Graphing Calculators: The Newest
Revolution in Mathematics

When I began teaching in 1970, however, I had never seen any hand-held calculator, much less used one to teach math. My first experience with a calculator was during the summer of 1975 when a student asked if he could use a calculator on the final examination. Since the rest of the class did not have access to one, I denied the student’s request. Several years later, I attended a workshop on using calculators in mathematics instruction, and shortly thereafter I began using a calculator in the classroom, mainly for trigonometry, because it eliminated the tedious four-digit decimal calculations necessary and it allowed more testing options for me and fewer arithmetic errors for the students. We were using the calculator to carry out "number-crunching" tasks as described in the first phase of the technological era.

These early hand-held calculators also began to change methods of instruction. They freed up enormous amounts of class time by eliminating the teaching of interpolation of tables, the algorithm for finding square roots, and other such skills. Teachers also became more willing to try more complicated problems since the calculations were easy to perform.

During the 1980s, calculators became more sophisticated; they were programmable and had more functions. In the mid-1980s, I saw my first graphing calculator. It had a small screen with limited memory and was quite slow, but it drew graphs. I was hooked. I began using the graphing calculator for classroom demonstrations, again changing my method of instruction. I could now use the graphing calculator to demonstrate relationships. For example, using the graphing calculator to examine the behavior of families of functions eliminates tedious point plotting and allows the students to see multiple graphs and compare them to discover patterns for themselves. However, students are still expected to know the basic shapes of polynomial functions, generalize about turning points and x-intercepts, and sketch some graphs without point plotting. The graphing calculator is used for more complicated functions that previously could not have been graphed efficiently. The use of graphing calculators goes beyond just "number-crunching"; now we are asking students to use the graphing calculator to draw conclusions about what they are learning. The graphing calculator can also help students see the connection between graphs (geometric representations) and algebra (symbolic representations). It allows for solutions of equations which cannot be solved algebraically, such as "solve sin(x) = x^2 + 1." Once students understand that roots of equations show up as x-intercepts of graphs, drawing a graph can easily approximate the solutions of equations.

The 1990s have seen the graphing calculator become much more powerful, culminating in 1997 with the TI-92 ¯ virtually a hand-held computer. It should be noted that even as the TI-92 came on the market, Texas Instruments was hard at work developing an improved model. The TI-92 has several features that set it apart from other graphing calculators: symbolic algebra, three-dimensional graphing, and symbolic calculus. In other words, the TI-92 can do any algebra or calculus operation currently taught in mathematics textbooks. It will also graph functions of two variables in three dimensions, a skill that, on a two-dimensional piece of paper, takes quite a bit of insight to accomplish.

There are definite advantages to using graphing calculators in the mathematics classroom. Instructional presentations are much more dynamic and are not restricted to examples that "work out nicely." By using the graphing calculator to discover properties for themselves, students can be more actively engaged in the learning process. And the graphing calculator connects the graphic, algebraic, and numerical approaches to many topics.

However, using graphing calculators, particularly the powerful TI-92, raises some related questions and concerns. One of my concerns is that students are not taught when it is appropriate to use a graphing calculator. These machines are not crutches to use for every mathematical computation; rather, they are tools to perform some computations more efficiently. A second problem arises because there are so many models of graphing calculators now available. How does an instructor, or a student, know exactly which one to use? This knowledge is especially crucial because of the rate at which new models are currently being introduced. Another concern is course content. If calculators can do everything that the textbook presents, how do instructors decide what topics to teach and what topics to eliminate? The major mathematical organizations (NCTM, AMATYC, and MAA) wrestle with this question on a regular basis. And, finally, there is the question of evaluation. Tests that just measure computational skills are out-dated if students have access to graphing calculators that do the work for them. Instructors must change the focus of their questions. For example:

In other words, testing must now evaluate higher-level thinking skills as well as computational ability.

As graphing calculators become more accessible and more sophisticated, professional mathematics organizations and reformers in mathematics education are constantly studying their usefulness in mathematics instruction. The National Council of Teachers of Mathematics (NCTM) and the American Mathematical Association of Two-Year Colleges (AMATYC) have both developed standards which provide guidelines for mathematics instruction in grades K-12 and in undergraduate precalculus mathematics. The guidelines for community college instruction are published in Crossroads in Mathematics. Reform efforts in calculus among four-year institutions also emphasize the "Rule of Three": every mathematical topic should be presented from the symbolic, algebraic, and graphic approach. The graphing calculator is stressed as a tool to help visualize concepts. Thus, we are not going through this revolution blindly; rather, we are striving toward a common goal: to educate our students and to enable them to have the mathematical power necessary to function in today and tomorrow’s society. As Eliot Masie stated in his keynote address at the 1998 New Horizons Conference, "The ability to create new technology is growing faster than our ability to train people to use it." Changes in technology will happen whether we want them to or not. We need to be ready to help our students develop the skills they will need to deal with whatever comes next.

Works Cited