Explaining WHY in Mathematics

Recently the mother of a prospective student told me that they had selected my College Algebra class because she and her son had interviewed former students, other teachers at my school, and advisers at my school.  What they learned from this due diligence was that I always explained the “WHY” in mathematics and did not simple show the “HOW”.  Of course I pleaded guilty to subjecting my students to that additional level of difficulty.

Later, while driving somewhere, I began thinking about that conversation and it all seemed quite clear.  The following three sentences explain why my teaching style is what it is now and what it has always been.  Don’t anticipate a change.

  • The “WHY” in college algebra is always the underlying algebraic structure.
  • The foundation of the underlying structure is the stipulative definitions used in the structure.
  • Because the definitions are stipulative, as opposed to the more common lexical (extracted) definition, learning these definitions must begin with memorization.

 

Here is an example of how those three points translate into a teaching style.

We begin by learning the definitions of

  • equation
  • solution of an equation
  • solution set of an equation
  • equivalent equations

We then learn two methods of generating equivalent equations.

We use those ideas and simple deductive reasoning to develop an unquestionable and foolproof method for solving linear equations in one variable.

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