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.
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.
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?
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.