1. “Procedural” is not enough B. Abramovitz, M. Berezina, A. Berman, L. Shvartsman 4 th MEDITERRANEAN CONFERENCE ON MATHEMATICS EDUCATION University of Palermo - Italy 28 – 30 January 2005

2. Department of Mathematics, Ort Braude College, Karmiel, Israel Department of Mathematics, Technion, Haifa, Israel

3. 1/29/20053 Teaching mathematics to engineering students has a twofold purpose We want the students to learn to use mathematics and we want them to understand mathematics. In other words we teach them to compute and we teach them to think. The procedural part of the teaching deals with computational algorithmic techniques: how to solve a system of linear equations, how to compute the gradient of a function, how to compute an integral. The conceptual part of the teaching deals with the understanding of the theory behind the computation: why is the theorem correct, what happens if some of the assumptions are changed, why does an algorithm work.

4. 1/29/20054 Many teachers and researchers agree that the procedural part is not enough We strongly believe that a good engineer should develop mathematical thinking, which will help him to solve problems he has not seen before. The conceptual thinking develops creativity in problem solving. The understanding of a proof improves the student’s logical thinking and communication abilities. Unfortunately, most engineering students are only interested in the computational part of the course and believe that solving many technical problems will be sufficient to succeed at the exam.

5. 1/29/20055 In order to encourage the students to pay more attention to the theoretical part of the course, we developed several problems that we used during the courses and in the exams. The student cannot solve such problems if you do not explain enough the theoretical material. We give examples of such problems in Calculus and Linear Algebra. In each example the problem is given in a “procedural form” and in a “conceptual form”.

6. 1/29/20056 An example in Calculus The concepts addressed in this section are: tangent line, composite function and in particular the Chain Rule. The procedural version can be used to check if the student knows the Chain Rule and how to find the tangent line. In the conceptual form a technical use of the Chain Rule is not sufficient, and the student has to fully understand the concept of the composite function and the theorem on its derivative. In this version the student also has to be able to use the tangent line to get information on the function.

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10. 10 An example in Linear Algebra We give two problems related to the well- known theorem on the relation between the solutions of a non homogeneous linear system and the related homogeneous one. In the procedural form the student has only to know the Gaussian Elimination method and Vector Addition. The aim of this problem is to give the student the main idea of the theorem without formulating and proving the result. The problem in the conceptual form can be readily solved if the student knows both the theorem and its proof and is able also to apply them.

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15. 1/29/200515 An additional example in Calculus

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19. 1/29/200519 An additional example in Linear Algebra

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