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Why Are Mathematical Concepts Difficult to Understand

Posted by Ravi Kumar at Thursday, July 7, 2011
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Mathematical concept means just about anything with a mathematical name. For example, some of the mathematical concepts we learn in high school are: constant, variable, polynomial, factor, factoring, equation, solving an equation, logarithm, sine, cosine, tangent, etc., point, line, triangle, square, and other geometric figures, area, perimeter of a geometric figure, etc., and many others. Among the mathematical concepts we learn in our first years of college mathematics are: set, operation, limit, function, and, specifically, continuous function, derivative, integral, theorem, proof, countable infinity, uncountable infinity, algebra, linear algebra, vector space, group, ring, field, and many others.

Now one thing that makes the understanding of these concepts difficult is that they are defined in terms of other concepts.

Thus, e.g., a vector space is defined in terms of the concepts of vector, set, function, abelian group, field, and others. How does the typical mathematics textbook, and mathematics course, deal with this fact? It attempts to teach the concepts in logical order, i.e., it assumes that, e.g., when you begin your study of vector spaces, you will already know — through having remembered what you learned in previous courses — the meaning of each of the concepts in terms of which a vector space is defined. And, indeed, one of the things that makes mathematics such a frightening subject to many students, is the grandiose manner with which these assumptions are set forth in the list of prerequisites for the course.

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