Graphing calculator

For the software tool of the same name, see NuCalc.

A graphing calculator (also known as a graphic calculator or graphical calculator) typically refers to a class of handheld calculators that are capable of plotting graphs, solving simultaneous equations, and performing numerous other tasks with variables. Most popular graphing calculators are also programmable, allowing the user to create customized programs, typically for scientific/engineering and education applications. Due to their large displays intended for graphing, they can also accommodate several lines of text and calculations at a time. Some graphing calculators also have color displays, and others may even include 3D graphing.

Since graphing calculators are readily user-programmable, such calculators are also widely used for gaming purposes, with a sizable body of user-created game software on most popular platforms.

There is also computer software available to emulate or perform the functions of a graphing calculator. One such example is Grapher for Mac OS X and is a basic software graphic calculator.

History

Casio produced the world's first graphic calculator, the fx-7000G in 1985. After Casio, Hewlett Packard followed shortly in the form of the HP-28C. This was followed by the HP-28S (1988), HP-48SX (1990), HP-48S (1991), and many other models. The current top-of-the line model, the HP 50g (2006), features a Computer Algebra System (CAS) capable of manipulating symbolic expressions and analytic solving. The HP-28 and -48 range were primarily meant for the professional science/engineering markets; the HP-38/39/40 were sold in the high school/college educational market; while the HP-49 series cater to both educational and professional customers of all levels. The HP series of graphing calculators is best known for their Reverse Polish Notation interface, although the HP-49 introduced a standard expression entry interface as well.

Texas Instruments has produced models of graphing calculators since 1990, the oldest of which was the TI-81. Some of the newer calculators are just like it, only with larger amounts of memory, such as the TI-82, TI-83 series, (including the TI-83, TI-83 Plus, and TI-83 Plus Silver Edition), and the TI-84 Plus series (including the TI-84 Plus and TI-84 Plus Silver Edition). Other models, designed to be appropriate for students 10–14 years of age, are the TI-80 and TI-73 series. Other TI graphing calculators have been designed to be appropriate for calculus, namely the TI-85, TI-86, TI-89 series, and TI-92 series, (including the TI-92, TI-92 Plus, and Voyage 200). TI offers a computer algebra system on the TI-89 and TI-92 series models with the TI-92 series having a QWERTY keypad. TI calculators are targeted specifically to the educational market, but are also widely available to the general public.

Graphing calculators are also manufactured by Sharp but they do not have the online communities, user-websites and collections of programs like the other brands.

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Graphing Calculators and Mathematics Learning

Graphing Calculators and Calculus Reform

The relationship between graphing calculators and calculus reform movements across the United States and around the world is one of co-evolution and bi-directional influence. The results of this integral relationship are evidenced in the prominent features common to most graphing calculators today and in the shifted emphasis on topics and approaches in calculus, especially as taught in secondary schools. For most students of the past, the predominant memory of calculus is of procedurally manipulating symbols.

Increasingly, educators are creating student experiences that are characterized instead by substantive mental connections between visual representations and conceptual understanding. Graphing calculators have been instrumental to enabling these critical connections. In his 1996 book chapter entitled, Much More than a Toy: Graphing Calculators in Secondary School Calculus, Thomas P. Dick explains why he believes graphing calculators will continue to catalyze important changes in calculus education.

To begin with, graphing calculators provide students with a sense of ownership in mathematics by giving them the power to personally create and manipulate graphs and model mathematical principles. This simple but important factor is contributing to a change in the dynamics of calculus classrooms.

Further, enhanced calculator functionalities have been instrumental in enabling a multi-representational approach to calculus education. Standard graphing calculators 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). Some of the most valuable supports for student learning derived from these new functionalities include depicting graphs at varying levels of magnification, illustrating slope fields, and rapidly calculating sequences of related expressions.

Take for example the concept of the derivative. Students can use a graphing calculator to numerically calculate approximate difference quotients quickly and easily, thereby strengthening their understanding of function approximations. Zooming capabilities within graphs enable students to explore the notion of local linearity—a key property of differential functions and the foundation of so many of the important concepts, results, and applications of calculus. By zooming in on the graph of a differentiable function, it is possible for students to actually see what happens to that function at a particular localization. This visualization helps students to understand the notion of a tangent line as a linear approximation of a function at a given point. Further, students can see that the tangent line is both a unique and optimal linear approximation of a function at a given point. In this way, graphs generated by the technology become a tool for illustrating the meaning of the derivative, rather than the mere output of symbolic calculations of the derivative.

While graphing calculators present an incredible opportunity for students to visualize math concepts, the fact that they are not infinitely precise does present some inherent limitations. Most notably, the discrete nature of calculators means that the visual display of a function partly depends on external factors. For example, in order for a hole in the graph of a function to appear, it must be within the parameters of the screen, and it must fall precisely on a pixel location. Nonetheless, graphing calculators offer truly new tools for teaching and learning calculus—tools that support both affective and cognitive aspects of student learning.

Instruction and some questions on the AP Calculus examination ^[1] require the use of graphing calculators which can perform interactive graphical analysis including numerical derivatives and integrals. Students who enroll in courses that require calculators should check which type will be used in their course or test.

Graphing Calculators in School Mathematics

The success of the graphing calculator in teaching and learning calculus has been followed by a much broader adoption of the technology for other topics in secondary school mathematics, such as algebra and trigonometry. Since the graphing calculator did not co-evolve with reforms in other aspects of mathematics teaching, it is reasonable to question the extent to which the broader use might have a positive impact. Connected Mathematics requires the use of graphic calculators in grades 7 and 8. One challenge is the cost of purchase these calculators which can cost over $100. The teacher may also need to learn how to use graphing calculators as such devices were not available when the teachers learned math.^[2] Such programs have also met opposition on the basis of problems associated with the reform mathematics approach in general.

Some educational research indicates that graphing calculators can benefits for mathematics learning. Research shows that students using graphing calculators develop flexible strategies for problem solving and a deeper appreciation of mathematical meaning than students who do not use graphing calculators (Ellington, 2003; Khoju, Jaciw, and Miller, 2005). In addition, students who use graphing calculators are better able to understand variables and functions, solve algebra problems in applied contexts, interpret graphs, and move among varied representations—that is from graphs to tables to equations—than students who do not have access to the technology.

Some question if students may become too reliant on calculators even for basic arithmetic, and they have been banned on some assessments such as TAAS in Texas. Countering this perception, some large-scale educational studies and meta-analyses associate graphing calculators and greater mathematics achievement. Research from the National Progress (NAEP) has consistently shown that at the eighth grade level frequent use of calculators is associated with greater mathematics achievement (Figure 1). Moreover, research shows that teachers and students who used graphing calculators most frequently learned the most. On the NAEP assessment, 8th graders whose teachers reported that calculators were used almost every day scored the highest (NCES, 2001). Similarly, in examining an implementation that aligned graphing technology with a comprehensive math curriculum, Heller found that daily use of graphing calculators is more effective than infrequent use (Heller, 2005).

Unlike many educational technologies, graphing calculators have gone beyond isolated implementations to achieve large-scale success. Two factors appear to contribute to the breadth of this success: a) graphing calculators are useful and usable in a wide variety of settings, and b) the form-factor, cost, and maintenance characteristics of the technology itself are conducive to wide-spread adoption. Research shows that the association between frequent graphing calculator use and high achievement holds true for a wide variety of grade levels, socio-economic backgrounds, geographic locations, and mathematical topics; the finding also holds across states with varied policies and curricula (National Center for Education Statistics, 2001; Ellington, 2003). As a technology, graphing calculators are simple, robust, and relatively inexpensive and they are aligned with curricula, instructional practices, and assessments. Relevant teacher professional development is widely available, and teachers can integrate graphing calculators into their classroom practice gradually, benefiting from concrete enhancements for teaching and learning math at each stage.

Non-mathematical uses of graphing calculators

The programming features of nearly every major graphing calculator on the market have been exploited to produce games of various sorts. Imitations of Tetris and Pacman are among the most popular. A variety of other non-technical applications have been written for graphing calculators as well. Among these include organizers, phonebooks and text editors. A software solution also exists for using the infrared port on the HP-48 series of calculators as a remote control for televisions, and those calculators with built-in speakers have been transformed into monophonic music sequencers. As a result of such programs, their use in schools has also received a great degree of criticism as it is extremely common to find that students have downloaded non-educational programs onto their calculators, presenting a potential distraction in the classroom.

Another major criticism of graphing calculators by school teachers is their ability to store large amounts of text in the same memory that is used to store programs. Such a feature presents a potential for students to cheat on examinations by storing notes and solutions on their calculators. While some enforce a rule by which students must perform a supervised memory clear of their calculator before an exam, this has become an increasingly difficult problem as the variety of available brands and models increases and false memory clear programs are released over the internet to deceive the proctor. In addition, many students use the calculator's memory to store useful programs, particularly those which improve the mathematical functionality of their calculators to be on par with other newer models, and requiring such students to clear their calculator memories would put them at a disadvantage. On the other hand, many courses have disallowed calculators on examinations altogether, and designing the assignment appropriately to purely test conceptual knowledge. Others argue that graphing calculators are too expensive. For example, if one compares a one hundred dollar graphing calculator (or any graphing calculator of arbitrary price) to a cell phone, GPS device, or PDA of equal price, one finds that the cell phone or other device outperforms the graphing calculator in terms of hardware (faster CPU and more memory). A new TI83+ typically costs $100 and has a 6 mhz processor. For $100 one can get a PDA with about 200 mhz and far more memory (and a color screen). Opponents of this view argue that graphing calculators are more reliable because they last longer and that they also use less energy allowing them to use alkaline batteries which are far cheaper than the lithium ion - batteries that PDA and other devices typically use. The next generation of graphing calculators (ie: the TI-Nspire) may also help alleviate this criticism.

References

Dick, Thomas P. (1996). Much More than a Toy. Graphing Calculators in Secondary school Calculus. In P. Gómez and B. Waits (Eds.), Roles of Calculators in the Classroom pp 31-46). Una Empresa Docente.

Ellington, A. J. (2003). A meta-analysis of the effects of calculators on students' achievement and attitude levels in precollege mathematics classes. Journal for Research in Mathematics Education. 34(5), 433-463.

Heller, J. L., Curtis, D. A., Jaffe, R., & Verboncoeur, C. J. (2005). Impact of handheld graphing calculator use on student achievement in algebra 1: Heller Research Associates.

Khoju, M., Jaciw, A., & Miller, G. I. (2005). Effectiveness of graphing calculators in K-12 mathematics achievement: A systematic review. Palo Alto, CA: Empirical Education, Inc.

National Center for Education Statistics. (2001). The nation’s report card: Mathematics 2000. (No. NCES 2001-571). Washington DC: U.S. Department of Education.

External links

casiocalc.org – A forum for discussing Casio calculators.
hpcalc.org – Another calculator program archive, but for HP calculators.
ticalc.org – A comprehensive archive of TI graphing calculator programs.
ticalcs.net – TI calculator forums, wiki, and downloads

[1] AP Calculus Sample Questions

[2] Connected Mathematics

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