Traditional K-12 Math Education

Chapter 1 of Understanding the Original NCTM Standards By Bill Quirk



Knowledge Transmission

This chapter outlines the traditional "knowledge transmission" philosophy of K-12 math education. The fundamental assumptions are:
  1. Math is a man-made abstraction that only exists in the human mind or in written form.
  2. There is an established body of math knowledge that  different people can understand in the same correct way.
  3. There is a stable foundational "K-12 math subset" that  can be understood by different K-12 students in the same correct way.
  4. K-12 math teachers can lead K-12 students to a correct understanding of K-12 math.

Why is K-12 Math Learned?

The traditional reasons are:

  1. For the practical math-related needs of daily life.
  2. To prepare for occupations that use math.
  3. To develop the power of the mind to think logically and abstractly.
  4. To experience the step-by-step process of building a remembered knowledge base, relative to a structured knowledge domain.
  5. As foundational knowledge for learning more advanced math.
  6. As foundational knowledge for more advanced learning in the many knowledge domains that use math to communicate the ideas of the domain.

How is K12 Math Traditionally Learned?

Learning Math Means to Build a Remembered Math Knowledge Base

Traditionalists believe that learning math is a process of building a personal math knowledge base that is stored in the brain. Conceptually, this knowledge base consists of math facts tightly linked to math skills. Math is a structured domain. "New" math facts are "built on" established math facts. For example, now we can use Theorem 1: 2 + 2 = 4 to establish the "new" Theorem 2: 2 + 3 = 5
  1. 2 + 2 = 4, by Theorem 1.
  2. (2 + 2) + 1 = 4 + 1, by the "add equals to equals" axiom.
  3. 2 + (2 + 1) = 4 + 1, by the "associative property of addition" axiom.
  4. 2 + 3 = 5, by the definitions of 3 and 5 and "substitution of equals for equals."
Math thinking involves the mind in a question and answer process. This process depends on remembered math facts and remembered math skills. This is not a simple matter of remembering a few facts. As the following example illustrates, it requires remembering and understanding a whole series of linked facts. If there are gaps in knowledge, the whole process breaks down.

Example:  Find the equation of a straight line that passes through the points (1,2) and (-1,4).


Understanding the Process of Building a Personal Math Knowledge Base


How is K-12 Math Traditionally Taught?


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Copyright 1997-2011  William G. Quirk, Ph.D.