Hand-held Calculators in Mathematics Education:
A Research Perspective
Penelope H. Dunham
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 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.
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).
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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