Using Calculators for Students in Grades 9-12: Geometry. Re-published with permission from American Institutes for Research

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1 Using Calculators for Students in Grades 9-12: Geometry Re-published with permission from American Institutes for Research

2 Using Calculators for Students in Grades 9-12: Geometry By: Center for Implementing Technology in Education (CITEd) Should high school students use calculators? Can calculators help students learn to think through geometry? The National Council of Teachers of Mathematics defines geometry as a capacity to "analyze characteristics and properties of two - and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships; specify locations and describe spatial relationships using coordinate geometry and other representational systems; apply transformations and use symmetry to analyze mathematical situations; and use visualization, spatial reasoning, and geometric modeling to solve problems." For more information about these standards, see Following are examples of how some researchers think calculators can be incorporated into math lessons to help high school students learn these geometry-related tasks. Before describing these works, a quick caveat is needed. None of the studies presented here used a research design that shows unambiguous evidence that calculators lead to better student achievement. Indeed, many of the articles summarized below simply describe ideas that some teachers think were helpful. Despite the limited evidence, it is important to let teachers know about what types of calculator practices are available, and judge for themselves if any of these fit with their style and specific classroom needs. Curriculum Studies The journal Mathematics Teacher offers a series of ideas for using calculators to teach geometry. For example, Barnes (1996) discusses how a graphing calculator can be used to teach students certain principals such as properties of polygons and the origin of Pi. The author notes that using a graphing calculator can not only demonstrate certain geometric principles, but can also teach students how to choose the appropriate functions/windows within the calculator. Another article by Hurwitz (1997) presents a lesson she uses to help students use a graphing calculator to understand the proof of the mean-value theorem by applying a

3 graphing calculator to teaching geometry. Although no data are presented Cyrus & Flora (2000) describe using the T1-92 geometry package to "discover a formula for determining the area of any given equilateral triangle" (p. 564). Embse & Engebretsen (1996) describe the use of the TI-82 and TI-92, combined with other technologies, to promote students' understanding of math concepts and problem-solving techniques. This article reviews how to apply a real-world phenomenon (a basketball free throw) to understand a math model. The article gives a description of exercises teachers could use with their classes and shows that technology allows students to "model problems long before they occur in the traditional secondary curriculum." Berthgold (2004/2005) uses a sewing concept and a graphing calculator to illustrate that the illusion of a curve can be made by stitching a sequence of strategically placed straight stitches. Students explore the relationship between straight lines and curves, develop connections between linear and quadratic functions, and produce equations of parabolas by using quadratic regression on a graphing calculator. Holliday & Duff (2004) describe how to use graphing calculators to model realworld data. They state that "students plot a data set in which the two variables have an apparent linear correlation with each other, compute and graph the line of best fit, and use the equation of best fit to determine interpolated and extrapolated data. They also identify within a data resource a real-world data set that approximates a linear relation and apply or transfer their understanding to a new condition, with decreasing teacher support" (p. 328). The article provides a teacher's guide with information about prerequisites, materials, objectives, and four sheets with actual questions and activities; solutions are also provided for each of the activity sheets. Of course, ideas for how to use calculators when teaching geometry can be found elsewhere. Farrell (1995) describes how to apply graphing calculators to algebra and trigonometry by rotating matrices. Felsager (2001) describes a practice using the TI-83 calculator in a science class to map stars using mathematical calculations. Focus is on content instruction in astronomy and introduction of graphing calculator functions. Another article provides ideas on how to rotate triangles, using a TI-92, to explore their various properties (Marty 1997). This article describes a series of exercises to explore the center of various triangles using the

4 Cabri Geometry on the TI-92 (including a listing of the geometry menus for the calculator in the appendix). For each exercise, there are instructions about which property of the calculator the student should explore, a definition of each term, and questions meant to stimulate further student exploration and thinking. DiCarlucci (1995) describes an activity he uses with secondary students to help them use a graphing calculator to analyze mathematical models. He first has them use matchsticks to solve a problem, and then use graphing calculators to understand the geometry, algebra, and trigonometry concepts also used to solve the problem. The author does not describe the characteristics of the students in the classroom in which he used this activity; however, he does mention that it is "accessible to every student regardless of mathematical background or talent" (p. 147). As we stated above, there appear to be many ideas for how to use calculators to teach geometry in grades 9-12, but few studies that produce evidence for whether calculators actually lead to better outcomes. There is, however, a study that used a case-study approach (i.e., intensive study of a single student, in this case a tenth-grader) to see if calculators make a difference in understanding trigonometry (Choi-Coh 2003). Specific research questions were: What patterns of thought processes did the student exhibit while working on technology-based explorations? What was the nature of the learning experience the student encountered during technology-based explorations? What was the role of the connections among multiple representations that the student used? What were the merits and limits of the graphing calculator used in trigonometry? Data "included observations of instructional tasks, clinical interviews with the student, and an audiotape of his learning processes [through a think-aloud process]" (p. 362). The study noted the student had lost interest in the subject since he began high school, but otherwise did not explicitly describe his math achievement. The main finding was that a series of graphing calculator questions helped the teacher facilitate conceptual understanding about trigonometry properties. The calculator served as a visualization tool that allowed the student to relate graphs to formulas. As he progressed to higher order thinking in math, he depended less on this technology to solve problems. Of course, CITEd will continue to seek further evidence about the benefit of calculator use in teaching the broad topic of geometry. In the meantime, we

5 hope some ideas presented here are useful, and encourage you to seek out the references below for more information. References Barnes, S. (1996). Perimeters, patterns, and pi. Mathematics Teacher, 89(4), Berthgold, T. (2004/2005). Curve stitching: Linking linear and quadratic functions. Mathematics Teacher. 98(5), Choi-Coh, S.S. (2003). Effect of a graphing calculator on a 10th grade student's study of trigonometry. The Journal of Educational Research, 96(6), Cyrus, V. F., & Flora, B. V. (2000). Don't teach technology, teach with technology. Mathematics Teacher, 93(7), DiCarlucci, J. A. (1995). Dynamic polygons and the graphing calculator. School Science and Mathematics, 95(3), Embse, C. V. & Engebretsen, A. (1996). A mathematical look at a free throw using technology. Mathematics Teacher, 89(9), Farrell, A. M. (1995). Matrices draw connections for precalculus students. Ohio Journal of School Mathematics, (30), Felsager, B. (2001). Mapping stars with TI-83. Micromath, 17(2), Holliday, B.W. & Duff, L.R. (2004) Using graphing calculators to model real world data. Mathematics Teacher, 97(5) Hurwitz, M. (1997). Visualizing the proof of the mean-value theorem for derivatives. Mathematics Teacher, 90(1), Marty, J. F. (1997). Centers of triangles exploration on the TI-92. Wisconsin Teacher of Mathematics, 48(2),

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