**ERIC Identifier:** ED433183

**Publication Date:** 1998-02-00

**Author: **Smith, Jeffrey P.

**Source: **ERIC Clearinghouse for
Science Mathematics and Environmental Education Columbus OH.

## Graphing Calculators in the Mathematics Classroom. ERIC Digest.

As predicted by the National Council of Teachers of Mathematics (NCTM) in
their series of reform documents ("Curriculum and Evaluation Standards," 1989;
"Professional Standards," 1991; "Assessment Standards," 1995), graphing
calculators have presented a challenging task for mathematics teachers in
today's classrooms. Faced with the power of a small and easy-to-use computer,
teachers must reconsider the mathematical content, educational methods, and
assessment strategies that they employ. In a world of accessible technology, the
very nature of mathematics and what it means "to do mathematics" is called into
question. Not unlike the painstaking infusion of four-function calculators into
the elementary curriculum, graphing calculators are redefining the notion of
demonstrated knowledge in secondary mathematics.

With calculators that can do so much more than simple arithmetic, increasing
attention has been devoted to developing materials that enrich learning
experiences in mathematics. Four distinct types of enrichment have been
identified, and this digest provides a sampling of references for each. Graphing
calculators can be used as: (1) tools for expediency, (2) amplifiers for
conceptual understanding, (3) catalysts for critical thinking, and (4) vehicles
for integration.

### USING GRAPHIC CALCULATORS AS TOOLS FOR EXPEDIENCY

It is not
surprising that students miss the main objective of a lesson when they are
caught in a trap of tedious computation or the point-by-point plotting of a
complex graph. Without argument, the graphing calculator reduces the time and
effort required to perform cumbersome mathematical tasks.

"A Look at Parabolas with a Graphing Calculator" describes an exploration
involving quadratic equations. The TI-85 graphing calculator is used to solve
routine problems associated with second-degree polynomials. In particular, the
menu and matrices capabilities of the calculator are emphasized, highlighting
the heuristic of solving a system of equations without the burden of multi-step
calculations.

Johnson, L.H. (1997, April). A look at parabolas with a graphing calculator.
"Mathematics Teacher," 90(4), 278-282.

"Sold on a New Machine" presents opinions of high school teachers on using
the graphing calculator in lower and upper level math courses. An underlying
theme centers on allowing students to graph equations that would, if plotted
manually, consume hours of important instructional time.

West, P. (1991, October). Sold on a new machine. "Teacher Magazine," 3(2),
18-19.

"Retaining a Problem-Solving Focus in the Technology Revolution" offers
guidelines for determining when to use mental math, paper-and-pencil, or
technology-assisted methods when attacking mathematical problems. Determining
the rational roots of a polynomial provides a means for comparing the graphing
calculator approach to other solution strategies. Suggestions for curricular
planning are included.

Duren, P.E. (1989, October). Retaining a problem-solving focus in the technology revolution. "Mathematics Teacher," 82(7), 508-510.

### USING GRAPHING CALCULATORS AS AMPLIFIERS FOR CONCEPTUAL UNDERSTANDING

Graphing calculators are capable of providing
multiple representations of mathematical concepts. By building tables, tracing
along curves, and zooming in on critical points, students may be able to process
information in a more varied and meaningful way. To enhance understanding, it
appears that working with representations that expose diverse aspects of a
concept is critical.

"Sharing Teaching Ideas: The Graphing Calculator and Division of Fractions"
suggests using a graphing calculator to demonstrate that dividing a fraction is
the same as multiplying by the reciprocal. Primarily intended for a middle
school audience, this article describes an activity that has students graphing a
division problem and an equivalent problem (multiplying by the reciprocal). The
use of a graphic representation for each of the problems provides convincing
evidence that these two arithmetic processes are identical.

Pelech, J., & Parker, J. (1996, April). Sharingteaching ideas: The graphing calculator and division of fractions. "Mathematics Teacher," 89(4), 304-305.

"The Effect of Graphing Calculators on Students' Conceptions of Function"
presents the results of a year-long study of topics in an Algebra II classroom.
Participating students were taking a course that made significant use of
graphing calculators, and the positive effects of graphing calculator usage are
presented, particularly on tasks that required graphical thinking.
Misconceptions resulting directly from the use of graphing calculators are also
discussed.

Slavit, D. (1994, April). "The effect of graphing calculators on students' conceptions of function." [ED 374 811]

"Student Understanding of Basic Calculus Concepts: Interaction with the
Graphics Calculator" describes the intuitive notions of five college and two
high school students regarding function and limit. Episodes where student
comprehension seemed to be influenced by the availability of graphing
calculators illustrate the power of multiple representations in deepening
mathematical understanding.

Lauten, A. D., et. al. (1994, June). Student understanding of basic calculus concepts: Interaction with the graphics calculator. "Journal of mathematical behavior," 13(2), 225-237.

"Multiple Representations: Using Different Perspectives to Form a Clearer
Picture" presents a teaching unit that intentionally introduces a variety of
methods for solving quadratic inequalities. During teacher-led instruction,
students were shown how to use cases and critical numbers as strategies in
finding solutions to quadratic inequalities. In addition to the symbolic method
demonstrated, students were able to view pictoral representations of the
inequalities on their graphing calculators. When students worked on their own,
most chose graphical methods for finding solutions.

Piez, C. M., & Voxman, M. H. (1997, February). Multiplerepresentations: Using different perspectives to form a clearer picture. "Mathematics Teacher," 90(2), 164-166.

"Explorations: Discovering Math on the TI-92" provides twelve innovative and
practical activities by practicing teachers that illustrate how to begin using
the TI-92 in high school classrooms. Many different content areas are covered,
from beginning algebra to advanced calculus. Each lab purposefully uses multiple
representations for learning about a topic, and most offer suggestions for
extending the investigations.

Brueningsen, C., et. al. (1996). "Explorations: Discovering math on the TI-92." Austin, TX: Texas Instruments Incorporated.

USING GRAPHING CALCULATORS AS CATALYSTS FOR CRITICAL THINKING

When wearisome computation and plotting tasks areminimized, students can
become engaged in answering "whatif" questions. The thought of changing the
premises of amathematical argument grows more attractive if the chore
ofexecuting those changes is easier. Moreover, the graphingcalculator promotes
autonomy in asking questions,encouraging students to pose their own problems.

"Problem-Based Mathematics -- Not Just for the College-Bound" contains an
abridged description of The Interactive Mathematics Program, a comprehensive,
standards-based high school mathematics curriculum. Aimed at replacing the
traditional sequence of mathematics courses, IMP emphasizes using graphing
calculators to enhance students' critical thinking skills. Produced by a
National Science Foundation initiative, this four year program integrates
content strands and focuses on problem-solving processes.

Alper, L., et. al. (1996, May). Problem-based mathematics - Not just for the college-bound. "Educational leadership," 53(8), 18-21.

"Investigating a Definite Integral - From Graphing Calculator to Rigorous
Proof" suggests that the graphing calculator can be used as a springboard for
discovery. While learning how to calculate definite integrals, students in an
advanced calculus class proposed their own theorem concerning integration. While
using the graphing calculators to investigate quick solutions to problems, the
students formulated conjectures that eventually led to a rigorous proof.

Touval, A. (1997, March). Investigating a definite integral - from graphing calculator to rigorous proof."Mathematics Teacher," 90(3), 230-232.

"Explorations: 92 Geometric Explorations on the TI-92" is an activity book
that motivates geometry students to look for patterns, form conjectures, and
justify arguments. Using the dynamic geometry capabilities of the TI-92,
students become interested in explaining why objects relate to each other in
specific ways. The underlying tenet of "proof as explaining" surfaces in each
lab, whether students are uncovering basic theorems or creating advanced
constructions. Although not written as a replacement for textbooks, Explorations
provides supplementary teaching material that strongly supports critical
thinking.

Keyton, M. (1996). "Explorations: 92 geometric explorations on the TI-92." Austin, TX: Texas Instruments Incorporated.

### USING GRAPHING CALCULATORS AS VEHICLES FOR INTEGRATION

Many
math educators view "integration" as an ill-defined term. Does it mean to bridge
the many content strands within mathematics? Is it the marriage of mathematics
to other disciplines? Can it be used to describe connecting mathematics to the
real world? Luckily, despite how the word is ultimately construed, the graphing
calculator supports each of these aspects of integration.

"Teaching Discrete Mathematics with Graphing Calculators" challenges the
belief that graphing calculators are most useful in classes built around
continuous topics, such as precalculus and calculus. In fact, rather than
looking at smooth curves, the author provides examples of graphing calculator
lessons that can help to connect ideas of algebra, geometry, measurement, and
probability and statistics. In addition to utilizing the power of being able to
graph functions, this article contains activities that help the reader learn
about the programming capabilities of graphing calculators.

Masat, F. E. (1994). "Teaching discrete mathematics with graphing calculators." ERIC Document: ED 380 282.

"Modeling Motion: High School Math Activities with the CBR and Math and
Science in Motion: Activities for Middle School" presents physics labs that
involve using the graphing calculator in conjunction with the Calculator-Based
Ranger (CBR). The CBR is a stand-alone motion data collection device that sends
information to a TI graphing calculator for analysis. Although the CBR works
with most Texas Instruments graphing calculators, these books provide detailed
instructions for analysis on either the TI-82 or the TI-83.

Antinone, L., et. al. (1997). "Modeling motion: High school math activities with the CBR." Austin, TX: Texas Instruments Incorporated.

Brueningsen, C., et. al. (1997). "Math and science in motion: Activities for middle school." Austin, TX: TexasInstruments Incorporated.

"Real-World Math with the CBL System" contains 25 activities that use
Calculator-Based Lab technology. After connecting the CBL to any Texas
Instruments graphing calculator, probes can be selected that collect data
related to a variety of scientific phenomenon (everything from light intensity
to pH levels). This workbook was designed to provide math students, from algebra
through calculus, with innovative ways to explore real-world applications of
mathematical concepts.

Brueningsen, C., et. al. (1995). "Real-world math with the CBL system." Austin, TX: Texas Instruments Incorporated.

### SUMMARY

The resources listed here constitute only a small
sample of those available. Readers may find that any article or book about the
use of graphing calculators in mathematics teaching could easily fit into more
than one of the four categories delineated (or perhaps an entirely new category
may be suggested). Regardless of how the materials are classified, however, it
is important to note that all of the works manifest a common philosophy, a
philosophy that the graphing calculator is an instrument for student
empowerment.

### REFERENCES

National Council of Teachers of Mathematics.
(1989). "Curriculum and evaluation standards." Reston, VA: Author.

National Council of Teachers of Mathematics. (1991). "Professional standards
for teaching mathematics." Reston, VA: Author.

National Council of Teachers of Mathematics. (1995). "Assessment standards
for school mathematics." Reston, VA: Author.

WORLD WIDE WEB RESOURCES

Math Forum: Internet Calculator Resources

http://forum.swarthmore.edu/mathed/calculator.search.html