Kastberg, S., & Leatham, K. (2005). Research on graphing calculators at the secondary level: Implications
for mathematics teacher education. Contemporary Issues in Technology and Teacher Education [Online serial], 5(1). Available: http://www.citejournal.org/vol5/iss1/mathematics/article1.cfm
Research On Graphing Calculators at the Secondary Level: Implications
for Mathematics Teacher Education
Indiana University Purdue University Indianapolis
Brigham Young University
This article focuses on three key factors that a survey of literature
indicated impact the teaching and learning of mathematics with graphing
calculators: access to graphing calculators, the place of graphing calculators
in the mathematics curriculum, and the connection between graphing calculators
and pedagogical practice. Access to graphing calculators is associated
with student achievement gains and a wide array of problem-solving approaches.
The research suggests students’ achievement is positively affected
when they use curricula designed with graphing calculators as a primary
tool. Studies of teachers’ use and privileging of graphing calculators
illustrate the impact professionals have on students’ mathematical
knowledge and calculator expertise. Implications of these research findings
for preservice and in-service teacher education are summarized.
Graphing calculators were first introduced in 1985 and within a
few years mathematics educators began to study the role and impact of this tool
on the teaching and learning enterprise. The field has amassed a significant
body of research on students’ performance and learning with graphing calculators
and a small, now growing, body of research on teachers’ use and knowledge
of graphing calculators. An analysis of research studies published in peer-reviewed
journals over the past 2 decades suggested a framework for summarizing the findings
and implications of this research. Three themes emerged that cut across the
existing literature: access to graphing calculators, the place of graphing calculators
in mathematics curricula, and the connection between graphing calculators and
pedagogical practice. This article addresses what this literature suggests about
teaching and learning mathematics with graphing calculators, as well as the
implications of these research findings for preservice and in-service teacher
Traditionally, mathematics has been taught as a collection of rules and procedures
that make computations more efficient. Thus, it comes as little surprise that
in a context where the focus of mathematical activity is computation access
to tools that can perform many of these computations has historically been restricted.
The studies discussed in this section illustrate how teachers’ beliefs
and knowledge influence access to graphing calculators and how, in turn, this
access influences students’ mathematical performance.
In a 1994 survey of close to 100 middle and secondary mathematics teachers,
just over 70% of those surveyed indicated that they had calculators available
for classroom use but had not used them in their own classes (Fleener, 1995a,
1995b). This study, and studies by others (Doerr & Zangor, 1999, 2000; Slavit,
1996), have consistently shown that merely providing teachers with access to
graphing calculators does not ensure that students then have access. Access
to graphing calculators, even for students who own them, is mediated by the
The question of whether students will be given access to graphing calculators
is often connected with teachers’ beliefs about the roles graphing calculators
should play in the learning process (Doerr & Zangor, 1999, 2000; Fleener,
1995a, 1995b; Leatham, 2002). Limited access is most often associated with a
belief that graphing calculators should be used only after students have mastered
a particular mathematical procedure by hand and then primarily as a means of
checking ones’ work. Frequent access is usually associated with a belief
that graphing calculators should be used to facilitate the understanding of
a mathematical concept. Leatham (2002) found that access to graphing calculators
was a critical concern for preservice secondary mathematics teachers and was
one of the primary dimensions of the participants’ core beliefs about
technology use in the classroom. These beliefs about access ranged from desiring
extremely limited access to desiring that students have access at all times.
These beliefs were similarly associated, respectively, with procedural and conceptual
In addition to demonstrating that teachers’ beliefs about graphing calculators
influence student access to graphing calculators, researchers have shown how
access to graphing calculators influences student performance. Harskamp, Suhre,
and van Streun (1998; 2000; van Streun, Harskamp, & Suhre, 2000) conducted
a study designed to compare the performances of students with differing levels
of access to graphing calculators (TI-81). They compared the performances of
calculus students in 12 classes who were randomly assigned to one of three groups:
those with no access to graphing calculators, those who had access during one
unit of instruction (approximately 6 weeks), and those with access for one year.
Although all students studied function and calculus concepts from a common textbook,
the graphing calculator groups received additional instructions on how to use
the calculators to perform several tasks. Graphing calculators were used to
check algebraic solutions, to find solutions graphically, and to graph functions.
More advanced operations were not explored. None of the 12 teachers assigned
to these classes had prior experience teaching with the graphing calculator.
Students were given pre- and posttests designed to assess their problem solving
strategies. All students were allowed to use scientific calculators on the pretest.
The control group was allowed to use scientific calculators on the posttest,
while the experimental groups were allowed to use graphing calculators. The
test was designed so that having a graphing calculator would not be an advantage.
Four problem solving strategies, drawn from the work of Kieran
(1992), were used to code the students’ work: heuristic, graphical,
algorithmic, and no solution or unknown. Heuristic strategies are those involving
trial and error. Graphical strategies depend on the creation of graphs. Algorithmic
strategies are based on algebraic procedures, such as computing the derivative.
Although scores on the pre- and posttests were not statistically significantly
different, the authors did see differences in the approaches used by students.
Results showed that students with the longest access to calculators used a
wider range of problem-solving approaches and “tended to attempt more
problems and obtain higher test scores than the students who had not”
(Harskamp et al., 2000, p. 37). In addition, students referred to as “below
average” by the researchers made more frequent use of graphical strategies
and “achieved a significantly higher mean posttest score (p <
0.05) than students in the control group” (pp. 47-48). Students using
the calculators for one unit also used more graphical strategies than they had
on the pretest (van Streun et al., 2000). These students tended to replace heuristic
and algorithmic strategies with graphical approaches.
This collection of reports suggests that even limited access to calculators
may have a positive effect on students’ approaches to mathematical problems.
This interpretation is supported by the findings of other studies (Adams, 1997;
Hong, Toham, & Kiernan, 2000) conducted with students whose access to calculators
was limited. Perhaps the shortest period of calculator access that produced
student performance gains was reported in Hong, Toham, and Kiernan (2000). In
the study, the performances of two groups of New Zealand calculus students were
compared on a series of tests. The experimental group was taught to use calculators
(TI-92) with computer algebra systems (CAS) in four 1-hour lessons. Students
in the control groups were taught integration using a traditional approach.
On a posttest consisting of traditional university entrance exam questions and
on which all students were allowed to use a non-CAS graphing calculator, students
in the experimental group outperformed those in the control group. Thus, access
to calculators, even limited access, appeared to result in improved student
Based on the findings from these two studies, one might be tempted to simply
supply calculators for students and assume that, provided teachers allow the
access, scores will increase. Additional findings of Hong et al. (2000), however,
suggest that short-term access may bring problems that are not apparent in these
comparisons. When Hong et al. (2000) compared students’ performance on
a calculator neutral test, designed to measure conceptual growth rather than
computation skill, students in the control group outperformed those in the experimental
group. In addition, Hong et al. (2000) investigated the impact of the calculators
on the performance of students they referred to as low achieving and found that
the calculators enabled students to complete computational problems that they
could not do on the pretest. This use resulted in large gains for these students
but, as the authors suggested, no new understanding of calculus. Based on these
study results, we learn that some students with short-term access may experience
hollow performance gains.
Additional problems associated with limited access to calculators were illuminated
in the report of a study conducted with eighth-grade general education students
(Merriweather & Tharp, 1999). The students used calculators (TI-82) in class
for 2 weeks to complete traditional algebra problems. The authors noted, “The
majority of the students felt they were not comfortable with the calculator
and did not use it” (p. 19). For these students the calculator caused
confusion; consequently, they tended to use problem-solving strategies that
were more familiar to them, such as working backwards. Although the performance
of some students may improve when they have short-term access to calculators
and are taught using traditional instructional materials, research indicates
gains do not appear to be conceptual. In addition, students with limited access
may become confused and overwhelmed as they attempt to integrate their knowledge
of mathematics with their developing understanding of a new tool.
In contrast to these negative findings associated with short-term access, Graham
and Thomas (2000) conducted a study in New Zealand in which algebra students
(13-14 years old) in treatment and control groups were taught for 3 weeks. The
treatment groups were taught using TI-82 or Casio FX7700GH calculators and a
3-week module, Tapping into Algebra, designed to provide opportunities for students
to develop their understanding of variables in algebraic expressions (symbolic
literals). The curriculum was designed “to use the graphic calculator’s
lettered stores as a model of a variable” (p. 269). The store function
allows a letter to be assigned a constant value. For example if A = 3, then
students would be asked to make conjectures about the result of A + 3. Later
students were asked to assign values to two letters and to predict results of
expressions such as AB, 2A + 2B, and 4(A + 5B). Also included in the unit were
investigations of “squares and square roots, sequences, formulas, random
numbers and function tables of values” (p. 270). The control groups were
taught the same topics with a focus on whole-class instruction and skill development.
Students were given one posttest designed to “measure understanding of
the use of letters as [a] specific unknown, generalized number and variable”
(p. 272) and another designed to measure procedural skills. Students in the
treatment groups outperformed those in the control group on the test of use
of symbolic literals and performed as well as those in the control groups on
the test of procedural skill.
Two factors in the Graham and Thomas study (2000) provide plausible reasons
why this short-term access had a more positive impact on students’ performance
than that reported in the previously discussed studies. First, the students
may have had access to either scientific or graphing calculators prior to the
treatment. Student comments reveal that at least some participants had experience
with scientific calculators. For example, one student noted he or she liked
“the way the screen showed all the numbers coming up. I found it much
easier than all the other calculators which don’t even show the number”
(p. 276). Experience with a scientific calculator may have had an impact on
the success of the module. Second, and more importantly, the calculator was
not simply added on to the standard curriculum. Instead, the authors of Tapping
into Algebra developed the content of the module with the tool in mind. Thus,
the results suggest that another critical factor in teachers’ use of and
student performance and learning with the graphing calculator is the role of
the tool in the curriculum.
Researchers in mathematics education assert that understanding a mathematical
concept, such as function, includes the ability to use and make connections
between multiple representations (Confrey & Smith, 1991; Hiebert & Carpenter,
1992). Although developing more than one representation for a mathematical object
can take a substantial amount of time, effective use of graphing calculators
allows quick and easy development of and translation between representations.
Curriculum developers were quick to realize the power of these tools and to
design investigations that made use of the device as a lever to learn mathematics
(Schwarz & Hershkowitz, 1999). This section discusses the results of studies
that investigated student learning with curricula designed to take advantage
of access to graphing calculators (Schwarz & Hershkowitz, 1999).
The curricula of the Core-Plus Mathematics Project (CPMP) and the second edition
of the University of Chicago School Mathematics Project (UCSMP) were developed
to take advantage of graphing calculators. The developers of CPMP worked from
the premise that graphing calculators provide access to mathematics that has
heretofore required proficiency in skills such as symbol manipulation (Huntley,
Rasmussen, Villarubi, Sangtong, & Fey, 2000). Graphing calculators afford
students the opportunity to explore problems using graphical, numeric, and symbolic
strategies and to make links between these strategies (Core-Plus Mathematics
Project, 2004). The UCSMP curriculum was developed to meet the mathematical
and technological needs of individuals and our society as suggested by mathematicians
and mathematics educators (Usiskin, 1986). While the first edition materials
required access to scientific calculators, the second edition materials were
“influenced by advances in technology, particularly the availability of
graphing calculators” (Thompson & Senk, 2001, p. 60).
|When a baseball is thrown straight up from a height of 5 feet with
an initial velocity of 50 ft/sec, its height h in feet after
t seconds is given by the equation h = -16t2
+ 50t + 5. Assuming that no one catches the ball, after how
many seconds will the ball hit the ground?
|Figure 1. Application problem (Thompson & Senk, 2001,
Studies designed to identify the effect of these curricula on student achievement
(Huntley et al., 2000; Thompson & Senk, 2001) found that students taught
using the curricula outperformed those taught using traditional approaches on
application problems (see example in Figure 1). However the findings of the
two studies differ with regard to student performance on procedural tasks such
as the one illustrated in Figure 2. On tests comprised of procedural items,
Huntley et al. (2000) found Algebra I students who had used traditional curricula
outperformed students who had used CPMP; Thompson & Senk (2001) found no
statistically significant difference between the performances of their groups
on a similar test.
|The solutions to 5x2 – 11x –
3 = 0 are…
|Figure 2. Procedural task (Thompson & Senk, 2001,
Although it is unclear from these studies which factors produced the performance
gains on application problems, interpretation of results from two other studies
(Boers & Jones, 1994; Ruthven, 1990) provide some plausible explanations.
By contrast, these two studies involved curricula not designed to take advantage
of graphing calculators. The results of these studies suggest that the intent
of curricular materials significantly influences the way graphing calculators
are used in the classroom.
Boers and Jones (1994) examined student work on a traditional calculus test.
The students and teacher used a traditional text; however, students were also
given instruction on the use of the calculator (TI-81) in “graphing, solving
equations and inequalities, the numerical evaluation of derivatives and integrals,
picturing and evaluating limits, and numerically checking analytically derived
mathematical solutions” (p. 492). Analysis of student solutions revealed
that students “made minimal use of the calculator outside of questions
requiring a specific graphical response” (p. 494) and had difficulty integrating
graphical information they generated into their solutions.
For example, when students were given a rational function and asked to find
the values for which the function was not defined, to find the limit of the
function as x approached a given value, and to graph the function,
only 5 of the 37 students integrated algebraic and graphical information to
produce their solution. Generally, students “treated questions as either
essentially ‘algebraic’ or ‘graphical’ (or as having
distinct algebraic and graphical parts), depending on what the questions asked
for” (Boers & Jones, 1994, p. 514). One plausible interpretation of
this finding is that students may have learned how to apply calculator techniques
as suggested in the supplemental lessons and how to apply algebraic techniques
as suggested in the text, but not how to integrate the information derived from
these two sources to solve problems. This disconnect between graphing calculators
and the curriculum impeded students’ ability to integrate the various
techniques they had learned.
Ruthven (1990) also studied the use of graphing calculators with curriculum
materials not designed to take advantage of those tools. He investigated the
effect of calculator access on upper secondary students’ ability to translate
graphs into algebraic form and to interpret graphics of contextual situations.
In this curricular context, Ruthven found that students who used graphing calculators
(fx-7000) outperformed students without access to the graphing calculators on
symbolization tasks (e.g., given the graph of a quadratic function, find an
algebraic representation). There was no statistically significant difference
between scores of the two groups of students on tasks that asked them to make
interpretations from graphical representations of contextual situations (e.g.,
given a time versus speed graph for a cyclist, determine when the speed will
be greatest). Ruthven suggested that this lack of distinction was due to the
fact that the curriculum materials were of little use in preparing students
to draw interpretations from graphs.
The results from these two studies suggest that students for whom the graphing
calculator is simply added on to traditional texts develop algebraic and graphical
approaches to mathematics problems. Unlike students using problem solving investigations
developed to be used with graphing calculators, students using calculators as
an add-on were unable to integrate mathematical information drawn from different
representations. Such students were no better than their peers with access to
traditional texts and scientific calculators at interpreting mathematics problems
in context. Inconsistent gains illustrated by Boers and Jones (1994) and Ruthven
(1990) are characteristic of instruction in which students are asked to make
sense of an inconsistent curriculum.
The results of these studies do not imply, however, that only large-scale
curriculum projects can produce healthy, connected student use of graphing calculators.
Indeed in each of the four studies discussed, the authors suggested that the
method of instruction was a critical factor in determining student learning.
This phenomenon, often referred to as the difference between the intended curriculum
and the “curriculum in practice” (Romberg, 1992, p. 750), leads
to a discussion of the influence of pedagogy on classroom teaching and learning
with graphing calculators.
In addition to providing materials designed to take advantage of access to
graphing calculators, UCSMP and CPMP curricula assumed instruction would consist
of investigations of novel problems by groups of students who explored and discussed
problem situations and worked cooperatively toward solutions. These solutions
were seen as the basis for the mathematics the students discussed as a whole
class and ultimately learned. This approach stands in contrast to the pedagogical
techniques used in the classrooms involved in the research of Boers and Jones
(1994) and Ruthven (1990). Although teachers in these latter studies welcomed
and used graphing calculators in their classrooms, they attempted to teach using
traditional lecture-style approaches. Students from this pair of studies appear
to have learned very different mathematics. This section discusses research
on the impact of pedagogy on student learning with graphing calculators.
To investigate how pedagogy might impact student learning, Kendal and Stacey
(1999) studied three teachers’ approaches to teaching calculus with the
graphing calculator (TI-92) and their Year 11 students’ approaches to
solving problems with the technology. The three teachers and several researchers
“designed a twenty-five lesson introductory calculus program, with a focus
on differentiation, that aimed to use CAS to enhance conceptual understanding,
connections between representations, and appropriate use of the technology”
(p. 234). Students and teachers were experienced users of the TI-83, but were
novice users of the TI-92.
|From the corners of a rectangular piece of cardboard, 32 cm by 12
cm, square sides of side, x cm, are cut out and the edges
turned up to form a box. Find the value of x if the volume
of the box is a maximum.
|Figure 3. Problem to introduce maximum/minimum (Kendal
& Stacey, 1999, p. 241)
Researchers found that, although the teachers had agreed to teach the lessons
using the same approach, three different emphases were apparent: Teacher A made
frequent use of the calculator and algebraic approaches; Teacher B “preferred
the traditional algebraic approach using graphs when essential” (p. 243);
and Teacher C emphasized the use of algebraic and graphical methods and the
connections between them. The difference in the three teachers’ approaches
was most obvious in the researchers’ descriptions of approaches they used
to introduce the same maximum/minimum problem (see Figure 3). Teacher A drew
a diagram, used a volume formula, and demonstrated how to solve the problem
using a CAS procedure. Teacher B guided the students to the development of a
function for the volume of the box and through the by-hand solution method.
Teacher C guided the students through the development of a function and used
student suggestions as the basis for graphical and CAS solutions. She also demonstrated
algebraic and graphical procedures using the projection unit linked to a graphing
The average test scores for the three classes were similar, but an item-by-item
analysis revealed “differences between classes with respect to the calculator
use and success on items” (p. 236). Although students in Class A attempted
more items, in general, and used the calculator more often, they had a lower
overall success rate on the items they attempted than did students in the other
two classes. Students in Class B “preferred to use by-hand algebra”
(p. 239) and applied these techniques with success; they failed to use CAS,
however, “when its use would have been advantageous” (p. 244). Students
in Class C were discriminating about their use of the calculator and “compensated
for their poorer algebraic skills by substituting a graphical for an algebraic
procedure” (p. 245). These students did not use CAS to differentiate polynomial
functions as students in Class A did. In an analysis of student conceptual errors,
errors that cannot be avoided by using CAS, the error rate for Class C was 4.9
errors per student as compared to 7.3 and 5.7 for students in classes A and
B, respectively. Thus, although the differential emphases in the classes did
not produce significant differences in overall student performance, a closer
look at students’ approaches to various problems revealed that the performance
of students in each class was closely aligned with the pedagogy of their teachers.
Similar results were found by Porzio (1999) in a study of students’ self-selecting
into three different sections of calculus: traditional text and lecture approach,
text developed to be used with a graphing utility and a graphing calculator,
and a Mathematica section taught without a text. The author found that “students
‘behave’ as they are taught” (p. 25), solving problems using
algebraic, graphical, or multiple representations depending on the representations
their teachers presented most often. Interpretation of these results suggests
that when access to graphing calculators and curriculum developed to use these
tools are controlled, the critical factor in student learning is pedagogy.
Implications for Mathematics Teacher Education
Thus far this paper has focused on three common themes in research on the
use of graphing calculators with secondary mathematics. As teachers largely
determine students’ access to graphing calculators, the curriculum that
is actually enacted, and the pedagogical landscape of the classrooms, the findings
of this research have serious implications for the role of teacher education
in ensuring success in the learning and teaching of mathematics with graphing
calculators. Organized around these three common themes, the following sections
illustrate ways in which teacher education programs can inform teachers of and
be informed by this research.
Access to Graphing Calculators
As has been discussed previously, mere access to graphing calculators does
not ensure an impact on teaching and learning. In order for teachers to be prepared
to take advantage of access to technology, teacher education programs must help
teachers feel knowledgeable and comfortable with technology. This can be done
in a number of ways. First, teachers need early and frequent access to graphing
calculators. They need experience both learning and teaching with graphing calculators
in constructive learning environments.
Second, the focus of such efforts should go beyond the functionality of the
tool to incorporate the potential and the constraints of the tool. In order
to make thoughtful decisions about when and how to use the tool, teachers need
to be given opportunities to discuss the benefits as well as the possible pitfalls
of using graphing calculators in their classrooms. It is through such discussions
that teachers can explore and develop their own beliefs about student access
to and use of graphing calculators and the implications of such access.
Third, teacher education should expose teachers to research on the effects
of access to graphing calculators. This opportunity will enable them to develop
beliefs that are both informed by current research and connected to their own
mathematical understandings. When experiences learning and teaching with graphing
calculators are coupled with exposure to research in these areas, teachers revisit
the instructional approaches they use or plan to use to teach mathematics. These
reflections may, in turn, motivate teachers to provide the kind of access to
graphing calculators advocated by current movements in mathematics education
reform (e.g. National Council of Teachers of Mathematics, 2000).
Teacher education programs need to address issues related to the place of graphing
calculators in the curriculum. Teachers often feel restricted by their curriculum—in
particular, by high-stakes testing. If teachers perceive that these tests value
by-hand procedures, then they feel obligated to prepare their students to perform
on these tests and, for teachers, this often means limiting the use of graphing
calculators. Teacher education programs should familiarize teachers with curricula,
such as UCSMP and CPMP, which have been developed to take advantage of graphing
calculators as a tool for developing mathematical understanding.
Exposure to research on the use of graphing calculators with various curricula
can influence teachers’ decisions with respect to curricular demands.
Although results from the studies examined herein are mixed, there is evidence
that students using graphing calculators in problem solving develop understanding
of mathematics that allows them to answer traditional test items successfully
(Thompson & Senk, 2001). Examples such as this suggest it is possible to
use all the tools at hand, including graphing calculators, to help students
gain deep understandings of important mathematics (National Council of Teachers
of Mathematics, 2000) while still successfully fulfilling curricular demands.
Research reports contain numerous suggestions for how teacher education programs
can effectively influence teachers’ pedagogical decisions regarding the
use of graphing calculators. We discuss here three main categories of these
suggestions: reflection, experience, and learning to manage inherent challenges.
Teacher education programs should give preservice and in-service teachers opportunities
to reflect on their personal philosophies and beliefs about graphing calculators
(Fleener, 1995a, 1995b), combined with reflection on their beliefs about mathematics,
its teaching and learning (Simmt, 1997). Professional development can facilitate
reflection by employing multiple strategies, such as participating in classroom
discussions, writing in journals (Tharp, Fitzsimmons, & Ayers, 1997), and
both sharing and listening to personal experiences learning and teaching with
graphing calculators (Simonson & Dick, 1997).
Opportunities to reflect should go hand in hand with opportunities to experience
teaching and learning mathematics with graphing calculators. One way for teachers
to have such experiences is for them to observe videotaped classrooms in which
teachers are teaching and students are learning with graphing calculators (Tharp
et al., 1997). Reading research reports such as those synthesized for this paper
can also provide teachers with opportunities to observe, through the written
word, examples of how teachers have effectively used graphing calculators in
their classrooms (Doerr & Zangor, 1999, 2000; Goos, Galbraith, Renshaw,
& Geiger, 2000; Slavit, 1996). Other studies (Drijvers & Doorman, 1996;
LaGrange, 1999a, 1999b; Zarzycki, 2000) provide illustrations of lessons teachers
developed that provide opportunities for their students to investigate mathematics
that is inaccessible without the use of graphing calculators. These studies
provide vivid examples of how technology “influences the mathematics that
is taught and enhances students’ learning” (National Council of
Teachers of Mathematics, 2000, p. 24). In addition, exploring the use of technology
in the classroom can provide a context for exploring other pedagogical decisions
(see Laborde, 1999). Observation of effective graphing calculator use should
be coupled with substantial mathematics explorations in which teachers experience
the benefits of learning with graphing calculators (Fleener, 1995a; Tharp et
al., 1997), in particular, the multiple representations graphing calculators
afford. Teachers should then be given opportunities to experiment with these
strategies in their teaching.
Research has documented and articulated a number of challenges inherent in
pedagogical approaches that involve the use of graphing calculators. Teacher
education programs should give teachers opportunities to explore implications
of these challenges and ways in which they can be addressed. Lessons involving
explorations with graphing calculators are often less structured, requiring
teachers to share control of the classroom with their students (Goos et al.,
2000; Tharp et al., 1997). Teachers should become familiar with the disadvantages
as well as the advantages of using graphing calculators in their classrooms
(Hong et al., 2000). Discussions about the limitations of graphing calculators
can be valuable learning experiences for teachers as well as for their students
(Mitchelmore & Cavanagh, 2000). Finally, Schmidt (1998) offered concrete
examples of how teachers can deal with external factors, such as standardized
tests, and pressures from parents and state and district mandates.
This paper has considered some common themes in the research on teaching and
learning with graphing calculators and implications of such research for teacher
education. There is a need, however, for more far-reaching research on the use
of graphing calculators, built on efforts to integrate research on the teaching
and learning of mathematics (e.g., Fennema, Carpenter, & Lamon, 1991). Research
on the use of graphing calculators in the classroom needs to move to a new phase
of complexity, where teacher education, teacher use, student use, and student
learning are all taken into account. This may be challenging, but it is necessary
if research is to produce the kind of knowledge base about teaching and learning
with graphing calculators that, in the face of constant technological advancement,
will continue to inform teacher education and practice effectively.
Adams, T. L. (1997). Addressing students' difficulties with the concept of
function: Applying graphing calculators and a model of conceptual change. Focus
on Learning Problems in Mathematics, 19(2), 43-57.
Boers, M. A., & Jones, P. L. (1994). Students' use of graphics calculators
under examination conditions. International Journal of Mathematical Education
in Science and Technology, 25, 491-516.
Confrey, J., & Smith, E. (1991). A framework for functions: Prototypes,
multiple representations, and transformations. Paper presented at the 13th
annual meeting of the North American Chapter of the International Group for
the Psychology of Mathematics Education, Blacksburg, VA.
Core-Plus Mathematics Project. (2004). Features of the CPMP curriculum.
Retrieved August 25, 2004, from http://www.wmich.edu/cpmp/features.html
Doerr, H. M., & Zangor, R. (1999). The teacher, the task and the tool:
The emergence of classroom norms. International Journal of Computer Algebra
in Mathematics Education, 6, 267-280.
Doerr, H. M., & Zangor, R. (2000). Creating meaning for and with the graphing
calculator. Educational Studies in Mathematics, 41, 143-163.
Drijvers, P., & Doorman, M. (1996). The graphics calculator in mathematics
education. Journal of Mathematical Behavior, 15, 425-440.
Fennema, E., Carpenter, T. P., & Lamon, S. J. (Eds.). (1991). Integrating
research on teaching and learning mathematics. Albany, NY: State University
of New York Press.
Fleener, M. J. (1995a). The relationship between experience and philosophical
orientation: A comparison of preservice and practicing teachers' beliefs about
calculators. Journal of Computers in Mathematics and Science Teaching, 14,
Fleener, M. J. (1995b). A survey of mathematics teachers' attitudes about calculators:
The impact of philosophical orientation. Journal of Computers in Mathematics
and Science Teaching, 14, 481-498.
Goos, M., Galbraith, P., Renshaw, P., & Geiger, V. (2000). Reshaping teacher
and student roles in technology-enriched classrooms. Mathematics Education
Research Journal, 12, 303-320.
Graham, A. T., & Thomas, M. O. J. (2000). Building a versatile understanding
of algebraic variables with a graphic calculator. Educational Studies in
Mathematics, 41, 265-282.
Harskamp, E. G., Suhre, C. J. M., & van Struen, A. (1998). The graphics
calculator in mathematics education: An experiment in the netherlands. Hiroshima
Journal of Mathematics Education, 6, 13-31.
Harskamp, E. G., Suhre, C. J. M., & van Struen, A. (2000). The graphics
calculator and students' solution strategies. Mathematics Education Research
Journal, 12, 37-52.
Hiebert, J., & Carpenter, T. (1992). Learning and teaching with understanding.
In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning
(pp. 65-100). Reston, VA: National Council of Teachers of Mathematics.
Hong, Y., Toham, M., & Kiernan, C. (2000). Supercalculators and university
entrance calculus examinations. Mathematics Education Research Journal,
Huntley, M. A., Rasmussen, C. L., Villarubi, R. S., Sangtong, J., & Fey,
J. T. (2000). Effects of standards-based mathematics education: A study of the
Core-Plus mathematics project algebra and functions strand. Journal for
Research in Mathematics Education, 31, 328-361.
Kieran, C. (1992). The learning and teaching of school algebra. In D. A. Grouws
(Ed.), Handbook of research on mathematics teaching and learning (pp.
390-419). New York: Macmillan.
Kendal, M., & Stacey, K. (1999). Varieties of teacher privileging for teaching
calculus with computer algebra systems. International Journal of Computer
Algebra in Mathematics Education, 6, 233-247.
Laborde, C. (1999). The integration of new technologies as a window on teachers'
decisions. Proceedings of the 1999 International Conference on Mathematics
Teacher Education, Taipei, Taiwan.
LaGrange, J.-B. (1999a). Complex calculators in the classroom: Theoretical
and practical reflections on teaching pre-calculus. International Journal
of Computers for Mathematical Learning, 4, 51-81.
LaGrange, J.-B. (1999b). Techniques and concepts in pre-calculus using CAS:
A two-year classroom experiment with the TI-92. International Journal of
Computer Algebra in Mathematics Education, 6, 143-165.
Leatham, K. R. (2002). Preservice secondary mathematics teachers' beliefs
about teaching with technology. Unpublished doctoral dissertation, University
of Georgia, Athens.
Merriweather, M., & Tharp, M. L. (1999). The effect of instruction with
graphing calculators on how general mathematics students naturalistically solve
algebraic problems. Journal of Computers in Mathematics and Science Teaching,
Mitchelmore, M., & Cavanagh, M. (2000). Students' difficulties in operating
a graphics calculator. Mathematics Education Research Journal, 12,
National Council of Teachers of Mathematics. (2000). Principles and standards
for school mathematics. Reston, VA: Author.
Porzio, D. (1999). Effects of differing emphases in the use of multiple representations
and technology on students' understanding of calculus concepts. Focus on
Learning Problems in Mathematics, 21(3), 1-29.
Romberg, T. A. (1992). Problematic features of the school mathematics curriculum.
In P. W. Jackson (Ed.), Handbook of research on curriculum (pp. 749-788).
New York: Simon & Schuster Macmillan.
Ruthven, K. (1990). The influence of graphic calculator use on translation
from graphic to symbolic forms. Educational Studies in Mathematics, 21,
Schmidt, M. E. (1998). Research on middle grade teachers' beliefs about calculators.
Action in Teacher Education, 20(2), 11-23.
Schwarz, B. B., & Hershkowitz, R. (1999). Prototypes: Brakes or levers
in learning the function concept? The role of computer tools. Journal for
Research in Mathematics Education, 30, 362-389.
Simmt, E. (1997). Graphing calculators in high school mathematics. Journal
of Computers in Mathematics and Science Teaching, 16, 269-289.
Simonson, L. M., & Dick, T. P. (1997). Teachers' perceptions of the impact
of graphing calculators in the mathematics classroom. Journal of Computers
in Mathematics and Science Teaching, 16, 239-268.
Slavit, D. (1996). Graphing calculators in a "hybrid" Algebra II
classroom. For the Learning of Mathematics, 15(1), 9-14.
Tharp, M. L., Fitzsimmons, J. A., & Ayers, R. L. B. (1997). Negotiating
a technological shift: Teacher perception of the implementation of graphing
calculators. Journal of Computers in Mathematics and Science Teaching, 16,
Thompson, D. R., & Senk, S. L. (2001). The effects of curriculum on achievement
in second-year algebra: The example of the University of Chicago School Mathematics
Project. Journal for Research in Mathematics Education, 32, 58-84.
Usiskin, Z. (1986). Translating grades 7-12 mathematics recommendations into
reality. Educational Leadership, 44(4), 30-35.
van Streun, A., Harskamp, E. G., & Suhre, C. J. M. (2000). The effect of
the graphic calculator on students' solution approaches: A secondary analysis.
Hiroshima Journal of Mathematics Education, 8, 27-39.
Zarzycki, P. (2000). Thinking mathematically with technology. International
Journal of Computer Algebra in Mathematics Education, 7, 181-192.
Brigham Young University
Indiana University Purdue University Indianapolis
The work described in this manuscript grew out of our work preparing Handheld
Graphing Technology at the Secondary Level: Research Findings and Implications
for Classroom Practice, a report prepared through a grant to Michigan State
University funded by Texas Instruments.
Acknowledgements: We wish to express our appreciation to our colleagues, Gail
Burrill (project director), Barbara Ridener, Jacquie Allison, Glenda Breaux
and Wendy Sanchez because their contributions to the project have helped shape
our thinking about this manuscript.