COMPUTER ALGEBRA SYSTEMS - TECHNOLOGY IN MATHEMATICS EDUCATION

Solving Equations:

Solve: 3x+5=9. The student enters the equation, then appropriately subtracts 5, then divides 3, arriving at the correct solution of x=4/3 or 1 1/3. The answer, of course, checks.

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PEDAGOGICAL CAS OR Pecas

Teaching tip:

The students who catch on quickly can be challenged to choose

the equivalence transformations in their head and enter as one step on the calculator

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Teaching tip:

As the students gain facility and familiarity with the key strokes and sequences of the algebraic procedures, challenge them to discover shortcuts for entering the expressions in the calculator.

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If, however, the student enters an incorrect equivalence transformation, e.g., subtract 3, she has the instant feedback to reflect on it and make the necessary corrections:

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Or perhaps the student chooses to divide by 4 as the last step, again the desired result of isolating the x is not achieved and hence the student can reflect on the process and select another equivalence transformation.

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In my classes I continue to witness countless ways in which this technology is helping my students gain greater understanding in exactly the skills the calculator can do for them, i.e., symbolic computation. The following examples will demonstrate how the calculator is a far more effective “trainer” than “chalk and board” methods. To solve equations, for example, the emphasis in using technology is in learning the equivalence transformations inherent in this basic algebraic topic. The numeracy, i.e., getting the correct answers throughout the equation solving process, is taken over by the calculator temporarily so that the student can concentrate on the underlying structures and principles of solving equations. Once the student has mastered the analytical skills necessary for solving equations the calculator can be “removed” and the student should be ready for “by hand” solving of equations.

Solving Systems of Equations:

There are two basic algebraic methods for solving systems of equations. The method of cancellation or elimination is shown here as one would perform by hand. Once again the “burden of numeracy” is temporarily removed so that the student can concentrate on the analytical manipulation.

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And the method of substitution using the above equations:

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White Box/Black Box Model

CAS as a “white box”

CAS as a “black box”

- Mathematical Modeling

- Picture mathematics