Explaining the Roles of Definitions in Mathematics V

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Role #5: They promote understanding.    See 21 Feb blog.

Personally, my first thought about the role of definitions in mathematics is that they are necessary to facilitate communication, but after a moment’s reflection I also realize that my understanding of a mathematics concept is quite dependent on the related definitions and so it must be for my students.

Years of teaching and carefully observing student behavior have convinced me that many of the common difficulties experienced by students when trying to learn mathematics is their lack of understanding of the requisite stipulative definition.

As I have stated in a previous blog, many students in beginning algebra classes refer to each of the following as equations: 3x +7 = 4, 3x +7 > 4, 3x +7 < 4, or 3x +7.  If their understanding of the word equation is that flawed, they will not understand discussions involving the word equation and will have difficulty understanding equations and related concepts.  This difficulty can and should be cleared up with explicit instruction of the definition of equation.

A common conceptual problem is understanding that an equal sign ( = ) refers to equality.  It is frequently interpreted as “put the answer here” or some other nonsense.  This concept becomes more difficult for students when we begin speaking of equality of complex numbers, equality of polynomials, equality of matrices, etc. This difficulty can and should be cleared up with explicit instruction of the definition of equality.  That definition must be adjusted to match the mathematics creatures under discussion.

Some astute thoughts about definitions promoting understanding in mathematics are presented by Barbara Edwards and Michael Ward in this PDF file.  For those of you who do not have time or inclination to read the PDF, I will present some of their major points.

Throughout this cited paper, and other papers by Edwards and Ward, they emphasize the difference between lexical and stipulative definitions and in fact one of their other studies established a key issue related to understanding mathematics is failure, on the part of the student, to understand the difference between lexical and stipulative definitions.  Edwards and Ward claim that the most common and obvious reason for including definition activities in mathematics instruction is to promote a deeper conceptual understanding of the mathematics.  This claim is in line with finding by Freudenthal in Mathematics as an Educational Task (1973).

Many students do not understand the role of definitions in mathematics and that prevents deep conceptual understanding. Therefore, that role of definitions should be explicitly taught.

In summary the authors advocate that:

  • Definitions should be taught.
  • Difference between lexical and stipulative definitions should be taught.
  • The role of definitions in mathematics should be taught.

Three final notes:

  1. Over the years I have read several papers by these two authors and find them to be trustworthy researchers.
  2. This paper includes an impressive reference list.
  3. I am particularly proud of the fact that work of one my former students (Dr. Anne Brown) is the fourth citation by Edwards and Ward.

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