Chapter 9.11
Graphing Calculators: Enhancing Math Learning for All Students
Jeremy Roschelle
Corrine Singleton
SRI International
jeremy.roschelle@sri.com
corrine.singleton@sri.com
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.
Keywords: Mathematics, Multiple representations, Handhelds, Conceptual Understanding
1. Introduction
Educators and policy makers want to improve students’ mathematics achievement,
particularly in middle and high schools. A consistent association between frequent calculator
use 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
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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, & Means, 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, 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.
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 [NCTM],
2000) and they are embedded in some curricula. Furthermore, research has identified best
practices for instruction (Burrill & Allison, 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.
2 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.
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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
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. 1996; Roschelle, Penuel, &
Abrahamson , 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, Kaput,
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Ares, Wilensky, Hegedus & Roschelle 2002). See also Penuel (2008) in this Handbook.
3 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 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.” (Texas Education Agency, n.d.)
While the NCTM and state agencies advocate 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
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skills, as with the classic “broken calculator” exercise (see for example, Collison, Collison, &
Schwartz, 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.” (NCTM, 2005)
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.
4 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
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”
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(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.
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 labouring 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, & 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
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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, & 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 &
Hershkowitz, 1999). Similarly, students who use calculators use graphs more often and
interpret graphs better than students who do not regularly use the technology (Hollar &
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.
5 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 largescale 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)
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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 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 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 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.
Researchers in different settings have investigated the effectiveness of graphing
calculators in relation to students, teachers, and schools with diverse characteristics.
Graham and Thomas, for example, examined the effectiveness of graphing calculators in
algebra classrooms in New Zealand (Graham & 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 student
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.
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Meta-Analyses Show the Effectiveness of Graphing Calculators
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
calculators) and control groups (not using calculators). 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 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 algebra. 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 algebra 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 algebra achievement.
Why have calculators been so successful?
A number of key features contribute to the success of calculators in bolstering math
learning. Calculators and graphing 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
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aspects of mathematics (Ellington, 2003).
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 & 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.
6 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.
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