USE OF THE GRAPHING CALCULATOR

IN MATHEMATICS INSTRUCTION

One preservice secondary mathematics teacher was interviewed about his perceptions about the use of the graphing calculator in mathematics instruction. One interview was conducted in February before the student had any field experience. The second interview was conducted in July after his student teaching experience. Long before the preservice teacher had the benefit of student teaching, he exhibited knowledge of the advantages and disadvantages of graphing calculator use and its possible role in the mathematics classroom.

In the last ten years, it has become common practice to incorporate the graphing calculator into the curriculum of Algebra II (in high school), pre-calculus, and calculus courses both in high school and in college. In the spring of 1997 in my own Algebra I classes, I engaged in some action research; the study was based on my belief that Algebra I students would benefit from the use of a graphing calculator. These classes were predominately composed of ninth graders.

The results from my study indicated that many students worked
harder at symbolic manipulative skills in order to get the calculator
to work for them. When working with the graphing calculator, it
is necessary to solve linear equations in x and y for the y variable
(i.e., in the form y = ). I saw students, who were previously
easily frustrated by symbolic manipulation, work to obtain the
correct expression needed for the calculator syntax. Also, assessment
scores for the topics in which students were allowed to use graphing
calculators appeared to be higher than in previous years. In light
of this limited study, I wanted to investigate the perceptions
of preservice teachers about using the graphing calculator in
mathematics instruction. I chose to investigate preservice teachers'
perceptions about the use of graphing calculators because of previous
research done with inservice mathematics teachers. For example,
Simmt (1997) found that inservice teachers used graphing calculators
to enhance the teachers' usual method of teaching. ìNo
new methods or approachesÖwere used by the teachersî
(p. 287). Simmt called for research with preservice and inservice
teachers to identify and address issues related to personal philosophies
about mathematics and mathematics education because her study
suggested that ìone's philosophy of mathematics is manifested
in one's instruction of mathematicsî (p. 287). Researchers
need to determine the malleability of preservice teachers' philosophies
of mathematics and mathematics education and how these philosophies
can best be modified (Thompson, 1992; Wilson & Krapfl, 1994).

This research study was formulated in order to better understand
the influences which affect the decisions teachers make to integrate
graphing calculator technology in mathematics instruction. Those
influences which appear to influence teachers' decisions, either
inservice or preservice, are recognition of advantages in use
(Bialo & Erickson, 1985; Ruthven, 1990), the individual's
position on concept mastery (Fleener, 1995), and whether the individual
views mathematics instruction as rule-based or non-rule-based
(Tharp, FitzSimmons & Brown Ayers, 1997). Therefore, the purpose
of this study was to identify and describe one preservice teacher's
perceptions about the use of graphing calculators in high school
mathematics instruction. To ascertain these perceptions, two research
questions were studied:

What do preservice teachers perceive as the role of the graphing calculator in high school mathematics instruction?What do preservice teachers perceive as the advantages and disadvantages of using a graphing calculator in high school mathematics instruction?

Regardless of the findings of recent research concerning the positive effect of using graphing calculators (DeMana, Schoen & Waits, 1993; Hembree & Dessart, 1993), technology has not had the impact on mathematics instruction as was once forecast (Fleener, 1995; Simmt, 1997; Strudler, 1993). Much has been said about what technology can do for learning, but very little is known about the consequences for teaching (Simmt, 1997). In her study, Simmt (1997) found that ìproviding a new tool is not sufficient to change instruction since one's philosophy of mathematics is manifested in one's instruction of mathematicsî (p. 287). To examine a teacher's philosophy, then, one must examine his or her belief systems and how those systems affect mathematics instruction.

Cooney, Shealy & Arvold (1998) posit three schemes for describing
the beliefs of preservice teachers: naÔve idealist, isolationist,
and connectionist. NaÔve idealists listen and give authority
to the voices of teacher educators and others with experience
whom they respect, many times ignoring their own voices, when
constructing a belief structure. An isolationist is one who cannot
reconcile contradictory belief structures based on evidence with
those beliefs held as nonevidential beliefs. A connectionist is
one who compares newly realized beliefs developed from contexts
with nonevidential beliefs and integrates both belief systems
into a philosophy. The authors note that students who have connectionist
views ì[set] the stage for becoming a reflective practitionerî
(p. 330). Although this study was not designed to look at the
beliefs of this preservice teacher, the work of Cooney, et. al.
(1998) provides a lens through which to interpret the changes
in the preservice teacher's thinking. His thinking mirrored that
of a naÔve idealist before the student teaching experience
and mirrored that of a connectionist after the field experience.
Initially, he gave authority to the mathematics educators (external
voices) in the methods course and then shifted to a combination
of internal and external authorities, as a result of the student
teaching experience.

Because I wanted to gain an in-depth understanding of a preservice teacher's perceptions about the advantages and disadvantages of using a graphing calculator and its role in mathematics instruction, I chose to do a case study of a preservice secondary mathematics teacher. Merriam (1998) stated that the case study is used to give ìintensive descriptions and analyses of a single unit or bounded system such as an individual, program, event, group, intervention or communityî (p. 19). The outstanding characteristic of the case study design is limiting the object of study to a bounded system, a unit of study where there is a finite number of people to interview or a finite amount of time for observations. The unit is considered to be one case among other cases.

**Participant**

The selection of the preservice teacher in mathematics education
was based mainly on his willingness to collaborate with me and
his having some knowledge of and experience with the use of graphing
calculators. Much of his knowledge and experience with the graphing
calculator came as a result of mathematics and mathematics education
courses taken at the university. I made no formal assessment for
knowledge of and experience with the use of the graphing calculator;
I simply relied on the word of the preservice teacher who volunteered.
To somewhat gauge the competency of the student, however, I designed
the first interview protocol (see appendix A) to assess, although
very informally and not very extensively, the calculator skills
of the student. To facilitate this assessment, I provided a graphing
calculator during the first interview. I chose the pseudonym of
ìJeffî for the preservice teacher.

Jeff was chosen, as a volunteer, from a class of preservice teachers
taking a secondary mathematics education methods course at the
University of Georgia. The course met two days a week for two
hours per session for ten weeks. In the methods course, the graphing
calculator was used as a problem-solving tool in many different
situations and with several mathematics concepts. The student
teaching course, during spring quarter, directly followed the
methods course.

The sample is a limitation of the research design. Future studies
may build on this one by including more preservice teachers who
were chosen purposively.

**Methods of Data Collection and Analysis**

Data Collection. The data for this study were collected over a
period of six months. I interviewed Jeff two times. One interview
took place in February before Jeff had any experience teaching
in a high school mathematics classroom situation but at the time
of the methods course when the graphing calculator was used as
a problem-solving tool. The second interview was conducted in
July after the student teaching experience during spring quarter,
in which Jeff taught algebra, general mathematics, and pre-calculus.
Both interviews were about 45 minutes in length and were recorded
and transcribed. I made summaries of the data from the interviews.
The data from the first interview were used to guide the construction
of questions for the second interview (see appendix B), thus creating
another level of refinement in the data.

The purpose of the first interview was twofold. I wanted to gain
information about Jeff's perceptions regarding the advantages
of using graphing calculators in mathematics instruction. Also,
I wanted to investigate his perceptions regarding the role of
the calculator in mathematics instruction. The purpose of the
second interview was again twofold. I used this interview to further
define the role of the graphing calculator in mathematics instruction
and to gain information about Jeff's perceptions regarding the
disadvantages of using the graphing calculator in the mathematics
classroom. The contents of both interviews contained problems
which I specifically chose to gain information about Jeff's knowledge
concerning uses of the graphing calculator. These problems ranged
from the traditional problem (e.g., graphing situations) to non-traditional
situations (e.g., solving equations and inequalities or factoring).
In these non-traditional problems, I asked Jeff to stretch his
thinking and identify a possible usage of the graphing calculator.

**Analysis.** The data were coded for categories and themes
which would serve to respond to the research questions studied.
I gave Jeff a copy of the summaries for both interviews and asked
him to verify my interpretation of the data from those interviews
in order to validate the information. He agreed with my interpretation
of what he had said. I used the constant comparative analysis
method to analyze the data (Creswell,1998; Merriam, 1998).

In the constant comparative method, there are three fundamental
types of coding: open, axial, and selective. The primary goal
of open coding is to compare similarities and differences in the
data. For axial coding the goal is to examine the emerging categories
related to their properties and then test this relationship against
the data. While working on this aspect of coding, I consulted
with a colleague several times about the meaning of the data.
Finally, selective coding places all the categories under the
umbrella of a core category so that there is a narrowing of all
data to this single, core category. Due to the nature of the data
collected, selective coding was not used in this study.

Two research questions guided this study. The first question was ìWhat do preservice teachers perceive as the role of the graphing calculator in high school mathematics instruction?î After analysis of the data, three themes or categories emerged concerning this preservice teacher's perceptions about the role of graphing calculator technology in high school mathematics instruction. Jeff's data suggested that graphing calculators were useful for generating alternative solution methods, showing complex graphs, and empowering students. The second question that guided this study was ìWhat do preservice teachers perceive as the advantages and disadvantages of using a graphing calculator in high school mathematics instruction?î Jeff was able to identify advantages and disadvantages for each of the three roles of the graphing calculator which emerged from the data. I organized the following section in three parts, by the roles, and discussed the data analysis in terms of advantages and disadvantages for each of the three categories, giving quotes from Jeff's interviews as evidence for my conclusions.

**Generating Alternative Solution Methods
Advantages.** Jeff described three functions of the graphing
calculator that define its advantages when its role is providing
alternative solution methods. The three methods are numeric solutions,
graphic solutions, and table-format solutions. With these different
solution methods students are able to verify their conclusions
and eliminate some of the guess work inherent in solving problems.
Jeff said it this way:

I guess it would do numbers [when factoring trinomials] if you're doing the first and last number and you find all the factors of it that add up to the middle one. In something like 48 where there are a bunch of different ones, you can find all the factors of 48 and see what they add up to. That takes care of the guess and check. (Interview #2)

Jeff thought that having so many solution methods available
provides students different approaches to solving problems which
help them better understand the solution they find.

Disadvantages. Jeff proposed disadvantages of using the graphing
calculator when its role is providing alternative solution methods.
He related these disadvantages as student dependence on the calculator,
mismatch between homework and instruction, and a detrimental effect
on basic or approximation skills. He phrased it in these terms:

They become dependent on it. Say with signed numbers. They just type it in and get the answer. They can't do the signed numbers in their head, then, because they have the calculator in front of them at the time. They haven't been doing the multiplication; they don't necessarily know what they did on the homework or why. They're just dependent on it. (Interview #2)

He was concerned that if students use the calculator too much they become dependent on the calculator, lack understanding of processes, and fail to develop a strong skill base. In both interviews developing strong basic or approximation skills emerged as very important to Jeff.

**Showing Complex Graphs
Advantages.** When the role of the calculator is showing complex
graphs, Jeff identified three advantages: allowing deeper analysis,
providing accuracy, and producing a large number of examples.
He thought that many examples helped cement concepts for the students.
He wanted students to have good, basic skills, but also strong,
critical thinking that develops from working with challenging
problems:

...after a certain point, it's just wasting their time and my time to continue doing hand graphs. At some point when they're going through some huge equation where finding the intersection is important, the calculator allows them to get the intersection and move on to other parts of the problem. (Interview #1)

Jeff indicated that the calculator was of great benefit when students were solving a multi-step problem involving a graph. The calculator alleviated the tedious parts of the solution process and allowed students more time to analyze and work out the solution.

**Disadvantages.** Jeff pointed out two disadvantages in calculator
use when the role is showing complex graphs. The disadvantages
he identified are glossing over ìwhyî things work
and prohibiting student thinking. His example was as follows:

They might push the inverse button; they need to know why that button causes that to happen. They should understand that is the 1/x button and what it will do. (Interview #1)

He was concerned that students would not make the effort to think
about the function being performed by the calculator since the
calculator ìdoes it for them.î He indicated strongly,
ìI would want them to know, without the calculator, what
should happen and why.

**Empowering Students
Advantages. **When the role of the calculator is empowering
students, Jeff proposed two advantages: creating a visual representation
and serving as a problem-solving tool. He stated several times
that being able to see an accurate graph quickly helps students
understand what the graphical representation shows them:

You can see what effect small changes have on an equation, like adding 1 to the whole thing or adding 1 to the x in the exponent of that equation. All of those changes can be seen quickly using the calculator. (Interview #1)

Jeff was aware of the problem-solving aspect of the graphing calculator. He related an example about an airline profit-loss problem using inequalities.

If the 4x + 3 represented the cost of a ticket and the 2x - 7 the break-even point for a company, then they could see where the numbers fit. (Interview #1)

He showed me how he would use the calculator to graph the lines
and help the students analyze the representation for the break
point in the profit line.

**Disadvantages.** When the role of the calculator is empowering
students, Jeff strongly articulated his feelings about the disadvantages
associated with the calculator. These disadvantages are cheating
and answer-production. He felt the technology is advanced enough
that students will find ways to hide unauthorized information
to use on assessment exercises. His concern was evident in this
statement:

If they put programs or files in [memory] that they can hide somewhere, formulas that they have when I want them to know the formulas. If they knew the formulas for the test, this makes things easier. At home it's going to be next to impossible to keep them from cheating, doing things that they're not supposed to...You can have them show their work and that helps a lot. (Interview #2)

Jeff stated that there was no way to control calculator usage away from the classroom setting. However, he thought that structuring the assignments in particular ways would cause students to think more about what they were doing.

In this study the data obtained from the two interviews addressed
the research questions under examination. The preservice teacher
identified three categories which define specific roles of the
graphing calculator in high school mathematics instruction. For
each of these categories he proposed advantages and disadvantages
for that role of the calculator.

The problem statement for this study was to understand the influences
which affect the decisions teachers make about integrating graphing
calculator technology in mathematics instruction. Reflecting on
the information gained in this study, preservice teachers have
many considerations to weigh in their decision to integrate graphing
calculator technology in mathematics instruction. In this comment
toward the end of our second interview, Jeff summed it up very
well, " [the graphing calculator] is not as ideal as I once
thought it was. I wouldn't want to be without it, but I'm going
to have to be careful about how I use it." From this statement
I suggest Jeff's developing belief system mirrored that of a connectionist
(Cooney, Shealy & Arvold, 1998) where he has integrated previously
held beliefs with new contextual evidence. Before Jeff's student
teaching experience, he was eager to use the graphing calculator,
an indication of the authority he placed in his mathematics educators,
evidenced in these two comments:

In [methods] class, the lesson Carl and I taught on translations and reflections comes to mind. I don't think that we could have done it without the use of the calculator. We did rotations around the axis and I know we couldn't have done nearly so many examples if we had to take time to draw the graphsÖThe calculator gives the graph immediately and then we can analyze it and figure out what is going on; it is much more accurate. (Interview #1)We've done problems in [the methods] class where the [number of] people on the flight changes the cost of a ticket and that sort of thing. You can see that one [line] is going up; the airline is making more money as the line goes up. The graph shows them this and they could see the two graphs together. (Interview #1)

However, after the student teaching experience, Jeff was more
cautiously optimistic about graphing calculator use, an indication
of the new contextual evidence seen during student teaching and
related in the second interview.

I've seen a lot more having been out in the schoolsÖthe kids are dependent on it. Makes it less of an ideal tool than I once thought it wasÖIt's frustrating how viciously they fought to use the calculators on things we didÖKids have the calculators and can find ways to do things that you don't realize they're doing and there are already so many other things to attend to. (Interview #2)

Jeff's thinking seemed to have changed; he appeared to give more
credence to his own voice of authority, developing from his student
teaching experiences. If Jeff continues to use his beliefs and
experiences to inform his instruction, he will likely be a reflective
practitioner who realizes the importance of graphing calculators
as a tool but understands there are cautions to which he must
attend.

Because teacher perceptions about graphing calculator technology
determine how those teachers will use this technology in classrooms
(Cooney & Wilson, 1993), it is imperative that more research
be done with preservice teachers. This particular preservice teacher
is ready to use the graphing calculator in mathematics instruction,
but student teaching in a high school classroom has taught him
that care must be taken about how much the teacher uses the calculator.

Colleges of education will be interested in this line of study
to plan better teacher education programs which specifically address
the integration of graphing calculator technology in mathematics
instruction. Secondary methods courses may need to provide specific
examples of content where the calculator is beneficial and introduce
new content areas which are less traditional for calculator usage.
It may also be helpful in placement of preservice teachers for
their student teaching assignments. By searching out mentor teachers
who foster the use of graphing calculators in their mathematics
instruction, preservice teachers could experience first-hand the
victories and challenges of calculator use in the mathematics
classroom.

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FIRST INTERVIEW PROTOCOL

Tell me about your experiences using the graphing calculator.

Describe for me the impact the graphing calculator has had on
your learning.

Would you use the graphing calculator in teaching?

What has influenced this decision to use the graphing calculator?

To what extent would you use a graphing calculator in your teaching?

Are there times when you would NOT use the graphing calculator?

How would you use the graphing calculator in Pre-algebra or Algebra
I?

Use the calculator to solve 4x + 3 = 2x ñ 7. Tell me what
you are thinking and what calculator operations you are using.

Using the same line equations and substituting the GREATER THAN
symbol for the EQUALS symbol, how would you use the calculator
to solve this problem?

You are teaching Algebra I in a local high school and have required
your students to purchase a graphing calculator. At open house,
John Q. Parent asks the following questions:

1) How can you justify requiring a $90 graphing calculator in
Algebra I? 2) How can my child learn algebra if the calculator
is doing all the work for her?

SECOND INTERVIEW PROTOCOL

How would you use the graphing calculator to teach addition or
subtraction of signed numbers?

How would you use the graphing calculator in factoring?

If you had this trinomial, x2 ñ 3x ñ 10, how could
you guide students to use the graphing calculator to help them
factor the trinomial?

How could you guide students to use the graphing calculator to
help them factor the trinomial 3x2 + 19x + 28?

What reservations do you have regarding the use of the graphing
calculator in mathematics instruction?

If you were using graphing calculators in your Algebra I class,
are there any issues you might be worried about?

How do you feel about equity issues?

Since our last conversation, have you thought any more about using
the calculator in your classroom? In what ways will you use it?

Author Note

Teresa G. Banker, doctoral student, Department of Mathematics
Education.

The research reported in this article was conducted as the pilot
study for the author's doctoral dissertation to be completed at
the University of Georgia under the direction of Denise S. Mewborn.
I wish to thank Dr. Mewborn for her helpful comments on an earlier
draft of the article and her enthusiastic encouragement throughout
this process.

I wish to express sincere thanks to Linda Crawford for her help
with making sense of the data from the interviews with Jeff.