Graphing Software in Understanding of Polynomial Functions



Graphing Software in Understanding of Polynomial Functions

Sadia M Syed

California State University, Northridge

College of Education

06/04/2008

Instructor, Dr. Brian Foley

TABLE OF CONTENTS

• Abstract

• Introduction

o Description of the Study

o Importance of the study

o The Research Questions

o Description of the Tool

• Literature Review

o Theoretical Framework

o Related Studies

• Methodology

o Description of the Participants

o Description of the Tool

o Description of how the Tool was used

o Research Instruments

o Procedure

o Analysis

• Findings

o Findings from post-test

o Findings from student survey

• Conclusion

• References

• Appendix A – Student Survey 1

• Appendix B – Post-Test

• Appendix C – Student Survey 2 - Experimental Group

ABSTRACT

This research was conducted to find out whether or not using the software, Graph (padowan.dk/graph/), during the instruction of polynomials can improve students’ understanding of function and their ability to create and interpret graphical representation of functions. This study was conducted over three weeks and with two groups (control group and experimental group) of students. Both groups were taught by the same teacher and were instructed the same lesson on polynomial function. After each lesson the control group was asked to work with graph-paper and the experimental group was asked to work with the software. Both groups were assessed by post-test, and a student survey was conducted after the post-test with the experimental group to gather information regarding students’ opinion on the software and its influence on their learning process. The t-test analysis of the post-test showed that there is no overall significant difference between the understanding of the concepts of the control group and the experimental group. But the software helped the students of experimental group as they did well on the question of analyzing and interpreting graphs. Moreover, the analysis of the ‘student survey’ conducted with the experimental group indicated that using software during the instruction improved students’ motivation, confidence and interest.

INTRODUCTION

Description of the Study:

In last several years functions and graphs have been a major focus of many researches in the field of mathematics and education. Several studies have been conducted to understand the influence of computers on students’ understanding of the concept of functions, and their ability to create and interpret graphical representation of functions (e.g., Asp, Dowsey & Stacey, 1994; Hollar & Norwood, 1999; Kaput, 1992; Manoucherhri, 1999; Ruthven & Hennessy, 2002; Simmt, 1997; Yerushalmy, 1991). The findings from these studies have provided strong indication that the use of computers as a thinking aid and an intellectual tool enrich learners’ mathematical understanding, facilitate students’ growth of mathematical explorations, and improve their problem solving skills and concept developments.

Most secondary schools in the USA are now equipped with computers and connected to the internet. According to Educational Technology Fact Sheet (2006), the ratio of students to computers in all public schools in 2003 was 4.4 to 1. The Mathematics 2000 report states that the availability of computers in classrooms increased by at least 20 percentage points from 1996 to 2000, although the use of computers in mathematics teaching increased at a lower rate than other subjects (Paulson, 2000).

Technology provides students an opportunity to use "hands-on" techniques in problem solving. Technology also helps students to develop an understanding of the processes and reasoning that are the heart of mathematical problem solving (Hudnutt, 2007). The NCTM also supports the use of technology to enhance student learning. As stated in one of the seven principles in the Principles and Standards, “Calculators and computers are reshaping the mathematical landscape, and school mathematics should reflect those changes. Students can learn more mathematics more deeply with the appropriate and responsible use of technology. They can make and test conjectures. They can work at higher levels of generalization or abstraction,” (NCTM, 2000, p. 25).

This research was conducted to find out whether or not using the software (Graph) during the instruction of polynomials can improve students’ understanding of function and their ability to create and interpret graphical representation of functions. The software that will be used in this study is ‘Graph’, a free graphing software from padowan.dk/graph/. The software can be used to draw mathematical graphs in various (i.e. Cartesian, Polar) coordinate systems. Users can easily draw graphs of functions and the program makes it very easy to visualize the functions. Students can readily enter a list of expressions using the graph editor, watch the graph, and explore a function through a numeric display of coordinates, intersection points, slopes, tangents and maxima. All these features are designed to be accessed through buttons and ‘Menu’ commands, and are extremely powerful for users of mathematics who are seeking data on a specific function or equation. I have chosen this software rather than other advanced graphing software, because it is easy to understand and manipulate (see the screen shots in chapter 3) by algebra beginners.

One of the important reasons for choosing this software is its ease of manipulation of the command bar. According to Yerushalmy (1999), one of the main problems of past and present graphing software is their complex manipulation and lack of user-friendly features. Using very simple commands this software allows students to easily enter a list of functions and explore those through various displays of coordinates, intersection points, slopes, tangents and maxima. Another problem identified by Guin and Trouche (1999) is the distinction between the function syntax of one set of commands and the solving syntax of the other commands that initiated difficulties for students who are at the early stage of developing conceptual understanding of functions. The software ‘Graph’ also meets the current focus of mathematics curriculum, which emphasize that the school should provide opportunities for students to construct knowledge and think mathematically through exploration and investigation (Manoucherhri, 1999).

Technology can have profound effect in the learning and teaching of functions and graphs. In their analysis of research on the teaching and learning of functions, Leinhardt, Zaslavsky, and Stein (1990) note that, “more than perhaps any other early mathematics topic, technology dramatically affects the teaching and learning of functions and graphs,” (p. 7). With the help of technology teachers can have students make observations and conjectures within a variety of function representations such as equations, graphs, and tables. According to Hudnutt (2007) “Students can then begin to make connections among the representations in order to develop a concept image without first having an in depth knowledge of function. With the use of technology, teachers can expose students at a much earlier stage in their cognitive development to the function concept. This, in turn, allows students to explore the connections among representations enabling the learning of functions to become investigative in nature.”

Importance of the Study:

Recent reform movements in mathematics education encourage the use of computer technologies in the classroom, and in particular, the use of computer supported explorations as contexts for mathematics instructions (Manoucherhri, 1999). The availability of computers and mathematics software (Freeware, Shareware etc.) has great potential to take a positive step towards engaging students more actively in a process of mathematical thinking and learning. Even though all aspects of a complex mathematical idea can not be expressed with a single representational system, Kaput (1992) argues that the ability to make translation from one representation of a function to another is a particularly important aspect of mathematical thinking which may be enhanced by technology. The convenient access provided by graphing software to numerical and graphical representations of a verity of functions may assist students to develop a broader and deeper understating of the concepts. Graphing software enables rapid and automatic translation between algebraic, graphical and numerical representations, whereas translation by hand is generally a slow and laborious process for students.

The study of polynomials of higher degree can become a fascinating part of school mathematics with the accessibility to graphical representations now available through computer software and graphing calculators. These representations can be incorporated and used to create a mental image of the functions as an aid to mathematical intuition needed to deal with functions (Movshovitz-Hadar, 1993). Since "seeing" the algebra has become possible through graphing technology, it can be used to make symbol manipulations more meaningful in operating the algebra.

Understanding the concepts of polynomial functions and their graphs using graphing software will provide students with insights that enable them to construct deeper and more coherent graphing concepts and it will help them develop important techniques needed to comprehend various concepts of polynomial functions. For example, when the students are learning translation and transformation of graphs, they can be asked to use the software to graph functions with different degrees and with different coefficients. Using this software the students can easily examine and discover how change in degree and coefficient of a function can change its shape and position. Following are some screen shot that display some of the transformations and translations created by the software.

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Figure 1

Graphing software provide more emphasis on graphs and their interpretation, both to help students understand key ideas of polynomial functions, their transformation and translation. According to the discussion of Kissane (1995), the ease with which calculators can draw graphs means that students can concentrate on the meanings inherent in graphs instead of the mechanics of producing them. Using graphing software in learning of polynomial concepts gives more importance on making the ‘use of’ graphs, rather than ‘producing’ a graph.

Another profound reason for conducting this study on effectiveness of graphing software in learning of polynomial concepts is to encourage teachers and students of mathematics education to take advantage of vast collection of freeware and shareware available on the internet. Schools in lower economic status can extend their instructional facility by combining their existing computer technologies with the graphing software. Many schools may not be able to provide enough graphing calculators for all students, but can probably use their computers equipped with this free graphing software, and thus create a significant connection between uses of computer and mathematics education. Through this research I think other researchers will be encouraged to study and evaluate the extensive collection of educational freeware and shareware available on the internet and also be able to make recommendations on their applications by using them with traditional instructions.

The Research Questions:

This study was conducted over three weeks and with two groups of students. Both groups were taught by the same teacher and they were equivalent in their academic background and knowledge in understanding functions. These groups were instructed the same lesson on polynomial function. After each lesson one group was asked to work with graph-paper and the other group was asked to work with the software. The research question of this study was to find out whether or not using the software (Graph) during the instruction of polynomials can improve students’ understanding of function and their ability to create and interpret graphical representation of functions. This question will be answered through the analysis of the students’ (both groups) score on their post-test.

LITERATURE REVIEW

Theoretical Framework:

Use of graphing technology in mathematics instructions is strongly influenced by Jean Piaget’s “Constructivism.” The perspective of constructivism on teaching and learning of mathematics “focuses on the provision of opportunities for students to engage in reflective mathematical thinking as they consider the viability of their extant understandings, strive to resolve creatively new cognitive perturbations, and test the viability of their tentative solutions strategies” (Forster & Taylor, 2000, p. 4). In the article “Calculators and Constructivism” by Wheatley and Clements (1990), it is cited from Glasersfeld (1990) that from a constructivist perspective, a calculator can aid mathematics learning when it “permits meaning to be the focus of attention, creates problematic situations, facilitates problem solving, allows the learner to consider more complex tasks, and lends motivation and boosts confidence” (p. 296). Graphing technology creates a constructive environment that helps students to explore mathematical concepts in an organized way. According to Yerushalmy (1999), graphing software helps to organize mathematical concepts by a limited collection of important terms, objects and actions. Graphing technology makes these organizations visible to user and thus becomes a strong mathematical thinking tool, tool for planning, and tool for problem posing.

The inquiry of using the graphing technology in mathematics instructions is also influenced by the socio-cultural view, described by Forster & Taylor (2000). According to Forster & Taylor, socio-cultural perspectives focus on the provision of opportunities for students to co-construct valid mathematical meaning through participation in rich mathematical conversations with the teachers, graphing calculators and fellow students. In their research with graphing calculators, Forster & Taylor (2000) found most of the students who used calculator during the instructions were able to successfully transfer their knowledge from a specific situation (without calculator) to another situation when they were permitted to use calculator. They described the transfer of knowledge of those students by Cobb and Bower’s (1999) social constructivist view, which explains that the use of graphing technology is social in a sense that students enter an interactive, intellectual partnership with it. From this perspective, the transfer of knowledge represents that students’ mathematics practices in one social context (an instructional setting without technology) were relevant to another social context (where instruction extended to using technology). From this research it is noticeable that when students experiment various concepts using graphing calculator, it can be directed as a transferable social practice of technology usage.

Research on Graphing Software:

A significant number of studies (Asp, Dowsey & Stacey, 1994; Hollar & Norwood, 1999; Kaput, 1992; Manoucherhri, 1999; Ruthven & Hennessy, 2002; and Simmt, 1997) have been conducted to examine the effects of graphing technologies on students understanding of algebraic functions. By generalizing the findings of the research, it is evident that students’ achievement is positively influenced by the graphing technologies when they are used to facilitate students’ higher cognitive skills (i.e. analyze, synthesize, evaluate). A number of studies (Heller & Curtis, 2006; Khoju & Miller, 2005; Ellington, 2003) show that the use of graphing technology positively impacts students’ performance in algebra, improves mathematics test scores – both with and without a calculator during testing, and establishes better student attitudes towards math.

Graphing technology helps increase students’ problem solving skills. According to Dunham & Dick (1994) the use of graphing technology when teaching problem solving strategies led to a significant increase in the achievement of the students. Graphing technology facilitates students to create relationships to new situations and to communicate solution strategies (Hubbard, 1998). Graphing technology lessens the amount of attention needed for algebraic manipulation, thus allowing more time for actual instruction. It also supplies more functions and can serve as a monitoring aid during the problem solving process (Dunham & Dick, 1994).

A meta-analysis of eight individual studies done by Khoju & Miller (2005) has found strong evidence of increased performance in algebra when students increasingly use graphing technology. On the other hand, Barton (2000) has expressed that when studies seeking to isolate the technology variable, by controlling curriculum, texts, homework, exams and teacher variables, did not find a significant difference in overall achievement between the treatment group and the control group. Barton (2000) also suggests that simply having access to technology does not ensure it will be used to enhance learning. A meta-analysis of 54 studies done by Ellington (2003) found that students who receive instruction using a graphing calculator perform as well or significantly better in conceptual problem solving and operational skills areas. One of the studies conducted by National Assessment of Education Progress (NAEP) has shown that frequent use of graphing technology is associated with greater mathematics achievement:

Eighth-graders whose teachers reported that calculators were used almost everyday scored highest. Weekly use was also associated with higher average scores than less frequent use. In addition, teachers who permitted unrestricted use of calculators and those who permitted calculator use on tests had eighth-graders with higher average scores than did teachers who did not indicate such use of calculators in their classrooms (National Center for Education Statistics 2001, p. 141).

Graphing technology can make designing, creating and using multiple representations easier. When using this technology during instruction, rather than using time for laborious and tedious calculations, students can have more time and mental energy to explore various underlying concepts. As an example, during instruction on quadratic equations, students can easily investigate the effects of changing the value of a, b and c on the graph of ax2 + bx + c, which can be very tedious when using paper-pencil graphing techniques. Specific research has shown that students can often reason best when they experience mathematics through related representation, such as equations, tables and graphs (Goldenberg, 1995; Kaput, 1992). Furthermore, technology can create connection between the representations, enabling students to make conceptual connections, such as, understating how a change in an equation links to a change in a graph (Roschelle, 2006).

In regards to the ease in which functions can be graphed and manipulated with software, Dugdale (1993) found that these tools have, “raised the possibility of visual representations of functions playing a more important role in mathematical reasoning, investigation, and argument. Relationships among functions can be readily observed, conjectures can be made and tested, and reasoning can be refined through graphical investigation,” (pg. 115). Doerr and Zangor (2000) analyzed a pre-calculus class that widely used the graphing calculator. They made in-depth observations and found that this technology was instrumental in facilitating analytical thinking.

Tall (1989) examined the potential of well-designed software to improve a learner’s concept image of function by allowing the learner to explore the complex structures of functions. The software allows students to experience higher level cognitive structures than they would be able to without such software. He suggested students can more readily explore complex concepts through the use of technology. Dugdale (1993) also agrees with Tall. In her own research on the use of technology to support student thinking with functions, as well as reviewing others’ research, Dugdale states, “Such tools have facilitated the movement away from a focus on calculating values and plotting points toward a more global emphasis on the behavior of entire functions, end even families of functions,” (pg 114). Therefore, using graphing software students can study the more complex global aspects of function prior to or in parallel with studying functions as input/output machines.

Graphing technology helps to improve students’ performance on visual and graphing tasks. According to Demana & Waits (1992), the visual impact of graphing calculators on students greatly enhances their learning of mathematics. Because graphing technologies enhance visualization and invite self discovery, students are able to relate to novel problem situations (Scariano & Calzaada, 1994, cited by Hubbard, 1998). Graphing technology also serves as a positive motivator among students because they seem to enjoy using it. In their research, Demana & Waits (1992) have shown that graphing calculators can make the study of mathematics fun and can give students excellent learning experiences.

Using dynamic software students can be benefited by the easy manipulation of a function through various representations to construct their own internal, flexible images of function. Yerushalmy and Chazan (1990) in their research with dynamic geometry environment found that students were able to reason more flexibly about geometric concepts than their counterparts who learned with static diagrams. Similarly, Moschkovich, Schoenfeld, and Arcavi (1993) explored student learning of functions with the dynamic software, GRAPHER, and found that such an environment “allows students to operate on equations and graphs as objects… in ways not possible before the existence of such technologies,” (pg. 98). In this research study, they also found that “it is not just the dynamics on the screen that makes the difference in student learning. Students must integrate what they see on the computer screen into their own conceptual structures for any learning to take place” (Moschkovich et al,1993).

A number of studies (Heller & Curtis, 2006; Khoju & Miller, 2005; Ellington, 2003) have shown that using graphing technology in the instruction of algebra improves students’ assessment scores and positively impacts students’ performance in algebra. Heller & Curtis (2006) conducted a study to look at the relationship between graphing calculator use and student standardized test scores in grades 9-11. One of the key findings of their study was increasing use of graphing calculators during instruction resulted in higher test scores even when students did not use graphing calculators during test taking. Ruthven (1990) in his study suggested the impact of technology in the secondary classroom might depend on the way in which the technology is used to mediate mathematics in the classroom.

Use of graphing technology in the classroom can be different depending on the curriculum and instruction. Simmt (1997) in the study “Graphing Calculators in High School Mathematics,” examines how mathematics educators used graphing calculators in their instructions and how their views of mathematics were manifested in the ways they choose to use this technology. In the study Simmt (1997) reported that ‘most of the teachers used the tool to facilitate one or two guided-discovery activities, and they did not use the tool to facilitate and/or encourage the students to conjecture and prove or refute ideas’ (p. 286). In the article “Graphing Calculators in the Mathematics Classroom” Smith (1998) stated, “When wearisome computation and plotting tasks are minimized, students can become engaged in answering "what-if" questions. The thought of changing the premises of a mathematical argument grows more attractive if the chore of executing those changes is easier. Moreover, the graphing calculator promotes autonomy in asking questions, encouraging students to pose their own problems (p. 1).” Touval (1997), in the research "Investigating a Definite Integral - From Graphing Calculator to Rigorous Proof," suggests that the graphing calculator can be used as a springboard for discovery. While learning how to calculate definite integrals, students proposed their own theories concerning integration. While using the graphing calculators to investigate quick solutions to problems, the students formulated conjectures that eventually led to a rigorous proof.

Teachers’ level of expertise in using graphing software has an impact on the proper implementation of the technology in the classroom, and teacher’s level of expertise has to be increased by proper training before conducting the research. Asp, Dowsey and Stacey (1994) conducted a study using ANUGraph software package to teach linear and quadratic functions. Their finding suggested teachers’ previous experiences of using graphing technology have a significant impact on successful implementation of these tools in mathematics instructions. One of the key findings of the research done by Heller & Curtis (2006) described that when the teachers participated in training on how to use graphing calculator or other computerized graphing technology, student achievement was significantly higher, compared to those teachers who learned to use those technology by reading manual. This finding suggests that students benefit when their teachers receive professional development that is specific to graphing technology in math instruction.

Even though there are many studies that deal with the positive impacts of graphing software and calculators, there are few studies that have found some shortcomings. Goldenberg (1988) identified student difficulties with graphing technologies due to issues of scale. Students unfamiliar with notions of window sizing, scaling on the axes, and global behaviors experienced difficulties in using the technology to understand graphs. When computers are used for simulation of mathematical constructs, there are still opportunities for misinterpretation. Moschkovich, Schoenfeld, and Arcavi (1993) found students interpreted the pixilation of a line represented on a graph as an actual property of the line.

METHODOLOGY

Description of the Participants:

The participants in this study were middle and high school students. The school is located in Eagle Rock, a suburb of Los Angeles. There were total thirty students in the study, and each group contained fifteen students. Among the participating students seventeen (57%) were female participants and thirteen (43%) were male participants; nineteen (63%) participants were Hispanic, seven (23%) were Caucasian, two (7%) were African American, and two (7%) were Asian. Approximately 34% of the participants were English Language Learners (ELL) and 47% of the participants were classified as economically disadvantaged. Among the participants there were twelve (40%) eighth grade students, fifteen (50%) ninth grade students and three (10%) tenth grade students. Approximately 85% of these participants had access to personal computer at home and 75% had Internet connection at home. According to a survey (See Appendix A), majority of the participants used their computers mostly to browse the internet, send and receive email and text massage, and perform word processing tasks. The participants had one 45 minutes math period everyday from Monday through Friday.

‘Matching’ process was used to create relatively similar “Control Group” and “Experimental Group” for the research, and each group had fifteen participants. The reason for using ‘Matching’ process for this study was to create two groups that contained participants who had similar scores in the Algebra Diagnostic Test. In this school, at the beginning of each academic year every algebra student takes a diagnostic test on algebra to determine their previous understanding about the prerequisite concepts. Based on the test scores, then the students are grouped for the respective level/class on algebra. The researcher had access to those diagnostic test scores of the participating students. Students in all levels of scores (i.e. participants who had more than 90%, had less than 90% and more than 80%) were equally distributed between the two groups (Control and Experimental). Ten participants had more than 90% on that test and they were equally divided between the two groups. Ten participants had scores between 90% and 81% on that test and they were also equally divided between the two groups. Ten participants had scores between 80% and 70% and they were also equally divided between the two groups.

Materials:

Description of the Tool:

The software that will be used for this study is ‘Graph’ (padowan.dk/graph/). This software can be used to draw mathematical graphs in various (i.e. Cartesian, Polar) coordinate systems. Students can readily enter a list of expressions using the graph editor, watch the graph, and explore a function through a numeric display of coordinates, intersection points, slopes, tangents and maxima. All these features are designed to be accessed through buttons and ‘Menu’ commands. According to the software publisher, this application allows students to do the following:

Draw functions: Graph can draw normal functions, parameter functions, and polar functions. Students can use a lot of built-in functions, e.g. sin, cos, log, etc. Students may specify color, width and line style of the graphs and the graphs may me limited to an interval. It is also possible to show a circle at the ends indicating open or closed interval.

Screen Shot # 1

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Draw relation: Graph can show any equation and inequality, for example sin(x) < cos(y) or x^2 + y^2 = 25. Students can choose line width and color for the equations, and color and shading style for the inequalities.

Interact with other programs: Students can save the coordinate system with graphs as an image on disk either as a bitmap (bmp), Potable Network Graphics (png), JPEG, metafile (emf) or Portable Document Format (PDF).

Evaluate: Given an x-coordinate the software will calculate the function value and the first two derivatives for any given function. Alternatively the function may be traced with the mouse.

Calculate: Graph can help students calculate the area under a function in a given interval and the distance along the curve between two points on the function. The program can also show the first derivative of a function, and create tangents and normal lines to a function at given coordinates.

Screen Shot # 2

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Point series and trend lines: Students can create series of points with different markers, colors and size. Data for a point series can be imported from other programs, e.g. Microsoft Excel. It is possible to create a line of best fit from the data in a point series, either from one of the built-in models or from a user specified model.

Shadings and labels: Graph can insert shadings used to mark an area related to a function. Shadings may be created with different styles and colors in a user specified interval.

Description of how the software used in the research:

The software that was used for this study is ‘Graph’ (padowan.dk/graph/). Using the software students can easily draw graphs of functions and the program makes it very easy to visualize the functions. Students can readily enter a list of expressions using the graph editor, watch the graph, and explore a function through a numeric display of coordinates, intersection points, slopes, tangents and maxima. Students can also use the images of the graphs in other programs by saving it as a bitmap (bmp), Potable Network Graphics (png), JPEG, metafile (emf) or Portable Document Format (PDF). All these features are designed to be accessed through buttons and ‘Menu’ commands.

The first concept that was introduced for this unit of instruction is ‘Polynomial Models’ where students learn the definition of polynomial as well as various types of polynomials (i.e. quadratic polynomials, cubic polynomials) that can arise in real life situation. During the lesson teacher used direct instruction at the beginning to clarify various vocabularies (i.e. degree, leading co efficient, quadratic and cubic polynomials, factoring, zeros of polynomial). In the second part of the lesson teacher used the software to demonstrate some graphs of polynomials. Simultaneously teacher demonstrated how to insert functions and create graphs using the software. Towards the end of this lesson students worked in groups of two in inserting functions and creating graphs using the software. Students were given a worksheet with various equations. Among those equations some were polynomial and some were exponential functions. Students were asked to create graphs of those equations and they were also encouraged to conjecture regarding the type of functions. At the end of the lesson students discussed and provided support for their conjectures.

The second concept was ‘Transformation and Translation of Quadratic and Cubic Polynomial’ where student learn how the graphs of polynomial functions change their size, shape, and position in respect to change in their degrees and coefficients. In this lesson teacher used guided discovery based instructions to encourage students explore ‘what-if’ types of questions. For example, to understand translation and transformation of quadratic graphs, teacher designed a set of functions for students to explore using the software. In one of the specific assignments, teacher asked the following: A) Use all integer values for ‘a’ where -10 ................
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