This paper was presented as an invited address at ENC
Technology and NCTM Standards 2000 Conference, Arlington, VA, June, 1998. It was reprinted in E. Laughbaum (Ed.), Hand-Held Technology in Mathematics and
Science Education: A Collection of Papers, pp. 39-47. Columbus, OH:
The Ohio State University (2000).
Copyright is held by NCTM.
Hand-held
Calculators in Mathematics Education:
A Research
Perspective
Penelope H. Dunham
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 understanding (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 (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 guide curriculum development and instruction
effectively, 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 answers. 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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