuring because it can provide the diversity in instructional methods necessary to reach all school children," according to Polin (1991). Papert (1996) described the important role technology can play in learning (Cherniavsky & Soloway, 2002; Papert, 1989, 1996; Schank, 2001; Schank & Cleary, 1995).
Synthesis
Upon discussing the related literatures, several realizations helped the researcher in the development of this study. Also, mathematics educators cannot separate the vision of how they should prepare high school teachers in mathematics from the vision of what and how students should learn mathematics in the high school. Prospective teachers should have the same kind of experiences integrating mathematics, and technology as their future students. One of the goals of the high school concept is the integration of mathematics with other areas. Teachers should experience how technology can be integrated in an authentic way, so that the integrity of mathematics is preserved. Different high schools incorporate to different degrees the ideal of the high school concept. Prospective teachers can also take part of the approach presented here to implement change and support the necessary reform in mathematics teaching over time, regardless of the degree of implementation of the high school concept in their placement school.
Studies and researches mentioned above were considered very significant because they help the researcher acquire a lot of knowledge and understanding regarding the concept of what modern technology gadgets (computer and scientific calculator) really are and how it affects the learning of the students in the present time. From this study, the researcher was given full view of modern technology gadgets, which will be the starting point in delving more with this research.
Fourth year high school students were the specific respondents of the present study since this was the time when they will be taking up Trigonometry and the use of scientific calculator is a must for them.
This study shares the same view on the aforementioned studies and researches conducted – that the use of modern technology gadgets and the frequency of use greatly affect the learning in mathematics of the students today.
CHAPTER III
RESEARCH METHOD AND PROCEDURE
This chapter deals with the research design, the subjects of the study, the data gathering procedure and the statistical treatment of the gathered data.
Research Design
This study used the descriptive type of research. Descriptive method was used in this study because it best describes the frequency of use of technology-aided teaching strategy, the effects of utilizing modern technology gadgets and the its effect of using in students learning. It is defined by Travers (1978) as the design which aims “to describe the nature of a situation as it exists at the time of the study and to explore the cases of particular phenomena”.
Respondents of the Study
The total population of the fourth year high school students of First Asia Institute of Technology and Humanities (FAITH) is one hundred thirty-nine (139) for the Academic Year 2008-2009 but one hundred thirty-six (136) answered the survey because three of them were absent. The Slovin’s Formula was used to get the number of respondents. From one hundred thirty-six (136), the researcher got one hundred two (102) students, which were composed of fifty-one (51) males and fifty-one (51) females. They were randomly selected using stratified sampling. Their ages ranges from fifteen (15) to seventeen (17) years old.
The researcher employed them as the respondents of the study since this was the time when they are taking up Trigonometry and the use of scientific calculator is a must for them.
Data Gathering Instrument
The researcher, in order to attain this objective, used a self-constructed, validated questionnaire as the main instrument in gathering data. The researcher looked for the unpublished and published materials available in the library and electronic sources for topics that are related to the present study. Items in the questionnaire were constructed based on the statement of the problem that was presented in the study such as the technology-aided teaching strategy that utilize modern technology gadgets and its frequency of use; the effects of utilizing modern technology gadgets in learning mathematics; and lastly the effects of using modern technology gadgets in students learning.
After constructing the questionnaire, the researcher had it validated, first with the thesis adviser by asking how to correct on the items and its format, then to two Mathematics teacher by checking the mathematical words used and finally to an English teacher by correcting the grammar. In validating the questionnaire, instead of using formula as a standard process of validation, a focused group was involved which composed of ten (10) college students of First Asia Institute of Technology and Humanities in the Tertiary Level. Since all items in the questionnaire were answered correctly by the focused group, it was assumed that they are valid – the items can answer the posted statement of the problems in this study. After the validation, it was made ready for actual administration to the respondents. It was constructed to gather responses from the fourth year high school students of First Asia Institute of Technology and Humanities.
The questionnaire consisted of three main parts. The first part contained the technology-aided teaching strategies and the frequency of their use wherein they were given scalar values and description such as: 4- Always; 3- Often; 2- Sometimes and 1- Never. There are twelve (12) items in this part.
The second part of the questionnaire contains fourteen (14) items about the effects of the utilization of modern technology gadgets in learning mathematics. The respondents were asked to use the scalar values and description such as: 4- Very Much; 3- Much; 2- Not so much; 1- Not at all, in answering each item.
Finally, the last part of the questionnaire is consisted of fourteen (14) items on the effects of using modern technology gadgets in students learning. Like in the first and second part, scalar values and its description were given such as: 4- Strongly agree; 3- Agree; 2- Disagree and 1- Strongly disagree, as the respondents guide in for answering the questions.
To further interpret the weighted mean, this scale was used.
Scale
Part I
Part II
Part III
3.49 – 4
Always
Very Much
Strongly Agree
2.49 – 3.48
Often
Much
Agree
1.49 – 2.48
Sometimes
Not Much
Disagree
1 – 1.48
Never
Not at all
Strongly Disagree
Data Gathering Procedure
When the researcher came up with the topic he wanted to study, he presented his chosen topic to the thesis adviser, and defended it. After some critical analysis and examination, the proposed topic was accepted. Thereupon, the researcher started to construct the questionnaire. With the help of the thesis adviser, grammarian and Mathematics teachers; the questionnaire was then approved and advised for a dry-run.
The researcher then prepared the questionnaire ready for dry-run administered to ten (10) college students, four first year Industrial Engineering students, one second year Secondary Education major in Mathematics, two third year Elementary Education major in Special Education and three fourth year Elementary Education major in Special Education. Since all items in the questionnaire were answered correctly by the focused group, it was assumed that they are valid – the items can answer the posted statement of the problems in this study. After tallying the results, the answers yielded were manifested that the items were clear because they were answered correctly by those students.
The researcher then ready for the actual administration of the questionnaire for the target respondents, made a formal letter to the high school principal of First Asia Institute of Technology and Humanities. The letter was duly signed by the adviser and the College Dean and was approved by the high school principal. He advised the researcher to administer the questionnaire to the whole population of the fourth year high school students. This is to keep records of all students to the perceived effects of the utilization of modern technology gadget in learning mathematics. After the approval of the letter, the data gathering instrument was reproduced and distributed to the respondents on August 15, 2008 from 1:30pm to 2:30 pm at room 30, 31, 40 and 41. The questionnaire was personally administered by the researcher with the assistance of the concerned high school teacher.
In tallying the answers the researcher used the number of respondents obtained through Slovin’s Formula. From one hundred thirty-six (136), only one hundred two (102) were tallied. The one hundred two (102) students were randomly selected to properly represent each section. This was personally analyzed by the researcher.
A one hundred percent (100%) retrieval of questionnaire was tallied and appropriate statistical tools were applied.
Statistical Treatment of Data
After the questionnaire was collected, the researcher made the appropriate tabulation of the responses and statistical tools were applied to the study which was follows:
Slovin’s Formula was used to determine the sample size of the respondents.
Formula:
Where:
N is the population size
e is the margin of error
Weighted Mean was used to determine the technology-aided teaching strategy that utilize modern technology gadgets and its frequency of use, the effects of utilizing modern technology gadgets in learning mathematics and lastly the effects of using modern technology gadgets in students learning.
Formula:
=
where:
w is the weights
is the mean.
is the sum of all score.
n is the total number of respondents.
Composite mean was used to determine the sum of the entire weighted mean in each part.
Formula:
Where:
is the weighted mean of the first item.
is the weighted mean of the second item.
is the weighted mean of the nth item.
n is the number of items.
c is the composite mean.
Pearson product moment correlation coefficient was used to determine the correlation between the frequency of the utilization of modern technology gadgets and learning of mathematics.
Formula:
)
Where:
x is the value of independent variable. is the mean of the values of x.
y is the value of dependent variable.
is the mean of the value y.
T-test was used to determine the significant relationship between the frequency of the utilization of modern technology gadgets and learning of mathematics.
Formula:
Where:
r is the value of the correlation (Pearson R)
N is the sample size
CHAPTER IV
PRESENTATION, ANALYSIS AND INTERPRETATION OF DATA
This chapter presents the analysis and interpretations of the data gathered on the perceived effects of the utilization of modern technology gadgets in learning mathematics of fourth year high school students of First Asia Institute of Technology and Humanities.
1. Teaching Strategy Utilizing Modern Technology Gadgets
Varied teaching strategy that utilizes modern technology gadgets were presented to the respondents as follows: (1) lecture method with power point presentation; (2) reporting with power point presentation; (3) independent research through the use of internet; (4) model-making through the use of related software; (5) computer hands-on activities; (6) problem solving with the use of scientific calculator; (7) cooperative learning with power point presentation; (8) use of graphic organizers in solving problems; (9) discussion with the use of power point presentation; (10) use of analogy (reasoning) with power point presentation; (11) discovery learning through the use of scientific calculators; and lastly, (12) guided discovery through the use of related software.
The abovementioned teaching strategies were assumed by the researcher to be utilized by the teacher in teaching mathematics as they are commonly used by the teachers in Mathematics across schools.
2. Frequency of Using Technology-Aided Teaching Strategy that Utilizes Modern Technology Gadgets
Table 1 shows the frequency of using the teaching strategy that utilizes modern technology gadgets. It was analyze by the researcher because it seems that the frequency of utilization has its effect in students learning.
Table 1. Frequency of Using Technology-Aided Teaching Strategy that Utilizes Modern Technology Gadgets
Technology-Aided Teaching Strategy
that Utilizes Modern Technology Gadgets
WEIGHTED
MEAN
VERBAL INTERPRETATION
1. Lecture method with power point presentation
3.12
Often
2. Reporting with power point presentation
3.15
Often
3. Independent research through the use of internet
3.00
Often
4. Model-making through the use of related software
2.88
Often
5. Computer hands-on activities
3.06
Often
6. Problem solving with the use of scientific calculator
3.07
Often
7. Cooperative learning with power point presentation
3.14
Often
8. Use of graphic organizers in solving problems
2.79
Often
9. Discussion with the use of power point presentation
3.21
Often
10. Use of analogy (reasoning) with power point presentation
2.85
Often
11. Discovery learning through the use of scientific calculators
2.69
Often
12. Guided discovery through the use of related software
2.73
Often
COMPOSITE MEAN
2.97
Often
It can be gleaned in the table that the three most commonly used teaching strategies used by the teachers in teaching Mathematics are: cooperative learning with power point presentation with a weighted mean of 3.14; second is reporting with power point presentation with a weighted mean of 3.15; and third is discussion with the use of power point presentation with a weighted mean of 3.21. All of them have a verbal interpretation, “often”.
It can be inferred from the results that the use of power point presentation is commonly used in different teaching strategy such as cooperative learning, reporting and discussion. This probably is made possible because in First Asia Institute of Technology and Humanities (FAITH), technology use is one of the focuses of teaching and learning. Students and teachers are encouraged to deliver their lessons through the use of computers and one effective means is through the use of power point presentation. As the school promotes technology, it is but natural that the lessons must be presented through the use of technology.
This is further explained by Sammons (1995) when he said that power point works best for things that are presented visually, not verbally. It helps when the pupil need to draw a picture. Communication delivered over multiple channels is more efficient than communication over a single channel. Multiple channels make it more likely that the whole message will be received. An appropriate picture adds another channel. A picture aids in memory by making a visual connection to an abstract idea like memory rests on connections and a vivid picture forms a solid connection. Lastly, power point makes it easy to create visuals, and, by using a template, make it easy to be consistent.
On the other hand, the three least used technology – aided teaching strategies are the following: (a) discovery learning through the use of scientific calculators with a weighted mean of 2.69; (b) guided discovery through the use of related software with a weighted mean of 2.73; and (c) use of graphic organizers in solving problems with a weighted mean of 2.79. All of them have a verbal interpretation of often.
These three seem to have lower score because calculator, graphic organizer and related software are not regularly used in the classroom. The teacher seems to use the computer through the power point presentation than the use calculator in solving, graphic organizer in the lesson and the related software because here in First Asia Institute of Technology and Humanities (FAITH), computer is the instrument in teaching and/or presenting the lessons while scientific calculator, graphic organizer and related software are already installed or programmed in the computer.
This is similar to Ray’s findings (2000), from her workshop where she concluded that there are rules when it comes to appropriate use of a calculator: (1) use a calculator as a tool in problem solving... sort of like a "fast pencil"; (2) use a calculator for complex computations but NOT for basic facts; (3) use a calculator to develop number concepts and skills; and (4) use a calculator in testing situations when not assessing computational proficiency. In general, she uses calculators to help teach mathematics matter, not to replace the teaching of mathematics.
Likewise Mitchel, (cited in Boyle, 1997), said that the use of graphic organizer can develop students’ visual skills. He added however that using graphic organizer must be based on the suitability of the lessons presented are therefore cannot be used regularly.
3. Effects of Utilizing Modern Technology Gadgets in Learning Mathematics
Table 2 shows the effects of utilizing computer in learning mathematics. It was analyzed in the research because it seems that the utilization of computer in teaching and/or presenting the lesson will enhance the learning of the students in mathematics.
Table 2. Effects of Utilizing Computer in Learning Mathematics
Computer
My learning is enhanced when…
WEIGHTED
MEAN
VERBAL INTERPRETATION
1. the teacher uses power point presentation in
presenting the lesson.
3.22
Much Enhanced
2. the teacher shows animation in the monitor
which is related to the topic.
2.93
Much Enhanced
3. the teacher gives an independent research
work using the internet.
2.81
Much Enhanced
4. we work in groups on a particular problem
when using the internet .
2.53
Much Enhanced
5. the lectures of my teacher were sent to my e-
mail.
2.22
Not so Much Enhanced
6. the teacher shows some presentation related
to our topic.
2.77
Much Enhanced
7. we always have computer hands-on
activities.
2.85
Much Enhanced
COMPOSITE MEAN
2.76
Much Enhanced
It is shown in Table 2 that the teachers up three ways of presenting and/or teaching the lesson with the use of computer wherein the students can enhance their learning are the following: (a) having computer hands-on activities with a weighted mean of 2.85; (b) the teacher showing animation in the monitor which is related to the topic with a weighted mean of 2.93; and (c) the teacher uses power point presentation in presenting the lesson with a weighted mean of 3.22. All of them have a verbal interpretation of “much enhanced”.
The three most commonly used ways of teacher in presenting and/or teaching the lesson with the use of computer that enhances students’ learning with the use of power point presentation is believed to be useful probably because through power point presentation, students can visibly see concepts in the simplest way – with bulleted information, readable and with interesting slide design that captures the attention of the students. It is user-friendly and its purpose is to really present concepts in a more interesting manner.
This is similar to Sammons (1995) statements saying that, that the teachers, in using power point presentation to enhance their students learning, must observed the following: (1) use it judiciously for a few key graphics or illustrations; (2) avoid text slides; (3) use text occasionally as a reference point for big ideas, (e.g. the three main objectives of a lesson); (4) remember other kinds of visuals; (5) handouts may be a more appropriate alternative. (6) Don’t be seduced by textbook publishers that offer canned presentations that go with a textbook; (7) the teacher should not be the publisher nor the textbook; (8) the teacher should make careful choices of what to use and what to avoid; (9) avoid using power point for discussion or coaching sessions.
He added that power point does not facilitate spontaneous discussion or discovery. The use of whatever media in the classroom must work to help students make connections. Making connections is the foundation of memory and ingenuity. Students learn as they make connections. An efficient use of visuals in the classroom can help students make connections between parts and the whole, between cause and effect, between problem and solution, between principle and practice.
On the other hand, the three least used ways in presenting and/or teaching the lesson with the use of computer wherein students can enhance their learning are the following: (a) the lectures sent to e-mail with a weighted mean of 2.22 which has the lowest score and verbal interpretation of “not so much enhance”; (b)working in groups on a particular problem when using computer with a weighted mean of 2.53; and (c) the teachers’ showing of some presentation related to the topic with a weighted mean of 2.77. The remaining two has a verbal interpretation of “much enhanced”.
The three least used in presenting and/or teaching the lesson with the use of computer wherein the students will enhanced their learning, are not preferable for the students. The sending of lectures to students’ email may not be preferred probably because of minimal contacts between the students and teachers. For example, students may still need more explanations and instructions that are lacking when the lecture is already sent through e-mail. Also, students may find it so taxing sending, downloading, and printing those lectures instead of understanding it with the help of the teacher. Similarly, not all students have their own e-mail and computer or internet at home.
Meanwhile, working in groups may not be preferred because most students want to discover for himself how a problem may be solved through computer hands-on activities. Computer hands-on may be one of the most interesting activities for students but since they will be working in group, it becomes minimal, for they only have a one-at-a time chance in using the calculator. Also in grouping some students become dependent to their leader.
TV showing of some presentations related to the topic may not be so attractive to students probably because they still have to develop the so-called integrative learning skills. Or it can also inferred that the teachers way lack the skills of integrative teaching, thus making students not interested to the presentations which really are not their lessons.
According to (Suber, 2008) note taking is the practice of writing pieces of information, often in an informal or unstructured manner. In taking down notes, the students can: (1) organized notes that help in identifying the core of important ideas in the lecture; (2) It serves as permanent records that help in learning and remembering lecture; (3) the lecture may contain information not available anywhere else, this will be the only chance to students to learn it; (4) lecture is where the students learn what the instructor thinks is important, and makes up the exams; (5) class assignments are usually given in the lecture; and (6) the underlying organization and purpose of the lecture will become clear through note taking.
Another gadget used in presenting Mathematics lesson is the scientific calculator. Table 3 shows the effects of utilizing scientific calculator in learning mathematics. It was analyzed by the researcher because it seems that the utilization of scientific calculator in teaching and/or presenting the lesson will enhance the learning of the students in mathematics.
It is shown in Table 3 that the up three ways of presenting and/or teaching the lesson with the use of scientific calculator wherein the students will enhanced their learning are the following: (a) use scientific calculator in finding the factors of a given large number with a weighted mean of 3.11; (b) use scientific calculator in solving problems involving trigonometric functions with a weighted mean of 3.12; and (c) evaluate, press the keys and see the results on the display screen of the scientific calculator with a weighted mean of 3.12. All of them have a verbal interpretation of “much enhanced”.
Table 3. Effects of Utilizing Scientific Calculator in Learning Mathematics
SCIENTIFIC CALCULATOR
My learning is enhanced when…
WEIGHTED
MEAN
VERBAL INTERPRETATION
1. I use scientific calculator in solving problems
involving trigonometric functions.
3.12
Much Enhanced
2. I evaluate, press the keys and see the results
on the display screen of the scientific
calculator.
3.24
Much Enhanced
3. the teacher encourages me to discover by
myself certain facts about numbers using
scientific calculator.
2.67
Much Enhanced
4. I challenge my classmate to a calculator
game like in being the first to solve the
problem and get the right answer.
2.24
Not so Much Enhanced
5. I learn how to minimize the number of
keystrokes in solving problems.
2.82
Much Enhanced
6. I use scientific calculator in finding the
factors of a given large number.
3.11
Much Enhanced
7. I use scientific calculator in doing my
homework.
3.03
Much Enhanced
COMPOSITE MEAN
2.89
Much Enhanced
It is observed by the researcher that the learning of the students are enhanced when they use scientific calculator in finding the factors, solving problems involving trigonometric functions and evaluating large numbers because they seems to have fun and enjoyment by means of pressing the keys of the scientific calculator and at the same time they are learning what their doing.
This also observed by Ediger (1997), who said that the use of scientific calculator is more enjoyable, excellent to notice fewer errors made in the computation, students have more fun and checking one’s work with a calculator could be done just as easily and accurately that’s why teacher preferred the students to use it to enhance their learning.
On the other hand, it is shown in Table 3 that the three least used ways in presenting and/or teaching the lesson with the use of scientific calculator with less effects on influence are the following (a) challenging classmate to a calculator game like in being the first to solve the problem and get the right answer with a weighted mean of 2.24 and has a verbal interpretation of “not so much enhanced”, then (b) the teacher encourages students to discover certain facts about numbers by themselves using scientific calculator with a weighted mean of 2.67 and lastly (c) learning how to minimize the number of keystrokes in solving problems with a weighted mean of 2.82. The last two have a verbal interpretation of “much enhanced”.
It is observed by the researcher that the students do not pay much attention to the other uses, functions and importance of scientific calculator. They only consider the use of calculators in solving problems that is why they do not use it for a calculator game; they do not notice the number of keystrokes; and also the facts about numbers.
According to Ediger (1997), the calculator programs allow individual students to work at their own speed through a set sequence of stages, problems or challenges. Where a student is consistently successful, the calculator program can progress to the next level of difficulty. This "responsiveness" of the exercise is something that cannot easily be emulated with a textbook. The programs typically generate problems with randomly selected numbers. As a result of this students can't copy answers from their neighbor, but rather they spend time discussing how to do questions and identifying methods that work. The motivational aspect of use of IT in the teaching and learning process cannot be undervalued - especially with the weaker, or younger, mathematician. Students typically enjoy lessons that are based around calculator programs - the desire to obtain a faster time, or a higher score, often adds an additional competitive element to the chosen exercise.
4. Effects of Using Modern Technology Gadgets in the Development of Students Skills
Table 4 shows the effects of using computer in the development of students’ skills. It was analyzed by the researcher because it seems that the using of computers can develop the skills of the students.
Table 4. Effects of Using Computer in the Development of Students Skills
ITEMS
By using computers…
WEIGHTED
MEAN
VERBAL INTERPRETATION
1. I become attentive to learn new concepts.
3.40
Agree
2. I understand the lesson easier.
3.28
Agree
3. I can visualize the underlying principles of
th topic.
3.21
Agree
4. I become more participative in class
discussion.
3.05
Agree
5. I learn while having fun.
3.37
Agree
6. I become interactive in group activities.
3.23
Agree
7. I am more receptive in the discussion.
3.03
Agree
COMPOSITE MEAN
3.22
Agree
It is shown in Table 4 that the up three behavior/skills mostly develop in learning mathematics with the use computer are as follows: (a) understanding the lesson easier with a verbal interpretation of 3.28; (b) learning while having fun with a weighted mean of 3.37; and (c) becoming attentive to learn new concepts with a weighted mean of 3.40. All of them have a verbal interpretation of “agree”.
It can be said that when using computer, the students will understand the lesson easier because it helps them to have a one click a way information in surfing the net, in terms of having fun, because of its graphics and because of that the (graphics and animation) it helps them to become attentive listener and they will pay more attention.
Similarly, the National Council of Teachers in Mathematics in USA (2000) stressed that technology can be used to learn mathematics, to do mathematics, and to communicate mathematical information and ideas. The Internet hosts a wealth of mathematical materials that are easily accessible through the use of search engines, creating additional avenues to enhance teaching and facilitate learning. Outside of class, students and faculty can pose problems and offer solutions through e-mail, chat rooms, or websites.
Technology provides opportunities for educators to develop and nurture learning communities, embrace collaboration, provide community-based learning, and address diverse learning styles of students and teaching styles of teachers. The integration of appropriately used technology can enhance student understanding of mathematics through pattern recognition, connections, and dynamic visualizations. Electronic teaching activities can attract attention to the mathematics to be learned and promote the use of multiple methods.
Learning can be enhanced with electronic questioning that engages students with technology in small groups and facilitates skills development through guided-discovery exercise sets. Using electronic devices for communication, all students can answer mathematics questions posed in class and instructors can have an instantaneous record of the answers given by each student. This immediate understanding of what students know, and don’t know, can direct the action of the instructor in the teaching session.
However, the three behavior/skills least developed in learning mathematics with the use computer are the following: (a) becoming more receptive in the discussion with a weighted mean of 3.03; (b) becoming more participative in class discussion with a weighted mean of 3.05; and (c) visualizing the underlying principles of the topic with weighted mean of 3.21. All of them have a verbal interpretation of “agree”.
The results can be associated to the observations that computer poses less interaction with the students and is useful to the visual learners only. In that case the other kind of learners which is not visual type cannot learn that much with computer.
According to Cole (2002), technology helps students document the validity of their mathematical/critical thinking process, facilitating and enriching the learning processes and the development of problem-solving skills. The use of technology should be guided by consideration of what mathematics is to be learned, the ways students might learn it, the research related to successful practices, and the standards and recommendations recommended by professional organizations in education. Technology can be used by mathematics educators to enhance conceptual understanding through a comparison of verbal, numerical, symbolic, and graphical representations of the same problem.
Consequently, Table 5 shows the effects of using scientific calculator in the development of students’ skills. It was analyzed by the researcher because it seems that the using of scientific calculator can develop the skills of the students.
It is shown in Table 5 that the up three behavior/skills mostly develop in learning mathematics with the use scientific calculator are the following: (a) skipping the step-by-step process on how to solve problems with a weighted mean of 3.32; (b) finding it easy to solve difficult problems with a weighted mean of 3.35; and (c) spending less time in solving large numbers with a weighted mean of 3.43. All of them have a verbal interpretation “agree”.
Table 5. Effects of Using Scientific Calculator in the Development of Students Skills
ITEMS
By using scientific calculator…
WEIGHTED
MEAN
VERBAL INTERPRETATION
1. I develop my skills in solving problems.
3.12
Agree
2. I reinforce my skills in computation.
3.04
Agree
3. I improve my reasoning to a higher-level of
thinking.
3.07
Agree
4. I enhance my analytical thinking skills.
3.07
Agree
5. I spend less time in solving large numbers.
3.43
Agree
6. I can skip the step-by-step process on how to
solve problems.
3.32
Agree
7. I find it easy to solve difficult problems.
3.35
Agree
COMPOSITE MEAN
3.20
Agree
It is then believe by the researcher that the using of calculators helped the students skip the step-by-step process, spend less time in solving large numbers, and find it easy to solve problems. Because of that they will have the time to review their answers, compare the results to others, helps to improve their solutions and they can track which is incorrect in their solutions.
According to Ediger (1997), the use of scientific calculator is more enjoyable, excellent to notice fewer errors made in the computation, students have more fun and checking one’s work with a calculator could be done just as easily and accurately that’s why students preferred to use it to develop their skills.
On the hand, the three behavior/skills least developed in learning mathematics with the use scientific calculator are the following: (a) reinforcing skills in computation with a weighted mean of 3.04, and then both (b) improving reasoning to a higher-level of thinking and (c) enhancing analytical thinking skills have a weighted mean of 3.07. All of them have a verbal interpretation of “agree”.
It is then believed by the researcher that the students don’t know the importance of scientific calculator, they are just compute a problem using scientific calculator without knowing that that their skills are to be developed because of that they become more dependent in using it. If they rely on the calculator, even the simple 1+1 will be solve using calculator.
According to Ediger (1997), when using paper and pencil, pupils have more time to deliberate in problem solving as compared with the use of calculator. Calculators provide the correct results if the user touches the proper keys. Pupils get frustrated if they see long answers and lastly, they preferred to use paper and pencil method when it comes to solve less complex problems.
5. Relationship between the Frequency of the Utilization of Modern Technology Gadgets in the Effective Learning of Mathematics
Table 6 shows the relationship between the frequency of the utilization of modern technology gadgets in the effective learning of mathematics. In this table it is also shows the variable, value, the decision and its verbal interpretation.
Table 6. Significant Relationship between the Frequency of the Utilization of Modern Technology Gadgets in the Effective Learning of Mathematics
Variables
Value
Decision
Verbal
Interpretation
Technology-Aided Teaching Strategies and the Frequency of their Uses And Effects of Utilizing Modern Technology Gadgets (computer and scientific calculator) in Learning Mathematics
r= 0.3824
Reject the null hypothesis
The frequency of using technology-aided teaching strategies relates to the effective learning of Mathematics.
The computed value of 0.3824 at 0.05 level of significance has a verbal interpretation of “moderately correlated” and therefore the hypothesis: there is no significant relationship between the frequency of the utilization of modern technology gadgets in the effective learning of mathematics, was rejected. Therefore, it could be said that the frequency of using technology-aided teaching strategies relates to the effective learning of Mathematics.
According to The National Council of Teachers in Mathematics in USA (2000), it says that technology can be used to learn mathematics, to do mathematics, and to communicate mathematical information and ideas. The Internet hosts a wealth of mathematical materials that are easily accessible through the use of search engines, creating additional avenues to enhance teaching and facilitate learning. Outside of class, students and faculty can pose problems and offer solutions through e-mail, chat rooms, or websites. Technology provides opportunities for educators to develop and nurture learning communities, embrace collaboration, provide community-based learning, and address diverse learning styles of students and teaching styles of teachers. The integration of appropriately used technology can enhance student understanding of mathematics through pattern recognition, connections, and dynamic visualizations. Electronic teaching activities can attract attention to the mathematics to be learned and promote the use of multiple methods. Learning can be enhanced with electronic questioning that engages students with technology in small groups and facilitates skills development through guided-discovery exercise sets. Using electronic devices for communication, all students can answer mathematics questions posed in class and instructors can have an instantaneous record of the answers given by each student. This immediate understanding of what students know, and don’t know, can direct the action of the instructor in the teaching session.
However, according to Cole (2002), technology helps students document the validity of their mathematical/critical thinking process, facilitating and enriching the learning processes and the development of problem-solving skills. The use of technology should be guided by consideration of what mathematics is to be learned, the ways students might learn it, the research related to successful practices, and the standards and recommendations recommended by professional organizations in education. Technology can be used by mathematics educators to enhance conceptual understanding through a comparison of verbal, numerical, symbolic, and graphical representations of the same problem.
CHAPTER V
SUMMARY, CONCLUSION AND RECOMMENDATION
This chapter presents the summary the findings, the conclusions and the recommendations of the study.
SUMMARY
The researcher conducted this study to find out the perceived effects of utilizing modern technology gadgets in learning mathematics of fourth year high school students of First Asia Institute of Technology and Humanities.
Specifically, this study sought answers to the following questions:
1. In what teaching strategies does the teacher utilize modern technology gadgets?
2. How often are these modern technology gadgets used?
3. How does the use of modern technology gadgets affect the respondents learning?
4. What are the perceived effects of using modern technology gadgets in learning mathematics?
5. Is there a significant relationship between the frequency of the utilization of modern technology gadgets in the effective learning of mathematics?
The study tested the null hypothesis of no relationship between the frequency of the utilization of modern technology gadgets in the effective learning of mathematics.
This study is limited to the responses of the one hundred two (102) fourth year high school students of First Asia Institute of Technology and Humanities (FAITH) enrolled in Academic year 2008-2009. The population was identified through the use of proportional stratified random sampling. This study includes only two kinds of modern technology gadgets – computer and scientific calculator.
The descriptive method of research was used and a self-constructed, validated questionnaire served as the basic instrument in gathering data.
Slovin’s formula, weighted mean, composite mean and Pearson product moment correlation coefficient were the statistical treatments used to qualify the data.
Findings
After the analysis and interpretation of the data gathered, the researcher was able to come up with the following findings:
1. Teaching Strategy Utilizing Modern Technology Gadgets
Varied teaching strategy that utilizes modern technology gadgets were presented to the respondents as follows: (1) lecture method with power point presentation; (2) reporting with power point presentation; (3) independent research through the use of internet; (4) model-making through the use of related software; (5) computer hands-on activities; (6) problem solving with the use of scientific calculator; (7) cooperative learning with power point presentation; (8) use of graphic organizers in solving problems; (9) discussion with the use of power point presentation; (10) use of analogy (reasoning) with power point presentation; (11) discovery learning through the use of scientific calculators; and lastly, (12) guided discovery through the use of related software.
The abovementioned teaching strategies were assumed by the researcher to be utilized by the teacher in teaching mathematics as they are commonly used by the teachers in Mathematics across schools.
2. Frequency of Using Technology-Aided Teaching Strategy that Utilizes Modern Technology Gadgets
The three most commonly used teaching strategies used by the teachers in teaching Mathematics are: cooperative learning with power point presentation with a weighted mean of 3.14; second is reporting with power point presentation with a weighted mean of 3.15; and third is discussion with the use of power point presentation with a weighted mean of 3.21. All of them have a verbal interpretation, “often”.
On the other hand, the three least used technology – aided teaching strategies are the following: (a) discovery learning through the use of scientific calculators with a weighted mean of 2.69; (b) guided discovery through the use of related software with a weighted mean of 2.73; and (c) use of graphic organizers in solving problems with a weighted mean of 2.79. All of them have a verbal interpretation of “often”.
3. Effects of Utilizing Modern Technology Gadgets in Learning Mathematics
a. Effects of Utilizing Computer in Learning Mathematics
The teachers up three ways of presenting and/or teaching the lesson with the use of computer wherein the students can enhance their learning are the following: (a) having computer hands-on activities with a weighted mean of 2.85; (b) the teacher showing animation in the monitor which is related to the topic with a weighted mean of 2.93; and (c) the teacher uses power point presentation in presenting the lesson with a weighted mean of 3.22. All of them have a verbal interpretation of “much enhanced”.
On the other hand, the three least used in presenting and/or teaching the lesson with the use of computer wherein students can enhance their learning are the following: (a) the lectures sent to e-mail with a weighted mean of 2.22 which has the lowest score and verbal interpretation of “not so much enhance”; (b)working in groups on a particular problem when using computer with a weighted mean of 2.53; and (c) the teachers’ showing of some presentation related to the topic with a weighted mean of 2.77. The remaining two has a verbal interpretation of “much enhanced”.
b. Effects of Utilizing Scientific Calculator in Learning Mathematics
The up three ways of presenting and/or teaching the lesson with the use of scientific calculator wherein the students will enhance their learning are the following: (a) use scientific calculator in finding the factors of a given large number with a weighted mean of 3.11; (b) use scientific calculator in solving problems involving trigonometric functions with a weighted mean of 3.12; and (c) evaluate, press the keys and see the results on the display screen of the scientific calculator with a weighted mean of 3.12. All of them have a verbal interpretation of “much enhanced”.
On the other hand, the three least used in presenting and/or teaching the lesson with the use of scientific calculator with less effects on influence are the following (a) challenging classmate to a calculator game like in being the first to solve the problem and get the right answer with a weighted mean of 2.24 and has a verbal interpretation of “not so much enhanced”, then (b) the teacher encourages students to discover certain facts about numbers by themselves using scientific calculator with a weighted mean of 2.67 and lastly (c) learning how to minimize the number of keystrokes in solving problems with a weighted mean of 2.82. The last two have a verbal interpretation of “much enhanced”.
4. Effects of Using Modern Technology Gadgets in the Development of Students Skills
a. Effects of Using Computer in the Development of Students Skills
The up three behavior/skills mostly develop in learning mathematics with the use computer are as follows: (a) understanding the lesson easier with a verbal interpretation of 3.28; (b) enjoying learning while having fun with a weighted mean of 3.37; and (c) becoming attentive to learn new concepts with a weighted mean of 3.40. All of them have a verbal interpretation of “agree”.
However, the three behavior/skills least developed in learning mathematics with the use computer are the following: (a) becoming more receptive in the discussion with a weighted mean of 3.03; (b) becoming more participative in class discussion with a weighted mean of 3.05; and (c) visualizing the underlying principles of the topic with weighted mean of 3.21. All of them have a verbal interpretation of “agree”.
b. Effects of Using Scientific Calculator in the Development of Students Skills
The first three skills to be developed in learning mathematics with the use scientific calculator are the following: (a) skipping the step-by-step process on how to solve problems with a weighted mean of 3.32; (b) finding it easy to solve difficult problems with a weighted mean of 3.35; and (c) spending less time in solving large numbers with a weighted mean of 3.43. All of them have a verbal interpretation “agree”.
On the hand, the last three skills to be developed in learning mathematics with the use scientific calculator are the following: (a) reinforcing skills in computation with a weighted mean of 3.04, and then both (b) improving reasoning to a higher-level of thinking and (c) enhancing analytical thinking skills have a weighted mean of 3.07. All of them have a verbal interpretation of “agree”.
5. Significant Relationship Between the Frequency of the Utilization of Modern Technology Gadgets in the Effective Learning of Mathematics
The computed value of 0.3824 at 0.05 level of significance has a verbal interpretation of “moderately correlated” and therefore the hypothesis: there is no significant relationship between the frequency of the utilization of modern technology gadgets in the effective learning of mathematics, was rejected. Therefore, it could be said that the frequency of using technology-aided teaching strategies relates to the effective learning of Mathematics.
Conclusions
With the major findings of the study as bases, the researcher had drawn the following conclusions:
1. Mathematics teachers are using varied technology-aided teaching strategies.
2. Power point presentation is the most commonly used strategy in presenting lessons in Mathematics while the use of scientific calculators, graphic organizers and related software are the least utilized.
3. The use of power point presentation, computer hands-on activities and animations were found to enhance students learning in Mathematics while sending lectures to e-mail, working in groups on a particular problem and showing of some presentation related to the topic were found not that much to enhance students learning. On the other hand, when it comes to utilizing scientific calculators, finding the factors of a given large number, solving problems involving trigonometric functions and evaluating, pressing the keys and seeing the results on the display screen were found to enhance students learning in Mathematics while challenging classmate to a calculator game, encouraging students to discover certain facts about numbers by themselves, and learning how to minimize the number of keystrokes in solving problems, were found not that much to enhance students learning.
4. The use of computers help the students to understand the lesson easier, to learn with fun, and to become attentive to learn new concepts while being receptive and participative in class discussion, and helping to visualize the underlying principles of the topic with the use of computer do not help that much in the development of students’ skills. However, when it comes to the use of scientific calculator, it helps the students to skip the step-by-step process in solving, to find it easy to solve difficult problems, to spend less time in solving large numbers while reinforcing skills in computation, improving reasoning to a higher-level of thinking, and enhancing analytical thinking, when using scientific calculator do not help that much in the development of students’ skills.
5. The frequent use of modern technology gadgets enhances the students’ learning in Mathematics.
Recommendations:
Based on the findings and conclusions, the following recommendations are offered:
1. Since it was found out that the students learn a lot in using the modern technology gadgets, the school should focus more on how to improve the learning of the students in terms of the utilization of modern technology gadgets (computer and scientific calculator) in learning Mathematics by providing many activities involving the use of modern technology gadgets.
2. As concluded, the frequency of utilizing varied technology-aided teaching strategies has an effect to the learning of the students, the Mathematics teacher should use different kinds of technology-aided teaching strategy which can lead the students to fully develop their skills and knowledge by being flexible enough in dealing with the students.
3. Since the study confirmed that there is a significant relationship between the frequency of utilizing modern technology gadgets and the learning of Mathematics, frequency of using technology-aided teaching strategy that utilizes modern technology gadgets should be based on the needs of the students which capture their interest towards the development of their knowledge and skills.
BIBLIOGRAPHY
A. Books
Bilbao, Purita, et.al., The Teaching Profession. Cubao, Quezon City: Lorimar Publishing Co., Inc. 2006
De Leon, Cecille. Integrated Mathematics IV. Sta. Mesa, Quezon City: Vibal Publishing House, Inc. 2000
Ediger, Marlow. Teaching Mathematics in the Elementary School, Kirksville, Missouri: Simpson Publishing. 1997
Ediger, Marlow. Teaching Mathematics Successfully. Kirsksville, Missouri, U.S.A: Simpson Publishing Company. 1997
Fabra, Norberto. Elementary Statistics for Secondary Schools. Las Pinas, Metor Manila: Fabra, N. F. Educational Books. 1979
Gagne, Robert (1985), The Conditioning of Learning and Theory of Instruction, New York: Holt, Rinehart and Winstonbrain teaser gamesmarble popper gamespc game downloadsmanagement gamescard gameshidden object gamessimulation gamesaction gamesmatch 3 games
Goldstein, I. (1947). Training: Program Development and Evaluation. Monterey, California: Brooks/Cole.
National Council of Teachers of Mathematics. Principles and standards for school mathematics. Reston, VA: Author. 2000.
Norton, Mary. Math Power I. Chicago, Illinois, U. S. A: World Book Inc. 1995
Papert, S.(1980). Children, Computers and Powerful Ideas. Basic Books, New York
Salandanan, Gloria, et.al., The Teaching of Science and Health, Mathematics, and Home Economics and Practical Arts. Quezon City: Katha Publishing Co., Inc. 1996
Zulueta, Francisco. Principles and Methods of Teaching. Mandaluyong City: National Bookstore. 2006
B. Unpublished Thesis
Acuna, Shirley, et.al., Perceived Influence of Rock Music to the Behavior of Third Year High School Students of San Pascual National High School. Batangas State University, Batangas City. 2003
C. Journals
Sammons, Martha C. (1995). Students Assess Computer-Aided Classroom Presentations, T.H.E. Journal 22 (May 1995): 66-69.
D. Electronic Data
Developing Mathematical skills (Loughborough University)
(Website: http://mlsc.lboro.ac.uk) Date Retrieved: July 16, 2008
Glossary of Mathematics Teaching Strategies
(Website: http://cehd.umn.edu/NCEO/Presentations/NCTMLEPIEPStrategies
MathGlossaryHandout.pdf) Date Retrieved: June 22, 2008
Implementation Standard: Instruction with Technology
(Website: http://www.beyondcrossroads.com/doc/CH7.html)
Date Retrieved: August 2, 2008
PowerPoint Presentations: The Good, the Bad and the Ugly
(Website: http://technologysource.org/article/use_of_powerpoint_in_teaching_ comparative_politics/) Date Retrieved: September 24, 2008
Ray, Susan (2000). The Use of Calculators Gets at the Heart of Good Teaching. Louisville, KY (Website: http://www.middleweb.com/CurrMathCalc.html)
Date Retrieved: September 24, 2008
Setzer, Valdemar W. (2000). Computers in Education: A Review of Arguments for the Use of Computers in Elementary Education
(Website: http://www.southerncrossreview.org/4/review.html)
Date Retrieved: August 3, 2008
Stiles, Mark (2002). Computers In Teaching And Learning.
(Website: http://www.staffs.ac.uk/cital/) Date Retrieved: October 2, 2008
Suber (2008). Importance of Notetaking.
(Website: http://www.edchange.org/multicultural/papers/math.html)
Date Retrieved: September 26, 2008
Teaching with Computers in Education
(Website: http://712educators.about.com/cs/technology/a/integratetech.htm)
Date Retrieved: October 1, 2008
Appendices
Appendix A
Letter of Request
First Asia Institute
Of Technology and Humanities
August 12, 2008
MR. ARNOLD CATAPANG
Principal
First Asia Institute of Technology and Humanities Unified School
Sir:
Greetings!
The undersigned BS Secondary Education major in Mathematics student from College of Education is currently undergoing a research entitled, “Perceived Effects of the Utilization of Modern Technology Gadgets in Learning Mathematics of Fourth Year High School students of First Asia Institute of Technology and Humanities.
In view of this, may I request for your approval to allow me distribute my questionnaire to the fourth year students of your institution.
Rest assured that the data and information that will be generated from this study will be utilized solely for academic purposes.
I’m hoping for your usual kind consideration.
Thank you!
Respectfully yours,
(SGD.) MARVELINO M. NIEM
(SGD.) ROSANNI DEL MUNDO, M.A.
Thesis Adviser
(SGD.) EVELIA S. ORBETA, Ed.D
Dean, College of Education
Approved by:
(SGD.) MR. ARNOLD CATAPANG
Appendix B
Cover Letter
First Asia Institute
Of Technology and Humanities
August 12, 2008
Dear Respondents,
Greetings!
The undersigned BS Secondary Education major in Mathematics student from College of Education is presently conducting a research entitled, “Perceived Effects of the Utilization of Modern Technology Gadgets in Learning Mathematics of Fourth Year High School students of First Asia Institute of Technology and Humanities.
In connection with this, may I request for your honest response to the attached questionnaire? This intends to gather information that would greatly enhance the quality of my study. Please do not leave any items unanswered. Rest assured that any information you will give will be treated with utmost confidentiality.
Thank you very much for your cooperation.
Respectfully yours,
(SGD.) MARVELINO M. NIEM
Noted by:
(SGD.) ROSANNI DEL MUNDO, M.A.
Thesis Adviser
(SGD.) EVELIA S. ORBETA, Ed.D
Dean, College of Education
Appendix C
Questionnaire
Name: _(optional)________________ Year &Section: __________
General Directions: Please accomplish the questionnaire very carefully and honestly. Rest assured that any information you will give will be treated with utmost confidentiality.
PART I: Technology-Aided Teaching Strategies and the Frequency of their Uses
Directions: Encircle the number that corresponds to the frequency of the use of such teaching strategy.
Always Often Sometimes Never
1. Lecture method with power 4 3 2 1
point presentation
2. Reporting with power 4 3 2 1
point presentation
3. Independent research through 4 3 2 1
the use of internet
4. Model-making through the 4 3 2 1
use of related software
5. Computer hands-on activities 4 3 2 1
6. Problem solving with scientific 4 3 2 1
calculator
7. Cooperative learning with power 4 3 2 1
point presentation
8. Use of graphic organizers in 4 3 2 1
solving problems
9. Discussion with the use of 4 3 2 1
power point presentation
10. Use of analogy (reasoning) with 4 3 2 1
the use of power point presentation
11. Discovery learning through the 4 3 2 1
use of scientific calculator
12. Guided discovery through the 4 3 2 1
use of related software
PART II: Effects of Utilizing Modern Technology in Learning Mathematics
Directions: Below are the possible effects in learning aided by modern technology. Using the scale, encircle the number that corresponds to your answer:
4- Very much 2- Not so much
3-Much 1- Not at all
Computer
My learning is enhanced when…
the teacher uses power point 4 3 2 1
presentation in presenting the
lesson.
2. the teacher shows animation 4 3 2 1
related to the topic in the monitor.
3. the teacher gives an independent 4 3 2 1
research work using the internet.
4. we work in groups on a particular 4 3 2 1
problem when using computer.
5. the lectures of my teacher were sent 4 3 2 1
to my e-mail.
6. the teacher shows some video 4 3 2 1
presentation related to our topic.
7. we always have computer hands-on 4 3 2 1
activities.
Scientific Calculator
My learning is enhanced when…
1. I use scientific calculator in solving 4 3 2 1
problems involving trigonometric
functions.
2. I evaluate, press the keys and see 4 3 2 1
the results on the display screen of the
scientific calculator.
3. the teacher encourages me to discover 4 3 2 1
by myself certain facts about numbers
using scientific calculator.
4. I challenge my classmate to a calculator 4 3 2 1
game like in being the first to solve
the problem and get the right answer.
5. I learn how to minimize the number 4 3 2 1
of keystrokes in solving problems.
6. I use scientific calculator in finding 4 3 2 1
the factors of a given large number.
7. I use scientific calculator in doing my 4 3 2 1
homework.
PART III: Effects of Using Modern Technology Gadgets in the Development of Students Skills
Directions: Below are the possible effects in learning which is aided by modern technology. Using the scale, encircle the number that corresponds to your answer:
4- Strongly Agree 2- Disagree
3-Agree 4- Strongly Disagree
Computer
By using computers…
1. I become attentive to learn 4 3 2 1
new concepts.
2. I understand the lesson easier. 4 3 2 1
3. I can visualize the underlying 4 3 2 1
principles of the topic.
4. I become more participative in 4 3 2 1
class discussion.
5. I learn while having fun. 4 3 2 1
6. I become interactive in group 4 3 2 1
activities.
7. I am more receptive in the discussion. 4 3 2 1
Scientific Calculator
By using scientific calculator…
1. I develop my skills in solving 4 3 2 1
problems.
2. I reinforce my skills in computation. 4 3 2 1
3. I improve my reasoning to a 4 3 2 1
higher-level of thinking.
4. I enhance my analytical thinking skills. 4 3 2 1
5. I spend less time in solving large 4 3 2 1
numbers.
6. I can skip the step-by-step process 4 3 2 1
on how to solve problems.
7. I find it easy to solve difficult 4 3 2 1
problems.
Appendix D
Tallied Data
PART I: Technology-Aided Teaching Strategies and the Frequency of their Uses.
Item Number
Always
Often
Sometimes
Never
1
39
47
16
0
2
35
41
24
2
3
27
48
25
2
4
17
58
25
2
5
37
37
22
6
6
32
39
22
6
7
24
63
15
0
8
12
55
33
2
9
34
51
15
2
10
14
54
30
4
11
17
36
46
3
12
13
57
30
2
PART II: Effects of Utilizing Modern Technology in Learning Mathematics.
a. Effects of Utilizing Computer in Learning Mathematics
Item Number
Very Much
Much
Satisfactorily
Not Satisfactorily
1
38
58
4
2
2
24
43
34
1
3
16
47
38
1
4
8
45
41
8
5
11
26
30
35
6
20
35
31
16
7
24
41
32
5
b. Effects of Utilizing Scientific Calculator in Learning Mathematics
Item Number
Very Much
Much
Satisfactorily
Not Satisfactorily
1
38
36
25
3
2
33
49
19
1
3
18
46
35
3
4
13
32
35
22
5
18
51
31
2
6
43
33
24
2
7
42
39
18
3
PART III: Effects of Using Modern Technology Gadgets in the Development of Students Skills.
a. Effects of Using Computer Gadgets in the Development of Students Skills.
Item Number
Strongly Agree
Agree
Disagree
Strongly Disagree
1
41
56
5
0
2
25
66
9
2
3
27
54
18
3
4
17
69
16
0
5
43
53
6
0
6
23
57
19
3
7
20
66
16
0
b. Effects of Using Scientific Calculator Gadgets in the Development of Students Skills.
Item Number
Strongly Agree
Agree
Disagree
Strongly Disagree
1
31
52
16
3
2
24
56
21
1
3
26
60
15
1
4
22
61
18
1
5
54
34
14
0
6
45
44
12
1
7
49
42
9
2
CURRICULUM VITAE
MARVELINO MANIPOL NIEM
H
151 San Felix, Sto. Tomas, Batangas
(
09195347768
*
vel_capcom@yahoo.com
PERSONAL INFORMATION
Nickname : Marvel
Birthday : April 20, 1988
Age : 20
Civil Status : Single
Nationality : Filipino
Religion : Roman Catholic
Height : 5’ 5”
Weight : 110 lbs.
Father : Mario Caponpon Niem
Mother : Maria Manipol Niem
Address : 151 San Felix, Sto. Tomas, Batangas
EDUCATIONAL BACKGROUND
2005 –2009
Bachelor of Science in Secondary Education
Major in Mathematics
First Asia Institute of Technology and Humanities
2 Pres. Laurel Hi-way, Darasa, Tanauan City, Batangas
2001-2005
St. Thomas Academy
Poblacion 3, Sto. Tomas, Batangas
Achiever
1995-2001
1994-1995
Sto. Tomas Central School
Poblacion 4, Sto. Tomas, Batangas
Achiever
San Felix Elementary School
San Felix, Sto. Tomas, Batangas
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