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Graphing Calculators: Enhancing Math Learning for All Students

Jeremy Roschelle & Corinne Singleton, SRI International

Keywords: mathematics, multiple representations, handhelds, conceptual understanding,

Abstract

Graphing calculators have become one of the most widely adopted technologies in education because they are a proven-effective, affordable handheld device with direct linkages to curricula. A robust and consistent body of research shows that graphing calculators can effectively support mathematics learning and democratize access to complex mathematical concepts. Graphing calculators enable students to explore connections across multiple representations. Recent extensions support formative assessment via a wireless network. Graphing calculators are aligned with existing standards and curricula and support valuable pedagogical practices, such as increasing attention to conceptual understanding and problem solving strategies by offloading laborious computations. Innovators who want to provide new technologies that will have a real impact on learning would do well to heed the lessons and successes of the graphing calculator.

1 I. Introduction

Educators and policy makers want to improve students’ mathematics achievement, particularly in middle and high schools. A consistent association between frequent calculator use [graphing calculators?] and higher average scores on the well- respected National Assessment of Educational Progress calls attention to calculators as a basis for interventions at these grade levels. It is rare for educational research to identify a tool for improving achievement that is inexpensive, ready for large-scale classroom use, and backed by strong research evidence. A closer look at the scientific evidence shows that graphing calculators are just such a tool.

A robust and consistent body of research shows that graphing calculators can effectively support mathematics learning and democratize access to complex mathematical concepts. Further, the story of the graphing calculator provides valuable information to guide the development and implementation of new educational technologies.

First and foremost, technologies that enhance learning for all students do so by changing how and what students learn (Roschelle, Pea, Hoadley, Gordin and Means et al., 2001). Graphing calculators change what students learn by making graphing a readily available representation. Graphing calculators change how students learn by reducing cognitive load, increasing opportunities for complex and multistep problem solving, and enabling teachers to emphasize mathematical reasoning not just calculation. In order to successfully transform learning in this way, educators must be able to transition from occasional, supplemental use of computers, to frequent and integral use of portable computational technology (Soloway & Norris et al., 2001; Tinker & Krajcik, 2001). Further, technologies must be integrated into the social practices of schools—a difficult but important challenge that requires integration with teaching practices, curricula, assessments and school leadership.

2 Graphing calculators have become one of the most widely adopted technologies in education because they are a proven-effective, affordable handheld device with direct linkages to curricula. Indeed, graphing calculators have reached far more K-12 mathematics learners than computers. Approximately 40% of high school mathematics classrooms use graphing calculators, whereas only 11% of mathematics classrooms use computers (National Center for Educational Statistics [NCES], 2001). Graphing products are now integrated with national and state standards (e.g., National Council of Teachers of Mathematics, 2000) and they are embedded in some curricula. Furthermore, research has identified best practices for instruction (Burrill & Allison, 2002; Seeley, 2006) (Burrill et al., 2002; Seeley, 2006) and teacher professional development offerings are widely available to further support the successful integration of graphing technology into classroom teaching and learning.

II. Features of Graphing Calculators

The consumer market for graphing calculators started with the introduction of the Casio graphing calculator in 1985. Other companies, including Texas Instruments, Hewlett Packard, and Sharp quickly joined in, bringing a variety of new capabilities to the market. Unlike earlier four-function calculators, graphing calculators could solve equations symbolically and use named variables; by the 1990s they could also perform symbolic algebra and do some calculus. Calculators with full “Computer Algebra System” (CAS) capabilities now cost less than $200; strikingly just 25 years ago, these capabilities were only available in massive research computers.

Importantly, graphing calculators also added display and visualization capabilities. In particular, standard models now provide three linked, canonical views of mathematical constructs: numeric views (using tables and lists), symbolic views (using algebraic expressions and function definitions), and graphical views (function graphs, charts, and scatter plots). Modern graphing calculators not only do calculations and create graphs, but they can also display geometric figures and integrative diagrams, opening the door for further innovations in classroom use.

Cutting edge research is exploring the latest new advance—graphing calculators that are connected via a wireless network. In simple uses, the wireless network can enable

3 teachers to engage in formative assessment. For example, a teacher can take a quick poll of students’ responses to a conceptual question and display the results instantly. Teachers can use this capability to give students feedback and to adjust instruction. Wireless networks can also be used to rapidly distribute teacher assignments and share student work. In some research studies, advanced use of these features increases student participation in classroom discussions by displaying student work in projected view at the front of the classroom (Dufresne, Gerace, Leonard, Mestre, & Wenk. et al. 1996; Roschelle, Penuel, & Abrahamson et al., 2004). For example, the classroom can use the network to engage in a participatory simulation, such that each student plays a unique role in an emergent mathematical phenomena (Stroup, Ares & Hurford, 2005). In another example, each student can be asked to create a slightly different algebraic function, such that a family of functions emerges when the functions are aggregated and displayed (Hegedus & Kaput, 2004). In designing these advanced uses, researchers aim to create classroom environments in which students learn through participation in a shared, social, mathematical space (Stroup et al, Kaput, Ares, Wilensky, Hegedus & Roschelle 2002). See also Penuel (2008) in this Handbook.

III. Alignment of Graphing Calculators with Standards and Practices

There is a growing consensus among state and national agencies that graphing calculators are a valuable tool for enhancing math learning for all students. The U.S. National Council of Teachers of Mathematics (NCTM) has advocated for the integration of calculators in mathematics education since the publication of their landmark Standards (a.k.a. Curriculum and Evaluations Standards for School Mathematics) in 1989. A recent scan of state education policies reveals that more than half of the states address appropriate calculator use as part of their mathematics standards. Indeed, a handful of states now require the use of calculators on high stakes accountability examinations. In Texas, for example, the use of graphing calculators is required in high school mathematics courses and on statewide assessments:

“The Texas Essential Knowledge and Skills (TEKS) mathematics curriculum implemented in 1996 requires the use of graphing calculators in high school mathematics courses. As a result, graphing calculators will be used on the statewide mathematics assessments for students in Grades 9, 10, and 11, providing alignment

4 among curriculum, instruction, and assessment. In May 2001 a letter was sent to districts stating that "a sufficient supply of calculators must be available so that each student in Grades 9, 10, and 11 has ready access to a graphing calculator, not only on the day of [TAKS] testing, but also for routine class work and practice.” (Retrieved from: http://www.tea.state.tx.us/taa/studass022502.html)

While the NCTM and state agencies advocate for the use of graphing calculators, they also recognize that calculators do not obviate the need for students to learn certain fundamental math skills. Educators and researchers alike firmly believe that non- calculator based skills—such as number sense, skill in estimation, and mental arithmetic —are important for both the workplace and further mathematics learning. Indeed, no one argues that calculators should be used to supplant learning of basic arithmetic, fractions, or other core aspects of mathematical fluency.

Nonetheless, calculators can both support students’ basic mathematical understanding and facilitate their success with more advanced mathematical concepts. In the early grades, calculators can be used in some creative ways to strengthen basic arithmetic skills, as with the classic “broken calculator” exercise (see for example, Collison, Collison, & Schwartz, et al. 2006). In the later grades, where most calculator use actually occurs, calculators can enable students to move beyond basic calculations toward a deeper understanding of math concepts. Accordingly, the NCTM recognizes the need for balance between manual computation and other problem solving techniques:

“[S]tudents no longer have the same need to perform these procedures with large numbers or lengthy expressions that they might have had in the past without ready access to technology…The teacher should help students learn when to use a calculator and when not to, when to use pencil and paper, and when to do something in their heads. Students should become fluent in making decisions about which approach to use for different situations and proficient in using their chosen method to solve a wide range of problems.” (From NCTM 2005 position paper http://www.nctm.org/about/position_statements/computation.htm)

5 A number of countries in Europe, including Austria and the Netherlands, as well as progressive states such as Ontario, Canada and Victoria, Australia are moving to a new level of the calculator debate. By incorporating calculators with “computer algebra systems” that can automatically transform symbolic expressions they are raising the question of how much effort students should devote to manipulating algebraic expressions versus focusing on meaning and problem solving with algebraic expressions.

IV. Pedagogical Affordances of Graphing Calculators

Graphing calculators have a powerful potential to help students master important concepts in mathematics because they provide specific functionalities that are valuable for math learning. They realize this potential well because they are because they are inexpensive, portable, and readily adaptable to existing classroom practices. Employed as an instructional technology, graphing calculators can enable teachers to foster a problem solving approach to mathematics and help students to reason mathematically. The unique contributions of graphing calculators to problem solving and reasoning can include:

 Increasing attention to conceptual understanding and problem solving strategies by offloading laborious computations;  Enabling students to hone their understanding by tackling more than “textbook examples” (those than can be completed in under 5 minutes with paper and pencil).  Examining the related meanings of a concept through the display of multiple representations, such as exploring rate of change (i.e. slope) in a function definition, a corresponding graph and a table of values;  Engaging students with interactive explorations, real world data collection, and more authentic data sets;  Giving students more responsibility for checking their work and justifying their solutions.  Providing a supportive context for productive mathematical thinking.

[add one or more references to support the above list]

6 Underlying these pedagogical affordances, we see several basic cognitive contributions of calculators to student learning. One prominent factor involves reducing cognitive load and allowing students to focus more attention on high-level thinking (Sweller, 1988). Students with calculators can take on traditional tasks in new ways and also tackle new topics that would otherwise be inaccessible. Rather than laboring over tedious calculations, classes that use calculators can devote more time to developing students’ mathematical understanding, their number sense, and their ability to evaluate the reasonableness of proposed solutions. Students can also use calculators to explore concepts and data sets that would otherwise be too complex or cumbersome. For example, students can easily investigate, graphically, numerically, and symbolically, the effects of changing a, b, and c on the graph of ax2 + bx + c, which can be quite tedious using paper and pencil graphing techniques.

A second basic cognitive factor is the complementarity of textual/linguistic, tabular, and graphical representations. According to a number of recent studies, students can often reason best when they experience mathematics through related representations, such as equations, tables, and graphs (Ellington, 2003; Khoju, Jaciw, &and Miller, 2005). Graphing calculators can make constructing and using multiple representations easier, allowing students to spend more of their time and intellectual energy exploring the underlying concepts. In addition, technology can link the representations, enabling students to make conceptual connections, such as understanding how a change in an equation links to a change in a graph. Standard mathematical representations can also be linked to other visualization aids, fostering further conceptual understanding.

A third cognitive factor is that calculators appear to enable students to engage in higher order mathematical reasoning. For example, because problem solving with a calculator saves time, students are more able to explore multiple solution strategies. Research has also shown that students using graphing calculators change their approaches to problem solving: they explore more and their attempted solution strategies are more flexible (Ellington, 2003; Khoju, Jaciw, &and Miller, 2005). In general, students who use calculators better understand variables and functions and are better able to solve algebra problems in applied contexts than students who do not use calculators (Schwarz &and Hershkowitz, 1999). Similarly, students who use calculators use graphs more often and

7 interpret graphs better than students who do not regularly use the technology (Hollar &and Norwood, 1999). Finally, students who use calculators are better able to move among varied representations—that is from graphs to table to equations—than students who do not have access to the technology (Ruthven, 1990). Clearly, students who regularly use calculators have an advantage over those who do not.

V. Research on Graphing Calculators

A strong body of literature provides evidence that graphing calculators can effectively improve mathematics achievement for a wide variety of students. Importantly, graphing calculator research also illuminates effective instructional practices and conditions of successful implementation. This research provides educators and policy makers with concrete guidance on how to achieve an effective implementation and confidence that large-scale implementations will also be successful. (Note: some of the research cited below does not discriminate between calculators and graphing calculators. We use the term “calculator” to include any form of calculator and reserve “graphing calculator” for those calculators that include graphing features.)

In the United States, the National Assessment of Education Progress (NAEP) samples both 4th and 8th graders throughout the country and measures how many students perform at proficient and advanced levels in mathematics. This research has consistently shown that frequent use of [graphing?] calculators at the eighth grade level (but not at the fourth grade level) is associated with greater mathematics achievement, stating:

Eighth-graders whose teachers reported that [graphing?] calculators were used almost every day scored highest. Weekly use was also associated with higher average scores than less frequent use. In addition, teachers who permitted unrestricted use of calculators and those who permitted calculator use on tests had eighth-graders with higher average scores than did teachers who did not indicate such use of calculators in their classrooms. The association between frequent graphing calculator use and high achievement holds for both richer and poorer students, for both girls and boys, for varied students with varied race and ethnicity, and across states with varied policies and curricula (National Center for Education Statistics, 2001, p. 144).

A study by Heller (2005) corroborates the NAEP findings. Heller examined a model implementation for high school students, which included a new textbook, teacher professional development, and assessment tools—all aligned with the graphing

8 technology by the theme of Dynamic Algebra. This study shows that daily use of graphing calculators is generally more effective than infrequent use, and establishes that the teachers and students who used graphing calculators most frequently learned the most.

An Example from New Zealand Researchers in different settings have investigated the effectiveness of graphing calculators in relation to students, teachers, and schools with diverse characteristics. Alan Graham and Michael Thomas, for example, examined the effectiveness of graphing calculators in algebra classrooms in New Zealand (Graham &and Thomas, 2000). The study compared pretest and posttest scores for year 9 and year 10 students in treatment and control group classrooms in two schools. In all of the classrooms, the regular classroom teacher taught the “Tapping into Algebra” curriculum module. In treatment group classrooms, each of the students received a graphing calculator to use throughout the module; in control group classrooms, students did not use graphing calculators. Students in all classrooms had similar background characteristics and math abilities. Graham and Thomas found that students in the treatment groups performed significantly better than students in the control groups on the posttest examination.

Meta-Analyses show the effectiveness of graphing calculators Meta-analysis is a technique that enables researchers to statistically summarize the results of a large body of experimental studies, yielding a robust estimate of true effectiveness. A meta-analysis by Ellington (2003) summarized 54 classroom experiments, of which 80% employed some form of random assignment of students to experimental groups (using [graphing?] calculators) and control groups (not using c[graphing?] calculators). Random assignment is a key component of true experimental studies, as it allows social scientists to make strong causal inferences with the fewest threats to experimental validity (Cook & Shadish, 1986). Ellington’s analysis shows a positive effect of graphing calculator-based interventions on student achievement. The effects are substantial, often increasing an average student’s achievement by 10 to 20 percentile points (Ellington, 2003). In addition, the studies suggest that when graphing calculators are allowed on tests, gains extend from calculations and operations to conceptual understanding and problem solving. Ellington’s summary includes a wide variety of

9 grade levels, socio-economic backgrounds, geographic locations, and mathematical topics, suggesting that the effectiveness of calculators holds true in a variety of contexts.

A second meta-analysis looked specifically at aAlgebra. Khoju, Jaciw, and Miller (2005) screened available research using stringent quality-control criteria published by the U.S. Department of Education’s What Works Clearinghouse. They found four suitable studies that examined the impact of graphing calculators on aAlgebra learning. Across a wide variety of student populations and teaching conditions, use of graphing calculators with aligned instructional materials was shown to have a strong, positive effect on aAlgebra achievement.

Why have [graphing?] calculators been so successful? A number of key features contribute to the success of graphing calculators in bolstering math learning. Calculators and gGraphing calculators are relatively simple, robust and cheap; they are also remarkably free of much of the complexity that accompanies full- featured computers. More importantly, there is a deep scientific linkage between the capabilities of the technology and how people learn. Students learn more when cognitive load is focused on the most important learning challenges (Sweller, 1988), when linked multiple representations make different features of a mathematical object available to students perception (Kaput, 1992), and when students have more time to focus on the strategic and problem solving aspects of mathematics (Ellington, 2003). [add reference].

Two less readily obvious factors also contribute to the success of graphing calculators. First, the adoption of the technology has been led by practicing teachers, who function as the key champions and influencers in a professional community (Ferrio, 2004). Second, efforts to integrate graphing calculators into classrooms did not begin with the expectation of a rapidly transformed classroom, but rather provided a context to support a long, steady trajectory of continuous improvement (Demana &and Waits, 1997). In this way, teachers can begin with one or two relatively simple applications of the technology, and gradually increase the depth and breadth of their calculator integration as they grow more comfortable with the technology. At each stage, graphing calculators can provide concrete enhancements for teaching and learning math.

10 VI. Discussion and Conclusion

Graphing calculators are an important case of successful large-scale adoption of IT in education with a robust and consistent research base that links technology use to increased student achievement. Further, new technological capabilities, including multiple, dynamic representations and connectivity, offer additional opportunities to transform teaching and learning in mathematics, making critical math concepts accessible to all students. If educators and policy makers want to improve students’ mathematics achievement, graphing calculators—in conjunction with aligned curricula, instructional practices and professional development—can provide an inexpensive and effective solution that is feasible and ready for large-scale implementation. Moreover, if members of the educational technology community want to provide new technologies that will have a real impact on learning, they would do well to heed the lessons and successes of the graphing calculator.

11 References

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Burrill, G., J. Allison, et al. (2002). Handheld Graphing Technology in Secondary Mathematics: Research Findings and Implications for Classroom Practice. Dallas, TX, Texas Instruments.

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