The Transitive Property Groups Many Problem Types

Share Cognitive Science Implications for Teaching Mathematics

Fundamental Premise: It is more productive to understand one logic process which solves a hundred different problem types than to memorize a separate method (gimmick) for each of the hundred problem types.

  • I believe the idea contained in the Fundamental Premise cannot be denied.
  • It is highly probable that understanding a logic process permits extension to new situations.
  • It is unlikely that a single-purpose gimmick can be extended to new situations.
  • A logic process which is understood is likely to be remembered.
  • Single-purpose gimmicks are forgotten immediately after the test.
  • A logic process helps the student understand how mathematics principles are used.
  • Each of The Big Four provide excellent illustrations of the Fundamental Premise.
  • Reread the posts

Study the posts of 29 and 31 January to see how The Law of Trichotomy and The Zero Factor Property illustrate the Fundamental Premise.  Study the remainder of this post to see how The Transitive Property illustrates the Fundamental Premise.

Beginning students need lots of help reading and understanding statements heavy with symbols. Instructors should help students come to understand that the Transitive Property, in simple language states,

If two things are equal to a third thing, then they are equal to one another.

The assumption that students will recognize an application of the Transitive Property is faulty.  Instructors must watch for, point out, and discuss every application of the Transitive Property.

From the time I was a high school student (and probably earlier) there was one class of problems which were dreaded by all students.  These were the so-called: word problems, application problems, real-life problems, or modeling problems.  Under the umbrella of Word Problems, we find many problem types each with some gimmick for solving that particular problem type.  A few familiar names are:

  • Distance, Rate, Time
  • Work
  • Percent
  • Present Value
  • Mixture

In these and all the other Word problems found in beginning algebra courses the solution process involves finding two different expressions for the same quantity.  The Transitive Property demands that these two expressions are equal.  The resulting equation is solved to provide the answer to the Word problem.

The book I expect to complete during 2019 will contain 18 – 20 chapters illustrating the use of The Transitive Property to solve many problem types.  Watch for some preview material in the Pages section of this blog.

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