Horton, R. M., Storm, J., & Leonard, W.H. (2004). The graphing calculator as an aid to teaching algebra. Contemporary Issues in Technology and Teacher Education [Online serial], 4(2). Available: http://www.citejournal.org/vol4/iss2/mathematics/article1.cfm
The Graphing Calculator as an Aid to Teaching Algebra
Robert M. Horton
Graphing calculators have been used in the mathematics classroom for
speed, to leap hurdles, to make connections among representations, and
to permit realism through the use of authentic data. In this study,
a graphing calculator tutorial provided on the Casio FX1.0 and FX2.0
PLUS models was found to serve a fifth purpose, improving manipulative
skills. Specifically, after using the tutorial, students in a beginning
college algebra course scored significantly higher on a test on solving
linear equations. Results concerning a change in attitudes were tentative,
although they suggest that the tutorial also may contribute to improved
Graphing calculator technology is recommended by national standards in mathematics
(National Council of Teachers of Mathematics, 2000). Even more significantly,
research has shown that such technology has a positive effect on student performance
(Ruthven, 1990; Smith & Shotsberger, 1997; Tolias, 1993). Reasons teachers
employ the technology, however, are varied. Many teachers may not have analyzed
why they use graphing calculators or how calculators can help students learn.
In NCTM’s Technology Standard (NCTM, 2000), several purposes for graphing calculators
and other technology are discussed, including the following:
- Speed: After students have mastered a skill, teachers allow the
use of graphing calculators to compute, graph, or create a table of values
- Leaping Hurdles: Without technology, it was nearly impossible for
students who had few skills and little understanding of fractions and integers
to study algebra in a meaningful way. Consequently, lower level high school
courses often became arithmetic remediation courses. With technology, all
students now have the opportunity to study rich mathematics. They can use
their calculators to perform the skills that they are unable to do themselves.
- Connections: A sophisticated use of graphing calculators is to
help students make connections among different representations of mathematical
models. Users can quickly maneuver among tabular, graphical, and algebraic
- Realism: No longer are teachers restricted to using contrived data
that lead to integral or other simplistic solutions. Graphing calculators
permit the creation of several types of best-fitting regression models. This
capability allows data analysis to become integrated within the traditional
curriculum; the tedium and difficulty of calculating a best-fit model are
no longer factors in introducing data analysis into the curriculum. (pp. 24-32)
Over the years, graphing calculators have become more sophisticated. One
relatively recent development is the inclusion of a tutorial on the calculator
to help develop skills. The Casio FX2.0 series of graphing calculators features
a student tutorial for four different types of algebraic problems: linear
equations, linear inequalities, simultaneous equations, and quadratic equations.
This tutorial can also be installed on the FX1.0 series. Consequently, a fifth
purpose for graphing calculators, facilitating the development of important
skills, becomes possible. For this study, we sought to determine if the tutorial
helps students learn to solve linear equations.
The linear equation tutorial on the Casio FX2.0 leads students step-by-step
through symbolic reasoning to solve a linear algebraic equation. In this study,
students used the tutorial to help them solve linear equations during a 3-week
unit in a college algebra class. The hypothesis was that this tutorial would
increase confidence in doing algebra and enable better understanding of by-hand
Although the degree to which students should be required to master the skills
of symbolic manipulation is often a topic of debate among educators, we were
convinced both by the literature (e.g., Nathan & Kroedinger, 2000a, 2000b;
NCTM, 1989, 1991, 2000; Usiskin, 1995; Waits & Demana, 1992) and the
researchers’ experiences that solving linear equations by hand is essential for
success in algebra, provides stimuli for higher order mathematics, and helps
students understand fundamental algebraic principles that serve as prerequisite
skills and concepts for future courses.
Palmiter (1991) studied the use of Computer Algebra Systems (CAS) in a
calculus class. Both the experimental and control groups used the same text.
However, the experimental group, which used a CAS system, covered the material
in 5 weeks; in contrast, the control group took 10 weeks to cover the same
material. Furthermore, the experimental group significantly outscored the
control group on both computational and conceptual exams; however, despite
efforts to ensure the same teaching style, the difference in conceptual scores
could be explained by teacher variation. Palmiter also claimed that the
experimental group “faired as well” as the traditional group in future classes.
Further, the CAS group overall had slightly more confidence in their success in
future mathematics courses, and a larger percentage of students in the
experimental group indicated that they had learned more in this class than in
any other mathematics class. Ninety-five percent of the experimental group
claimed they would sign up for another class using a CAS system.
O’Callaghan (1998) studied the effects of a computer intensive algebra (CIA)
system on university students in a college algebra course. CIA focused more
on concepts, employing symbolic manipulators to perform most of the skills.
Three of four hypotheses for greater conceptual understanding were supported,
with significant gains found in the ability to model functions, interpret functions,
and translate functions. No difference was found in manipulative procedures.
Hembree and Dessart (1986) found that, when calculators are integrated with
regular instruction, students at all achievement levels show an improved attitude
toward mathematics, improved test scores in basic operations, and improved scores
in problem solving.
These findings address Bartow’s (1983) fear that students would depend too
heavily on the calculator and that their individual skills would “atrophy.”
Instead, the use of CAS allows students to generate symbolic, graphical, and
numerical representations, to reason with these representations, and to improve
students’ work with symbols (Heid, 1997; Heid & Edwards, 2001). These
results certainly support the use of technology in the classroom. However, no
data have been found regarding a tutorial such as that featured in the Casio
|Figure 1. Screenshot from the Casio FX2.0 tutorial solving
the equation –3x + 5 = –8x – 30 in Automatic mode.
In addition to a built-in CAS, the Casio FX2.0 has an additional tutorial menu
that demonstrates a step-by-step procedure for solving equations. Not only will
the Casio FX2.0 solve an equation through the tutorial, but it permits the user
to solve the problem without aid in the same step-by-step fashion. It can, in
a sense, teach as well as allow practice of symbolic manipulation. Figure 1
shows the screens the tutor shows in solving the equation –3x + 5 =
–8x – 30 in Automatic mode. A description of each step appears at the top of
the screen, and the results of the particular step are shown below the line.
By working through examples such as this, users can then try solving equations
on their own, selecting the manual option provided on the calculator.
The central question of this study sought to determine if the Casio FX2.0
tutorial would help students improve their skills in solving linear equations by
The study was conducted at a small 2-year liberal arts college with a focus
on preparing students for a 4-year college or university. Of the students
attending the college, 47% were enrolled in developmental studies and 35% in
developmental algebra courses. Ninety-four percent of the students enrolled were
traditional students, attending college immediately after high school
graduation. Caucasian, African American, and foreign students comprised 44%,
49%, and 7% of the student population, respectively.
The sample was taken from the enrollment in two beginning algebra classes. To
control for instructor variability, two sections taught by the same person were
chosen. One of these was randomly selected as the experimental group and the
other as the control group. Each class had an initial enrollment of 25 students.
Information provided by the registrar’s office and a test of SAT scores
confirmed that the two classes were roughly equivalent. The instructor also
detected no difference between the abilities of the two classes, both of which
she perceived as weak. However, she did think that the early start time (8:00
a.m.) for the experimental group caused them to take longer to focus on tasks.
Use of the Casio FX2.0 tutorial was implemented in the experimental class,
and students were shown how to use the tutorial early in the semester. The
control group used the TI-83 or TI-83 Plus, the calculator required of all
students. Although students were allowed to use the Solve feature on the TI-83
or 83 Plus to check their work, use of this feature was not taught. The section
on solving linear equations through symbolic manipulation lasted approximately 3
weeks. Students in the experimental group used the tutorial on a daily basis in
class as the teacher lectured on step-by-step manipulation. They were also
encouraged to use the tutorial while doing homework.
Skill in solving linear equations was measured by a posttest given at the conclusion
of the section on solving linear equations. The posttest consisted of 14 linear
equations and two word problems that were solved with pencil and paper only;
calculators were not permitted (see Appendix A).
Scoring of the problems was based on a rubric provided by the Educational
Testing Service Network. Possible scores ranged from 0 to 4, with categories of
algebraic knowledge, communication of this knowledge, and demonstration of the
skills comprising the score. Scores of 3 or 4 represented successful responses,
a 2 represented some conceptual understanding with inconsistent proficiency and
incomplete explanations, and scores of 1 or 0 were deemed unsatisfactory.
Scoring was done “blindly.”
Two group interviews, one with students in the experimental group and one with
students in the control group, were conducted near the conclusion of the unit.
Six students in each class, 3 students from the top and bottom quartiles as
determined by SAT scores, were randomly selected for the interviews. The questions
for both groups included attitudes toward calculators and solving linear equations
and general feelings toward mathematics. The experimental group had additional
questions concerning the use of the Casio FX 2.0 tutorial. See Appendix B for the interview protocol.
A Likert questionnaire adapted from the Fennema-Sherman scale to inquire about
students’ attitudes toward algebra and solving linear equations was administered
at the beginning and end of the 3-week study. This scale, originally created
in the 1970’s, has been tested and modified several times, and is available
in the public domain (see, for example, http://www.woodrow.org/teachers/math/gender/08scale.html).
Excerpts from the questionnaire used for this study are provided in Appendix C.
Finally, an anonymous homework questionnaire was administered to the
experimental group on five different days during the linear equation unit to
determine the extent of use and the degree to which students found the tutorial
The results of a one-tailed t-test showed that the experimental class
significantly outperformed the control class on the posttest (t = 2.09,
p = 0.021, df = 47). Overall, mean scores for the two groups
were 2.74 and 2.35 out of a maximum of 4, respectively. Posthoc statistics were
performed to determine on which questions there was a significant difference
between the two groups. The experimental group outperformed the control group
on six questions; the control group did not outperform the experimental group
on any questions. Interestingly, of the six questions on which the experimental
group bested their counterparts, four contained fractions (numbers 2, 4, 8,
and 11), with the experimental group average score above 3 on the first three
of these. (On Problem 11, the experimental group averaged 1.96, compared with
the control group’s 0.52.) On the test, only five problems contained fractions;
in fact, the experimental group showed a significant gain on every problem that
contained a fraction on both sides of the equation. Further, the experimental
group outperformed the control group on multistep problems.
Whether or not these differences occurred for all levels of students in the
classes was also investigated. Students were sorted into low, medium, and high
groups based on their SAT scores; only students in the medium and high groups
had gains on individual questions. The high attrition and failure rate in the
lowest performing group might suggest that these students were either unprepared
or unwilling to put forth the necessary effort for success.
In addition to the improved performance, there was also a significant increase
in attitude from the prequestionnaire to the postquestionnaire for the experimental
group (t = 2.11, p = 0.020, df = 47), with the mean
attitude increasing from 3.49 to 3.74 on the 5-point scale. There was no significant
increase found for the control group (t = 1.35, p = 0.091,
df = 47), though their average increased from 3.40 to 3.62. However,
upon checking for differences between groups, there was no significant difference
between the two groups on either the prequestionnaire or the postquestionnaire
(t = 0.81, p = 0.21, df = 44 for the post-questionnaire).
Consequently, the results concerning the increase in attitudes are tenuous.
Post hoc analysis showed that gains in attitude occurred for students in the
medium group, but not for students in the low or high groups. This result may be
due to the low group’s overall poor performance and commitment to their work and
to the high group’s overall positive attitudes with which they began the course,
leaving little room for gain.
Although the time students in the experimental class spent on homework varied
daily, there were some detectable patterns. For the first assignment of the
unit, most students spent the majority of their time with the tutorial; all but
one student reported it was useful. As the unit progressed, those who perceived
the assignments as being difficult spent more time with the tutorial than did
those who found the assignments relatively easy. Some students found that the
tutorial was difficult to learn, but once learned, it was helpful.
The tutorial clearly helped students master the process of solving linear
equations. This confirms an important and fifth potential use of graphing
calculators in the mathematics classroom, that of developing algebraic skills.
Further, the gains appeared on problems that involved fractions and on problems
that took multiple steps, areas that often impede students’ progress through
upper-level mathematics courses.
The interviews and observations suggest that this additional benefit may have
occurred because, to use the tutorial, students had to understand and
distinguish coefficients. This understanding apparently transferred to students’
ability to solve linear equations, especially those containing fractions.
In regard to calculator usage, higher-level students took advantage of the
calculator, while students in the lower group may have used the calculator as a
crutch. One young lady from the lower quartile commented, “It makes it so we
don’t have to think.” In contrast, one young man in the upper quartile
mentioned, “The tutorial is wonderful!!! It gives us an opportunity to check our
answers so we have extra confidence. It is also very useful because it shows us
the steps if we forget.”
Although the results concerning improved attitudes are more tentative,
students in the experimental group showed a statistically significant
improvement in attitudes between the pre-intervention survey and the
post-intervention survey, but the control group did not. Further analysis
revealed that this difference occurred because of changes in attitudes among
students in the middle achievement group; those in the high and low groups did
not change attitudes significantly. The ceiling effect may have impeded gains
for the high group, and the low group may have been unprepared either
academically or motivationally to take advantage of the tutorial. Further
research is needed to determine if the immediate availability of “explanations,”
the privacy which the tutorial affords, and the endless supply of examples that
the tutorial allows contribute to increased confidence, in addition to improved
skills, and for which students any benefits are found.
Interviews conducted with students in the experimental class supported the
conclusion that the tutorial was beneficial. One student in the medium group
summarized his experience as follows: “Before this, I felt scared. I still don’t
like algebra, but I feel more comfortable trying to solve these things
[equations] because I know I have the answer.”
The gain in understanding of coefficients through work with the tutorial may
play a role in helping students in additional, unanticipated areas. During her
classes the next semester, the instructor noted that she could identify the
students who had been in the experimental class just by their quickness in
understanding synthetic division and the factoring process. Interestingly, one
student from the experimental group who was doing poorly suddenly blossomed when
studying the factoring process. Changes in the performance of other students who
had been in the experimental group were also noted. Unfortunately, due to the
high attrition rate, the number of returning students was insufficient to
conduct any quantitative tests of the results. Nevertheless, this increased
ability to use their understanding of coefficients in other areas seems
promising and is something that should be explored more fully.
Overall, the evidence suggests that a calculator-based tutorial may have
significant benefits for the mathematics classroom. It proved to be an
influential learning tool in solving linear equations and perhaps beneficial in
improving attitudes. Further, the benefits of strengthening this skill may
extend into other unexpected areas.
As supported by the findings here, the tutorial expands the potential
benefits of the graphing calculator. In addition to the four purposes stated at
the beginning of this article, there is now evidence that the graphing
calculator can help accomplish a fifth purpose, that of skill development.
Further, though future studies are needed to investigate this more
systematically, a sixth purpose seems possible, even likely: that of making
connections among different algebraic skills. Although graphing calculators have
been used to help students make connections among tabular, graphical, and
algebraic representations of functions (e.g., Heid, 1997), the connection among
skills is a new, uncharted area. In fact, the tutorial may help in areas that
teachers and researchers have not considered or with ideas that teachers may
have assumed students already know. While teaching the solution of linear
functions, perhaps many teachers emphasize the importance of coefficients and
the distinctions between variables and coefficients. In this study, however, it
was not until students used the tutorial to solve linear equations that this
conceptual connection was recognized and then made.
With forethought, teachers can employ technology to help students learn. Even
more exciting, this learning help may extend into areas that are not
anticipated. Of course, an inappropriate use of calculators may cause students
not to learn what they should, so teachers must always observe and assess their
students’ achievement carefully and consistently. Nevertheless, the results of
this study demonstrate that there is cause for confidence and even excitement
about the benefits that technology can bring to the mathematics classroom.
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Robert M. Horton
William H. Leonard
Post-Test Used to Measure Skills
Solve for x. Work must be shown to receive
credit. Calculators are not permitted.
(Note: Room was provided on the test for students to solve the
Specific interview questions were based on the
responses of the students but the following general format was used. Students
were expected to explain their responses.
Questions were asked to both groups surrounding their use of technology in
the past and interest in Algebra.
- Have you ever used any technology, like graphing calculators, in a math
class? If so, when and what technology did you use?
- Would you like (or did you like based on answer above) using this
technology in your class?
- Do you like algebra and solving linear equations?
- Do you feel confident solving linear equations?
For students involved in the experimental group, follow up questions were
asked if the tutorial seemed helpful.
- Did you use the tutorial to help you solve linear equations?
- Did you like using the tutorial?
- Was the tutorial helpful?
- What is your feeling on using this tutorial for other math classes?
- Think back to before you started the class on solving equations as
compared to afterward. Have you changed your opinion? Was the tutorial one of
the reasons your opinion changed, if there is a change?
Questions from Attitudinal Survey
Likert Scale Questionnaire
Indicate the extent to which you agree or disagree with each statement. There
are no correct answers for these statements.
||Generally, I have felt secure about
attempting to solve linear equations.|
||I think I could handle solving more
difficult linear equations.|
|Neither Disagree or Agree
||I’m no good at solving linear equations.
|Neither Disagree or Agree
||For some reason, even though I
study, solving linear equations seems hard for me.|
|Neither Disagree or Agree
||I’d be proud to be the outstanding
student when it comes to solving learn equations.|
|Neither Disagree or Agree
||If I had good grades when solving
linear equations, I would try to hide it.|
|Neither Disagree or Agree
||Gender: male or female