Hand-held Calculators in Mathematics Education:

A Research Perspective

Penelope H. Dunham

pdunham@muhlenberg.edu

Muhlenberg College

Department of Mathematical Sciences

Allentown, Pa 18104

The *Curriculum and Evaluation Standards *(NCTM, 1989) for Grades
K-4 state:

** • Integrating calculators and computers into school
mathematics programs is critical in meeting the goals of a redefined
curriculum. **(p. 19)

For Grades 5-8:

** • All students will have a calculator with functions
consistent with the tasks envisioned in this curriculum. **(p. 19)

For Grades 9 to 12:

** • Scientific calculators with graphing capabilities will be
available to all students at all times.** (p. 124)

The National Council of Teachers of Mathematics has long advocated the use of calculators at all levels of mathematics instruction, as indicated by the position statements above. After nearly three decades of availability, calculators have gained a foothold in classrooms across the country (Futch & Stephens, 1997; Porter, 1991; Spath, 1990; Tan, 1995). Spurred by recommendations of national organizations like NCTM, by increased acceptance on standardized tests such as the SAT and AP Calculus exams, and by issues of price, portability, and ease of use, hand-held devices are now more prevalent than other forms of technology in mathematics education. A recent study by Burke (1996) indicates the difference: 19% of Alabama secondary teachers surveyed use microcomputers in mathematics instruction while 83% use calculators. Despite the prevalence of calculators, however, their role in mathematics instruction has not reached the level of NCTM's goals stated above. Porter (1991) states that, although 60% of elementary teachers in a California district report using calculators with students, the amount of time allotted to calculator activities and the types of activities are very limited. Spath's (1990) survey of fifth-grade teachers in Colorado indicates that only 20% use calculators at least once a week and that 53% have as many as three students sharing a calculator.

Research provides strong empirical evidence to support the
*Standards* view that hand-held technology can and should play an
important role in mathematics instruction (Dunham & Dick, 1994; Heid,
1997; Hembree & Dessart, 1986, 1992; Smith, 1997). Why, then, have
calculators yet to reach their full potential in education? Studies point
to a host of reasons: shortages of calculators and curricular materials,
lack of training and inservice opportunities, little planning time, few
incentives, and limited administrative support (Hope, 1997; Johnson, 1991;
Porter, 1991; Schmidt & Callaghan, 1992; Spath, 1990). Such lists beg
the question, though. Why haven't teachers and parents, with research
results and national recommendations in hand, demanded that school
districts correct the deficiencies? Part of the answer is that parents and
classroom teachers often are not aware of research supporting the benefits
of calculator-based instruction (Fine & Fleener, 1994); but the most
important reason is that a complex web of beliefs about the nature of
mathematics and the goals of mathematics education works against the full
inclusion of technology (Fleener, 1995; Graber, 1993; Schmidt &
Callaghan, 1992; Terranova, 1990). Teacher fears that students will lose
computational skills, use calculators as crutches, and not master basic
concepts, play an important role in limiting calculator usage (Payne, 1996;
Simonsen & Dick, 1997; Smith, 1996; Zand & Crowe, 1997) -- despite
evidence to the contrary.

In this paper, I will review the research evidence supporting the
case for fully integrated hand-held technology at all levels of mathematics
instruction. First, I'll outline the general results for three types of
calculator: non-graphing machines, scientific graphics models without
symbolic computation, and graphics calculators with symbolic computation.
Next, I will highlight findings relative to several themes in calculator
research: problem solving, concept development, computation skills, errors,
student and teacher roles, and effects on special populations. The last
section will feature research on attitudes and beliefs about mathematics
and technology and present some suggestions for inservice and education
programs to promote better implementation of the *Standards'* vision
for technology-enhanced mathematics instruction.

Research Overview

Research supplies ample evidence of positive benefits in
computation and problem solving for students who use non-graphics
calculators (*i.e*., four-function, fraction, and scientific models).
The definitive report on non-graphics calculators in school mathematics is
Hembree and Dessart's (1986) meta-analysis of 79 studies from a 15-year
period. Analyzing effect sizes for studies of students' achievement and
attitudes in calculator-enhanced settings, Hembree and Dessart conclude
that students who use calculators possess better attitudes and have better
self-concepts in mathematics than non-calculator users and that testing
with calculators produces higher achievement scores at all grades and
ability levels. For all but one grade level, average-ability students who
use calculators in conjunction with traditional mathematics instruction
perform better on paper-and-pencil tests of basic skills and problem
solving. For fourth graders, there is evidence that repeated calculator
use may hinder the computational skills of average students. High- and
low-ability students displayed no significant difference in skill
acquisition with calculators; but, in an update of the original
meta-analysis, Hembree and Dessart (1992) cite new studies showing that
calculator-enhanced instruction can improve paper-and-pencil performance
for these two ability groups just as it did for average-ability students.
In another meta-analysis of 24 studies, Smith (1997) reports significant
achievement differences in problem solving, computation, and conceptual
understanding favoring students who use calculators *vs.* those who do
not. Recent studies show students using non-graphing calculators perform
as well (Malloy, 1996; Riley, 1993) or better (Bridgeman *et al.*,
1995; Cronin, 1992; Frick, 1989; Glover, 1992; Liu, 1994) on several
measures of achievement than students who do not use calculators.

In the dozen years since graphing calculators were introduced in 1986, we have seen a steady flow of research on graphing calculators in mathematics classrooms. Research reviews by Dunham (1993, 1995), Dunham and Dick (1994), Heid (1997), Marshall (1996), and Penglase and Arnold (1996) indicate mostly positive benefits for achievement in algebra, trigonometry, calculus, and statistics. The consensus of the reviews is that students who use graphing calculators display better understanding of function and graph concepts, improved problem solving, and higher scores on achievement tests for algebra and calculus skills. In particular, in a precalculus curriculum that fully integrates graphing calculators, there can be a strong positive impact on achievement and understanding (Demana, Schoen, & Waits, 1993; Harvey, Waits, & Demana, 1995; Waits & Demana, 1994) together with significant improvement in calculus readiness (Harvey, 1993). Moreover, several studies indicate that graphing technology may have even greater benefits for some special populations -- in effect, leveling the "playing field" for women (Dunham, 1995; Nimmons, 1998; Smith & Shotsberger, 1997), nontraditional college students (Austin, 1997; Zand & Crowe, 1997), low-ability students (Owens, 1995; Shoaf-Grubbs, 1994), and students with less spatial visualization ability (Galindo-Morales, 1995; Shoaf-Grubbs, 1993; Vazquez, 1991). The reviews also point to positive changes in classroom dynamics and pedagogy (Farrell, 1990, 1996; Kaplan & Herrera, 1995; Slavit, 1994, 1996). Along with the benefits, however, new types of errors related to graphing technology are emerging (Slavit, 1996; Steele, 1995; Tuska, 1993; Ward, 1997; Williams, 1993).

While there are too few studies of so-called "supercalculators" to
draw general conclusions about the effects of graphing calculators with
symbolic manipulation (*i. e.*, devices that combine numeric and
graphic features with a computer algebra system [CAS]), the studies we do
have echo the results for graphing calculators. For example, students
using a TI-92 to solve word problems in college algebra had greater
achievement compared to students solving problems by hand (Runde, 1997).
Hart (1992) reports that students using HP28 and HP48S models better
understood the connections between multiple representations (numeric,
graphic, symbolic). Keller and Russell (1997) note that calculus students
using the TI-92 CAS technology for problem solving were more successful,
exhibited more metacognitive behaviors, and had greater confidence in their
problem solving ability than did students without access to CAS technology.
Because the symbol manipulation software on the TI-92 is Derive, we might
get some insights on its impact by looking at studies of Derive and other
CAS-based instruction with computers. Landmark computer-based studies by
Heid (1988), Judson (1990), and Palmiter (1991) indicate: greater
understanding of concepts for CAS users; effective resequencing of content
to teach concepts before manipulation skills; and no difference in
achievement on manipulation skills for CAS and non-CAS users when CAS
students learn skills after concepts.

Trends in Calculator Research

*Problem solving. * What effect does calculator use have on
problem solving? Dick (1992) claims that calculators can lead to improved
problem solving because they free more time for instruction, provide more
tools for problem solving, and change students' perception of problem
solving as they are freed from the burden of computation to concentrate on
formulating and analyzing the solution. Research supports these
observations. Hembree and Dessart's meta-analysis (1986) shows that using
a calculator in problem solving creates a computational advantage and more
often results in selection of a proper approach to a solution. Moreover,
calculator use produces a greater positive effect size for high- and
low-ability students than average-ability students. Dunham and Dick
(1994) and others report that students using graphing technology (a) were
more successful on problem solving tests (Frick, 1989; Keller &
Russell, 1997; McClendon, 1992; Runde, 1997; Siskind, 1995; Wilkins, 1995);
(b) had more flexible approaches to problem solving (Boers-van Oosterum,
1990; Slavit, 1994); (c) were more willing to engage in problem solving and
stayed with a problem longer (Farrell, 1996; Mesa, 1997; Rich, 1991); (d)
concentrated on the mathematics of the problems and not the algebraic
manipulation (Keller & Russell, 1997; Rizzuti, 1992, Runde, 1997); and
(e) solved nonroutine problems inaccessible by algebraic techniques (Rich,
1991).

*Concept development.* Irwin (1997) reports that calculators serve
as catalysts for acquiring fraction concepts, in that most learning in her
study resulted from students' reconsidering their ideas after finding
conflicts between their expectations and the calculator results, while
Cronin's (1992) study shows no significant difference in concept learning
with fraction calculators. Graphing calculator use can significantly
improve students' understanding of functions and graphs (Hollar, 1997;
Kinney, 1997) . According to Dunham and Dick (1994), students who use graphing calculators: (a) place at higher levels in a hierarchy of graphical unders
tanding (Browning, 1989); (b) are better able to relate graphs to their
equations (Rich, 1991; Ruthven, 1990); (c) can better read and interpret
graphical information (Boers-van Oosterum, 1990); (d) obtain more
information from graphs (Beckmann, 1989); (e) are better at "symbolizing"
(Rich, 1991; Ruthven, 1990; Shoaf-Grubbs, 1992); (f) understand global
features of functions better (Beckmann, 1989; Rich, 1991; Slavit, 1994);
(g) increase their "example base" for functions by examining a greater
variety of representations (Wolfe, 1990); and (h) better understand
connections among graphical, numerical, and algebraic representations
(Beckmann, 1989; Browning, 1989; Hart, 1992). In the few instances where
calculator use produced negative results on conceptual understanding (e.g,
Giamati, 1991; Upshaw, 1994), we find that those studies involved
treatments of very brief duration so that learning the calculator may have
interfered with learning the content.

*Computation Skills.* A persistent theme in surveys of teachers',
parents', and students' attitudes is the fear that calculator use will
adversely affect computational skills (Fleener, 1995; Futch & Stephens,
1997; Johnson, 1991; Payne, 1996; Schmidt & Callaghan, 1992; Simonsen
& Dick, 1997; Smith, 1996; Zand & Crowe, 1997). Yet, the research
evidence is to the contrary. Students who learn paper-and-pencil skills in
conjunction with technology-based instruction (from simple four-function
calculators to the most sophisticated CAS software) and are tested
*without* calculators perform as well or better than students who do
not use technology in instruction (Heid, 1997; Hollar, 1997; Kinney, 1997;
Liu, 1994; Wilkins, 1995). Hembree & Dessart (1986, 1992) express
concern for negative results with sustained calculator use at one grade
level (4) and urge special attention to skill development at that level. A
number of teachers believe that calculators should be withheld until
students have mastered basic skills (Fleener, 1995; Johnson, 1991; Spiker,
1991), despite evidence that concept learning can take place before skills
are mastered or even taught (Heid, 1988, 1997). Research indicates that
teachers' beliefs about mathematics affect their beliefs about calculator
use (Simmt, 1995); those who support "mastery first" often view mathematics
merely as computation rather than a process for patterning, reasoning, and
problem-solving (Fleener, 1994,1995). Teachers with a rule-based view of
mathematics are more likely to believe that calculators will hinder rather
than enhance learning (Futch & Stephens, 1997; Tharp *et al.*,
1997).

*Calculator-induced errors*. Although research supports the claim
that calculator use improves student performance in computation, concept
development, and problem-solving, a growing number of studies show that
there may be a class of errors and misconceptions that are induced by
calculators. Tuska (1993) identifies eight types of errors made by
students using graphing calculators, such as considering every number as
rational, assuming "solve" means "find zeros," and thinking of the domain
as a subset of the range. Students' difficulties with scale (Goldenberg,
1988) are compounded by the flexible scaling required when using different
window settings on graphing calculators (Dunham & Osborne, 1991).
Recent studies mention continued difficulties with scaling and with domain
and range concepts (Adams, 1994; Kaplan & Herrera, 1995; Ward, 1997;
Wilson and Krapfl, 1994); however, Steele (1995) reports that adding units
on scale issues to the curriculum can alleviate scale misconceptions for
calculator users. Lauten, Graham and Ferrini-Mundy (1994) note a loss of
distinction between variables "*x*" and "*y*" for graphing
calculator users. They suggest an emerging pattern of "equal" treatment of
"*x*" and "*y*" --wherein the dependence of *y* on *x*
and the height interpretation of *y* are lost -- results from GRAPH
and TRACE commands presenting both coordinates simultaneously. (Dunham and
Osborne (1991) suggest ways to combat this error.) Slavit (1994) observes
that graphing calculators aid "objectification" of functions but notes that
a steady diet of graphing calculators supports students' faulty views of
functions as always continuous, with infinite domains and symbolic
representations of the form *y = f(x)*. Thus, Slavit claims graphing
calculators actually restrict students to a smaller variety of function
types instead of expanding their example base (Dunham & Osborne, 1991;
Wolfe, 1990).

*Classroom dynamics.* One of the most profound impacts that
graphing calculators may have is in changing the climate of the classroom,
creating learning environments like those envisioned by the NCTM
*Standards* (1989) . Farrell (1990, 1996) reports students become more
active in classrooms with graphing technology in use, and do more group
work, investigations, explorations, and problem solving; she says graphing
calculators act almost as a third agent in the classroom as students
consult with both the technology and the teacher. Simonsen (1992),
Beckmann (1989), Rich (1991), Dick and Shaughnessy (1988), and Slavit
(1996) all note a shift to less lecturing by teachers and more
investigations by students in graphing calculator classrooms, although some
educators express concern that such explorations place an overemphasis on
induction vs. deduction (Quigley, 1992). Hylton-Lindsay's (1998) claims
that graphing calculator use enhances metacognition and encourages students
to self-regulate thought processes, and Slavit (1996) reports higher levels
of discourse and an increase in analytic questions when calculators are in
use.

*Special populations.* As the body of research on hand-held
technology grows, we begin to see clusters of studies pointing to positive
benefits for groups of students who traditionally do less well than the
general population. In effect, calculators level the playing field
(Dunham, 1995) so that the special groups perform as well or better than
the main group. The "leveling" effect of calculator use is evident for a
variety of groups traditionally disadvantaged because of different
cognitive styles, learning disabilities, or special circumstances. Studies
show that calculator use benefits non-visualizers (Galindo-Morales, 1995),
low-ability and at-risk students (Ferraro, 1997; Hembree & Dessart,
1986, 1992; Owens, 1995); non-traditional college students (Austin, 1997;
Zand & Crowe, 1997), students with learning disabilities (Glover,
1992), and those with low mathematical confidence (Dunham, 1995). One
cluster of studies indicates gender differences in the effects of using
calculators; there is evidence that with calculators female students
perform as well as or better than males (Dunham, 1995). That is, in some
instances, women and girls made greater gains with calculators than males
did, and females who performed at lower levels than males without
calculators reversed the pattern when calculators were in use (Austin,
1997; Bitter & Hatfield, 1993; Bosche, 1998; Nimmons, 1998; Ruthven,
1990; Jones & Boers, 1993; Wilkins, 1995). Christmann and Badgett
(1997) report that, in a study of statistics achievement, males outperform
females using computers, but the pattern reverses in favor of females when
calculators are used. Explanatory factors may include reduction of anxiety
and increased confidence for female students (Bitter & Hatfield, 1993;
Dunham, 1995; Ruthven, 1990). Jones and Boers suggest, however, that
calculators do not give women an edge; rather, men are "deskilling" in
algebra in the presence of calculators. Some studies show improvement in
spatial visualization skills when instruction is calculator-based (Nimmons,
1998; Shoaf-Grubbs, 1993; Vazquez, 1991), and spatial ability is sometimes
a significant predictor for mathematics achievement in women.

Recommendations

*Future Research*. What areas should researchers be
investigating now to better inform our use of calculators in mathematics
instruction? Most of the studies mentioned in this article have been
descriptive, telling us *what * happens when calculators are in use.
For research to effectively guide curriculum development and instruction,
we need to find out *why *calculators make a difference. As Bright
and Williams (1994) note, we could use more true educational research that
attempts to explain relationships among variables, as opposed to evaluative
studies which say, "We used calculators and they worked." We need studies
that document the way calculators are used by individual students, studies
that ask: who uses calculators; how often and when are they used and on
what kinds of tasks; whether there are ethnic, gender, or social
differences in calculator uses, and whether calculators evoke different
effects among these various groups? For graphing calculators, we should
ask what aspect of the grapher brings about improved understanding: the
presence of a graph, the dynamic creation of the graph, the ability to
manipulate graphs, or the ability to generate many graphs quickly and
easily. There is critical need for research in instructional design to
create curricula that use calculators to their best advantage, to find
effective materials to combat calculator-induced errors, and to evaluate
programs that incorporate calculators. There should be long term studies
that look at the effects of prolonged exposure to calculator-based
instruction and studies that follow-up calculator users to measure
retention of benefits. We don't know what happens if students have ready
access to calculators (four-function through symbolic manipulators)
throughout their mathematics career, what paper-and-pencil skills are still
important, whether students need some paper/pencil manipulation for concept
development, and whether the quality of mathematics they learn is the
same.

*Implications for Inservice and Training.* If research shows that
calculator use benefits students across grade levels and ability levels, as
well as acting as a "leveler" by increasing performance for special
populations, why are so many teachers still reluctant to adopt hand-held
technology? This is a crucial question because the best curriculum in the
world won't do any good if it is not properly implemented in classrooms.
Part of teachers' avoidance of technology is based on lack of knowledge
about the research findings, about the capabilities of the machines, about
ways to use calculators effectively, and even about how to operate some
calculators (Porter, 1991; Terranova, 1990). Inservices can help educate
teachers, but that means there must be appropriate professional development
opportunities and funds available to supply the knowledge teachers lack.
Spath (1990) notes that innovations cannot replace existing curricula
easily; training must be on-going and materials must be recommended
repeatedly. Very few teachers report learning about calculator methods or
related research in their education courses (Terranova, 1990); therefore,
it is imperative that college and university faculty give preservice
teachers information about and experiences with calculator-enhanced
instruction. Education programs should not be restricted to teachers
(Graber, 1993); there is a need to educate parents, administrators, and
school boards to ensure funding for sufficient equipment and materials.
(The cost of a single computer could supply several classrooms with
calculators!) Finally, it is important that calculator education efforts
do not focus solely on "how to"; it is necessary to explain *why*
calculator use is important and to address teachers' and parents' beliefs
about mathematics that lead to fears and misconceptions about calculators
(Tharp *et al.*, 1997).

*Curricular implications.* If we assume that all students will
have access to grade-appropriate hand-held technology in the next decade, what changes should we anticipate in mathematics curricula? Research gives some a
nswers. First, curriculum writers may resequence the order of concepts and
skill development. Research shows that paper and pencil skills can be
taught in a shorter time, after developing concepts, without a loss of
achievement on skills (Hollar, 1997; Heid, 1988, 1997; Kinney, 1997; Liu,
1994; Wilkins, 1995) and that students tested with calculators perform as
well or better than non-calculator users on computation tests (Hembree
& Dessart, 1992). Second, it is important that curricular materials
fully integrate calculators -- not just as add-ons or enrichment, but as
standard tools available to all students as a part of regular instruction.
The greatest gains from technology use occur with materials and instruction
designed for the technology (Heid, 1997). Short-term interventions and
infrequent use may actually hinder students who have not had time to learn
how to use the calculators (Giamati, 1991), whereas long-term use
establishes a classroom "culture" that has a positive impact on
achievement, metacognition, problem solving, and teacher and student roles
(Dunham & Dick, 1994; Farrell, 1996; Slavit, 1996). Third, curricula
should give greater emphasis to some topics because of increased calculator
use. Mental arithmetic and estimation are more important now to evaluate
the correctness of calculator answers; emerging errors with graphical
displays require more instruction on scale issues (Steele, 1996; Tuska,
1993; Ward, 1997); reduced attention to by-hand algebraic manipulation
leaves more time for developing better symbol sense among CAS users (Heid,
1997); graphing can be an larger part of mathematics instruction at a much
earlier stage because of graphing calculators (Demana, Schoen, & Waits,
1993); curricula can feature more problem solving -- and more interesting
problems -- because calculators provide a wider range of problem solving
tools (Rich, 1991; Slavit, 1994).

Conclusion

Despite almost three decades of research showing the benefits of
calculator-enhanced curricula and endorsements from every major mathematics
education organization, there is not universal acceptance by parents,
teachers, and administrators of the role of calculators in mathematics
education. We continue to see contradictions such as teachers who fear
that students using calculators will lose basic computation skills
(Johnson, 1991; Spiker, 1991) and studies claiming that that won't happen
(Heid, 1997; Hembree & Dessart, 1992). As we consider the
*Standards 2000* recommendations for hand-held technology, we must
find ways to ensure that the new recommendations will be accepted and
implemented as intended. If, as Futch and Stephens (1997) report, a group
of Georgia middle school teachers rejected almost one-third of a set of
statements underlying the original NCTM *Standards*, we face a
challenge as we try to reach the same teachers with new *Standards*.
Three ways to meet that challenge are: (1) to make a better case for our
side by making sure the public knows that research underlies and supports
recommendations for calculator use; (2) to design inservice and education
programs that not only prepare teachers to teach with calculators but that
also challenge their beliefs about mathematics and mathematics instruction;
and (3) to offer training and support continually.

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