The Texas Instruments TI-92 as a vehicle for the teaching and learning of functions, graphs, and analytic geometry.

Background

The C2PC project, now some 10 years old, is an ongoing curriculum-development and teacher-enhancement effort directed by Franklin Demana and Bert K. Waits. The teacher-enhancement component is based on intensive week-long summer inservice courses that have been attended by thousands of high school teachers at locations throughout the United States. These courses are now taught by a cadre of high school teachers as a part of the Teachers Teaching with Technology (T3) program, headquartered at the University of Texas at Arlington. The curriculum-development component has led to a series of high school and college textbooks. The first regular edition of the high school text was Demana and Waits (1990). From the beginning, the curriculum has been designed to help students acquire the skills and understandings necessary for the successful study of calculus and science. The text materials focus on functions and graphs because students' lack of understanding of graphing and functions accounts for a major portion of their difficulties in calculus and because focusing students' attention on graphs and functions improves their readiness for calculus.

Demana, Waits, Clemens, and Foley (1997) is a significant revision and reworking of prior editions. It reflects the experience gained from two and a half years of pilot- and field-testing together with the seven-year maturing process during the use of earlier editions. The maturing process has been influenced by American Mathematical Association of Two-Year Colleges (1996) and National Council of Teachers of Mathematics (1989) recommendations and the calculus reform movement in the U.S. To provide balance and support for the focus on functions and graphs, the curriculum develops skills and understandings in algebra and trigonometry and helps students learn how to model real-life problems.

Numerous research studies have been conducted related to the project. Some examples are Browning (1989), P. Dunham and Osborne (1991), and Quesada and Maxwell (1994). Several other related investigations are summarized in P. Dunham and Dick (1994).

The distinctive features of the C2PC curriculum include realistic applications and genuine data (à la Freudenthal, 1991); the linking of verbal, algebraic, numerical, and graphical representations; the use graphing calculators and associated technology; a systematic approach to problem solving; and the development of the conceptual underpinnings of calculus. Use of the graphing calculator is integrated throughout. This technology is used to help visualize and solve problems and develop students' graph viewing skills. The table-building, matrix, and statistical utilities are exploited as well. The mathematical emphases embraced are viewing windows and scale, local behavior of functions, end behavior of functions, graphical-numerical solutions, geometric transformations, and multiple linked representations.

The systematic problem-solving process that recurs in the solutions to many of the examples throughout the curriculum materials relies on the linkage of the verbal, algebraic, numerical, and graphical representations of problems. Students are expected to read the problem statement for understanding, to determine the problem situation, what answer is being sought, and the relevant data or conditions, and often to draw a figure. The second step is to model the problem. Next the problem is solved algebraically, or if this is difficult or impossible, the problem is solved graphically or numerically. Often a second method is used to support or confirm the solution. If the second method is algebraic, we use the verb confirm to emphasize that the method logically and firmly establishes the result. On the other hand, if the second solution method is numerical or graphical, we use the verb support to suggest that these grapher methods are not exact but only approximate and that they do not establish the result but only provide corroborating evidence. The final step of interpretation calls to mind Pólya's (1957) step of looking back. To interpret means to put the mathematical result back in the context of the verbal problem situation.

Throughout this problem-solving process, students make directional links among verbal, algebraic, numerical, and graphical representations. Figure 1 shows the 12 directional links. The problem situation hovers over the others as the overarching context. The algebraic representation is at the center of the diagram to indicate the central and determining role of algebra. The three mathematical representations--algebraic, numerical, and graphical--are closely linked as suggested by the tight triangle they form in the figure. It is within this triangle that the solving, confirming, and supporting occur. The numerical and graphical nodes are beneath the algebraic to convey that numerical and graphical methods support the algebra. The three inward links drawn to the algebra node suggest the modeling phase. The five upward links suggest the various steps that may be involved in making an interpretation of the solution; often we must pass through the algebra to get meaning from tables and graphs.
Figure 1. Links between representations of a problem

Foley - 2 OCT 1996



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Gómez, P. & Waits, B. (Eds.) (1996). Roles of calculators in the classroom.

Mail comments to Pedro Gómez: pgomez@uniandes.edu.co