Graphing Software
Graphing Software in Understanding of Polynomial Functions
Sadia M Syed
California State University, Northridge
College of Education
06/04/2008
Instructor, Dr. Brian Foley
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Graphing Software
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
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Graphing Software
ABSTRACT
This research was conducted to find out whether or not using the
software, Graph (www.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.
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Graphing Software
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
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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
www.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
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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.”
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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.
Graphing Software
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
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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.
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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
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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’
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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:
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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 paperpencil 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
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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.
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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.
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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
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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.
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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
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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’ (www.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.
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Screen Shot # 1
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
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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
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’ (www.padowan.dk/graph/).
Using the software students can easily draw graphs of functions and the program
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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
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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<a<10 in the given function f(x) = ax2 to create graphs.
What can you conclude about the effect of the value of ‘a’? B) Use all integer values for
‘b’ where -10<b<10 in the given function f(x) = x2+bx to create graphs. What can you
conclude about the effect of the value of ‘b’? C) Use all integer values for ‘c’ where 10<c<10 in the given function f(x) = x2+x+c to create graphs. What can you conclude
about the effect of the value of ‘c’? For this assignment students worked in groups of
two where they used the software to explore various changes in the graphs and
developed their understanding. The software created a dynamic environment that
helped the students to observe the translations and transformations of the graphs.
The third concept that was discussed in this unit was ‘Factoring Polynomials’
where students learned four common ways of factoring: 1) factoring the largest common
monomial factor, 2) factoring following a pattern, 3) quadratic polynomial factoring, 4)
using the Factor Theorem. This lesson was delivered using direct instruction and group
discussions. Students worked individually to work out exercises that were presented
through worksheets. The use of the software was minimal during this lesson.
The forth concept was ‘Factor Theorem and Identifying Zeros of the Polynomial
Function’ where students learned the Factor Theorem to identify the zeros of a
polynomial function. In this lesson teacher designed the instructional tasks involving a
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various degrees of polynomials. In these tasks student were required to use the
software to create the graphs of those given polynomials and by analyzing the xintercepts of those graphs they further developed the understanding of relations
between zeros of the polynomials. For example, after analyzing the graphs student
would recognize and generalize about even/odd degree polynomial and multiplicity. As
an example, following is a screen shot of a task where students had to produce the
graph and after analyzing the graph they had to identify the four zeros of the function
f(x) = 2x4-5x3-x2 using the concept of even function and multiplicity.
y
4
f(x)=2(x)^4-5(x)^3-(x)^2
2
x
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
10
-2
-4
-6
-8
-10
-12
Figure 2: Screen shot of a fourth degree polynomial
Research Instruments:
In this research, students’ understanding was measured by post-test. The
post-test (See Appendix B) contained five questions, among which one question was
knowledge based, one question was comprehension based, one question was
application based (Plotting graph), and two questions were based on analytical thinking.
After the post-test a student survey was conducted. The survey had six questions and
each question had five answer choices. The survey was only conducted to the
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experimental group to gather information regarding students’ opinion on the software
and its influence on the learning process. The questions of the survey were related to
the students’ level of motivation in using the software, helpfulness of the software in
learning the concepts, level of difficulty in using the software, students’ level of
confidence about the concepts, and students’ level of interests in future use of the
software. Another survey was also conducted at the beginning of the study to gather
information about participants. Through this survey the researcher collected information
regarding student demographics and their habit of computer use.
Procedures:
In this study the research design used was the Posttest only Experimental
design. The study took three weeks to complete. At the beginning of the study the
participants were grouped into two different groups (Control and Experimental) through
‘Matching’ process. The teacher who was delivering the instruction to both groups was
also trained during the first two days of the study. Both the control and experimental
groups included students from grades 8th, 9th and 10th. Each group was consist of
fifteen participants who are currently taking Algebra I. Both groups of students were
taught using lecture, classroom discussion, and text book readings and activities, but
only the experimental group had access to the graphing software. Participants of the
experimental group had access to eight desktop computers equipped with the graphing
software during the instruction. The groups were also given twelve instructional days to
learn the material and one day to prepare for the test. Both groups were given forty
minutes to take the post-test towards the end of the research study. During this study
the groups explored concepts related to Polynomial Functions (i.e. Graphing,
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Translating, Transforming, Factor Theorem and Analyzing solutions) from the book
Advanced Algebra, published by Scott, Foresman.
Analysis:
In this study statistical analysis was done by using SPSS computer program. The
statistical analysis was included both descriptive statistics and inferential statistics. The
descriptive analysis was incorporated by mean and standard deviation of the data. The
inferential statistics included t-test to determine the significance of the difference
between the two different means. Scores from the post-test was compared using an
independent sample t-test of statistical significance. The t-test measured the difference
between the mean of the scores of the post-test from the control group (who did not use
graphing software during the instruction) and the experimental group (who used the
software to analyze polynomial functions). The student survey that was conducted to
the experimental group was also analyzed by using MS Excel.
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FINDINGS
This chapter presents and analyzes the findings of the research project. The
findings will be described in two major sections. The first section will contain an analysis
of the post-test scores of the two groups (Control and Experimental) and the second
section will contain an analysis on the ‘Student Survey’ (See Appendix C) which was
conducted after the post-test. In this study the t-test analysis of the post-test scores was
not statistically significant (t(28)=-0.306, p>.05). Interestingly though, in two (analysis
base questions) out of the five questions in the post-test, students in the experimental
group showed significantly higher understanding of the concepts, and in one
(application base question) out of the five questions, students in the control group
showed significantly higher understanding of the concepts. After the post-test, the
researcher conducted a student survey with the experimental group in order to gather
information regarding students’ opinion on the software and its influence on their
learning process.
Findings from post-test:
A post-test was administered to both the experimental and control groups. All
thirty students from the two groups had participated in the test. The test had five
questions and among those questions one question was knowledge based, one
question was comprehension based, one question was application based, and two
questions were analysis based. Students were not allowed to use any calculators or the
computer software. It was a paper-pencil based test, where the students had to write
their response and show their work to receive full/partial credit. Each question was
worth 20 points, thus the total adding up to 100 points.
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29
Table 1 below shows the analysis of the scores of the first question (see
Appendix B) in the post-test. This question was knowledge based and it required
students to use the definition of polynomial in order to find the values of given terms.
The difference between the means (%) of the control and experimental groups is less
than 2%. Students in the experimental group scored slightly better in this question, but
the t-test (t(28)=-0.235, p>.05) shows that the difference between the means (%) of the
two groups is not significant. This indicates that using the software did not provide any
significant advantage to the experimental group in the knowledge based question.
Table 1
Question 1: Knowledge Base
Groups
N
Mean (%)
Std. Dev.
1(Ctrl)
16.32
15
2(Expr)
15
86.67
88.00
t
df
Sig. (2-tailed)
-0.235
28
0.816
14.74
Table 2 below shows the analysis of the scores of the second question (see
Appendix B) in the post-test. This question was comprehension base and it asked the
students to identify whether the given functions were polynomials or not, and to state
the degrees and the leading coefficients of those functions. The difference between the
means of the scores of the control and experimental groups is less than 1%. Students in
the experimental group performed slightly better in this question, but the t-test (t(28)=0.314, p>.05) shows that the difference between the means of the two groups is not
significant. This indicates that using the software did not provide any significant
advantage to the experimental group in comprehension based questions.
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30
Table 2
Question 2: Comprehension Base
Groups
N
Mean (%) Std. Dev.
1(Ctrl)
15
2(Expr)
15
89.33
90.67
t
df
Sig. (2-tailed)
-0.314
28
0.756
10.33
12.80
Table 3 below shows the analysis of the scores of the third question (see
Appendix B) in the post-test. This question was application based and it asked the
students to find the equation of a fourth degree polynomial, and to sketch the graph of
that polynomial. The difference between the means of the control and experimental
groups is approximately 14%. Students in the control group scored higher in this
question, and the t-test (t(28)=3.28, p<.05) shows that the difference between the
means of the two groups is significant. This indicates that using the software did not
provide any significant advantage to the experimental group in application based
questions, and the control group was able to learn the application concepts better
without the use of the software. Since the students in the control group used paperpencil to graph during the instruction, it was expected that they would score higher than
the experimental group in this question.
Table 3
Question 3: Application Base
Groups
N
Mean (%)
Std. Dev.
1(Ctrl)
7.24
2(Expr)
15
15
96.67
82.67
14.86
t
df
Sig. (2-tailed)
3.280
28
0.003
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Table 4 shows the analysis of the scores of the fourth question (see Appendix B)
in the post-test. This question was analysis base and the students were given a graph
of a polynomial function. After analyzing the graph students were supposed to find the
degree and the zeros of the function. They were also asked to find the equation of the
given graph. The difference between the means of the control and experimental groups
is approximately 8%. Students in the experimental group performed better in this
question, and the t-test (t(28)=-2.245, p<.05) shows that the difference between the
means of the two groups is significant. This indicates that using the software did provide
a significant advantage to the experimental group in analysis based questions.
Table 4
Question 4: Analysis Base
Groups
N
Mean (%)
Std. Dev.
1(Ctrl)
9.90
15
2(Expr)
15
84.67
92.67
t
df
Sig. (2-tailed)
-2.245
28
0.033
9.61
Table 5 below shows the analysis of the scores of the fifth question (see
Appendix B) in the post-test. This question was also analysis base and it tested
students’ understanding of translation and transformation, as well as the analysis of the
Factor Theorem. The difference between the means of the control and experimental
groups is approximately 7%. Students in the experimental group performed better in this
question, and the t-test (t(28)=-2.323, p<.05) shows that the difference between the
means of the two groups is significant. This indicates that using the software did provide
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a significant advantage to the experimental group in analysis based questions.
Table 5
Question 5: Analysis Base
Groups
N
Mean (%)
Std. Dev.
1(Ctrl)
8.28
15
2(Expr)
15
76.00
83.33
t
df
Sig. (2-tailed)
-2.323
28
0.028
9.00
Table 6 below shows the analysis of the total scores of the five questions in the
post-test. The difference between the means of the control and experimental groups is
less than 1%. Students in the experimental group performed diminutively better in the
average scores in the post-test. Even though the t-test analysis (t(28)=-0.306, p>.05)
shows that the difference between the means (%) of the two groups is not statistically
significant, the analysis of the individual questions show that using graphing software
provided students with a greater achievement in analysis base questions.
Table 6
Questions 1 – 5: Total Scores
Groups
N
Mean (%)
Std. Dev.
1(Ctrl)
5.79
15
2(Expr)
15
86.67
87.46
t
df
Sig. (2-tailed)
-3.06
28
0.762
8.30
This finding concurs with Tall (1989), who examined the potential of welldesigned software to improve a learner’s concept image of function by allowing the
learner to explore the complex structures of functions. According to Tall (1989), the
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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.
Findings from student survey:
The student survey had six questions and each question had five answer
choices. The survey was conducted only to the experimental group (the group that used
graphing software during instruction). It was conducted to gather information regarding
students’ opinion on the software and its influence on the learning process.
The first question of the survey dealt with the students’ level of motivation when
learning the concepts using the software. Students could choose an answer from the
five choices provided in the survey. The analysis (Figure 3) of the responses shows that
60% of the students were above average motivated or highly motivated when the
software was integrated with the daily lessons.
Number of Responses
Q1: How motivated were you to learn the concepts
using the software?
7
6
5
4
3
2
1
0
6
4
3
2
0
Not motivated
at all
Sort of
motivated
Average
motivated
Above average
motivated
Highly
motivated
Level of Motivation
Figure 3: Responses to question # 1
The second question of the survey was regarding to how helpful the software
was to the students when they used it during the lessons. Students could choose an
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answer from the five choices provided in the survey. The analysis (Figure 4) of the
responses shows that thirteen out of fifteen (87%) of the students responded that the
software was above average helpful or highly helpful.
Number of Responses
Q2: How helpful was the software in understanding
the concepts?
8
7
6
5
4
3
2
1
0
7
6
1
1
Sort of helpful
Average
helpful
0
Not helpful at
all
Above average Highly helpful
helpful
Figure 4: Responses to question # 2
The third question dealt with the software’s level of difficulty when the students
were using it during the lessons. The analysis (Figure 5) of the responses shows that
almost half the students (47%) felt that the software had average level of difficulty when
using it during instructions. On the other hand, about 27% of the students felt the
software was above average or highly difficult, and the rest (about 27%) of the students
found that the software was sort of difficult or not difficult at all.
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Number of Responses
Q3: What is the level of difficulty in using the
software?
8
7
6
5
4
3
2
1
0
7
3
2
2
1
Not difficult at Sort of difficult
all
Average
difficult
Above average Highly difficult
difficult
Figure 5: Responses to question # 3
The fourth question of the survey dealt with the level of confidence of the
students in their learning of the concepts using the software. The analysis (Figure 6) of
the responses from the students shows that 60% of the students’ level of confidence
was above average or high. 27% of the students felt their confidence level was average,
and the rest of the students (about 13%) felt they were sort of confident.
Number of Responses
Q4: How confident do you feel about the concepts
that you have learned using the software?
8
7
6
5
4
3
2
1
0
7
4
2
2
0
Not confident
at all
Sort of
confident
Average
confident
Above average
confident
Figure 6: Responses to question # 4
Highly
confident
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The fifth question asked the students about their level of interest about the
concepts when using the software in their learning. The analysis (Figure 7) of the
responses shows that among the 15 students in the group, 8 students (About 53%)
responded that they had above average or high interest in the concepts. Only 27% of
the students had average interest and the rest of the 20% students had less than
average interest.
Number of Responses
Q5: How interested were you in learning the concepts
by using the software?
8
7
6
5
4
3
2
1
0
7
4
3
1
0
Not interested
at all
Sort of
interested
Average
interested
Above average
interested
Highly
interested
Figure 7: Responses to question # 5
The last question of the survey was related to the future use of math software by
the teachers. The analysis (Figure 8) of the responses shows that 100% students
recommended that their future teachers should use math software during the
instructions. Among the students, 67% had above average or high recommendation for
using math software during instruction.
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Number of Responses
Q6: Would you recommend your future teacher to use
software in teaching mathematics?
9
8
7
6
5
4
3
2
1
0
8
5
2
0
0
Not at all
Sort of
Average
Above average
Highly
Figure 8: Responses to question # 6
The analysis of the ‘student survey’ conducted with the experimental group
indicated that using software during the instruction improved students’ motivation,
confidence and interest. The high level of motivation among many of the students in the
experimental group was noticed by the researcher during the instruction. Students were
eager to share their understanding and experience with the class when they were
working with the software. During a guided discovery based activity in one of the lesson
involving ‘Translation and Transformation’, students spontaneously generated various
types of polynomial graphs using the software and share their results with their peer. In
her observation the researcher found that the students in the experimental group were
very confident about the contents and they were open about asking clarifying questions
to instructor. From the survey it is also evident that majority of the students were very
interested to use the software. Some of the students were highly fascinated and wanted
to get a copy of the software to use it at home.
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CONCLUSION
The purpose of this research was to find whether or not using graphing software
during the instruction of polynomials can improve students’ understanding of function
and their ability to create and interpret graphical representation of functions. In this
study, using the graphing software, the experimental group investigated concepts
related to Polynomial Functions (i.e. Graph, Translation, Transformation, Factor
Theorem, and Analyzing Solutions). From the analysis of the findings it is evident that
the experimental group was confident in their understanding of function. They also
scored higher in questions that tested their ability to analyze and interpret graphical
representation of functions. The software helped the students to readily observe the
relationship among the functions, and they were able to make conjecture and test those
conjecture through graphical investigation. The software allowed students to spend
more time making conjectures and creating their own reasoning to either prove or refute
those conjectures. Rather than using time and mental effort to calculate values and
plotting graphs, through the use of this software students were able to focus more on
the overall behavior/pattern of different polynomial functions. All the activities that were
designed and implemented using the software facilitated students to gain higher
analytical thinking skills, and consequently they performed better on the questions that
required higher level thinking and analyzing.
In the conclusion of his research Simmt (1997) indicated that most of the
teachers used calculators to facilitate one or two guided discovery activities. They did
not use the calculators to facilitates and/or encourage the students to conjecture and
prove or refute ideas. In this research the students were not only conducted discovery
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based activities, but also regularly encouraged to use the graphing software to
conjecture and discuss their results in class. From the findings of this research, it is
evident that these activities helped students to apply higher level thinking skills on those
questions in the post-test that required analysis and evaluation of graphs.
In his study Ruthven (1990) found that students developed three distinct
approaches to symbolizing a graph: an analytic-construction approach that exploits
mathematical knowledge, a graphic-trial approach that compares successive expression
graphs with the given graph, and a numeric-trial approach guided by the coordinates of
the given graph. My findings also confirm the results found by Ruthven (1990). In the
fourth and fifth questions of the post-test, students applied ‘graphic-trail’ approach
where they had to identify the zeros and the function of a given graph. The third
question required students to construct the graph and find its equation using the given
roots. Students applied ‘numeric-trail’ approach in their class activity which helped them
to answer this question. Moreover, an ‘analytical- construction approach that exploits
mathematical knowledge’ is a skill that students achieved through out the lessons which
incorporate the graphing software.
Interestingly, through this study the researcher observed that when graphing
tools were used during the instruction, a spontaneous shift in the teachers’ role enriched
the learning process. When paper-pencil was used to learn about graphing, the
teachers’ role was a task setter and an explainer. On the other hand, when students
used the software in their learning, teachers’ role shifted to a consultant, a facilitator and
a fellow investigator. Farrell (1996) also described that these shifts occurred within each
class as the teachers moved from not using the graphing calculator and computer
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technology to the use of it; the shifts were also accompanied by a decrease in lecture
and an increase in group work.
It is also interesting that the statistical analysis of the students’ over all scores of
the post-test did not show a significant advantage in understanding the concepts of
polynomial functions when the students used graphing software during the instruction.
Two possible reasons for not attaining significant statistical differences are (1) the
sample size (total thirty participants) was too small to analyze the data and to get a
statistically significant result, (2) length of the treatment (total three weeks) was too
short to get useful information regarding students understanding using the software
during instructions. Therefore, it is recommended that future research needs to be
conducted with a significantly larger numbers of participants and over a longer period of
time.
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STUDENT SURVEY 1
Please choose ONE response from each of the following question (from question # 1 to
question # 5):
1.
What is your current grade level?
[ ] 8th
2.
[ ] Male
Which race do you associate yourself the most?
[ ] African American
4.
[ ] Asian
[ ] Caucasian
[ ] Hispanic
Do you have access to a personal computer at your home?
[ ] Yes
5.
[ ] 10th
What is your gender?
[ ] Female
3.
[ ] 9th
[ ] No
Do you have internet connection (dial-up, DSL, cable etc.) at your home?
[ ] Yes
[ ] No
Please choose all that applies for the following question:
What activities do you perform on a regular basis (at least once a week) using your
computer/internet?
[ ] Browse the internet for news
[ ] Browse the internet for homework help
[ ] Send and receive email
[ ] Send and receive text Messages
[ ] Do homework using computer
[ ] Word Processing tasks (Writing papers, essays etc.) using computer
- Thank you for your responses -
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POST-TEST
1. Refer to the definition (in your math journal) of the nth degree polynomial and the
polynomial 5x7 + 4x6 – 8x3 + 1.3x2 – x
[5 X 4 = 20 points]
State the value of each of the following:
a. n
b. an
c. an-1
d. a0
e. a3
2. In a – d, tell whether or not the expression is a polynomial. If it is, state its degree and
leading coefficient:
[4 X 5 = 20 points]
b. 7x2 + 13x – 8
a. 3x + 7
c. 5x6 + 4x3 – 15
3. 6, 4, – 1 and – 3 are the four roots of a fourth degree polynomial.
d. 2x3 + 3x
[2 X 10 = 20 points]
a. Write an equation of the polynomial
b. Sketch the graph of the polynomial in an X-Y Plane
4. Analyze the following graph.
a.
b.
[2 X 10 = 20 points]
Find the degree of this polynomial function and indicate the zeros
Write the equation of the graph
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5. Graph of cubic function may have any one of the following four types of shapes below.
[2 X 10 = 20 points]
Type 1
Type 2
Type 3
Using the Factor Theorem find the equation for a cubic function
a. with three x-intercepts whose graph looks like ‘type 2’
b. with one x-intercepts whose graph looks like ‘type 3’
Type 4
48
Graphing Software
STUDENT SURVEY 2 – EXPERIMENTAL GROUP
1. How motivated were you to learn the concepts using the software?
a.
b.
c.
d.
e.
Highly motivated
Above average motivated
Average motivated
Sort of motivated
Not motivated at all
2. How helpful was the software in understanding the concepts?
a.
b.
c.
d.
e.
Highly helpful
Above average helpful
Average helpful
Sort of helpful
Not helpful at all
3. What is the level of difficulty in using the software?
a.
b.
c.
d.
e.
Highly difficult
Above average difficult
Average difficult
Sort of difficult
Not difficult at all
4. How confident are you about the concepts you have learned using the software?
a.
b.
c.
d.
e.
Highly confident
Above average confident
Average confident
Sort of confident
Not confident at all
5. How interested were you in learning the concepts by using the software?
a.
b.
c.
d.
e.
Highly interested
Above average interested
Average interested
Sort of interested
Not interested at all
49