Math Wars in Massachusetts

The Battle Over The Mathematics Curriculum Framework

by Bill Quirk   (E-Mail:

The Constructivist Philosophy of The 1995 Massachusetts Mathematics Curriculum Framework

To understand the battle over the 2000 Massachusetts Mathematics Framework (2000 MMCF), with its specific math content standards, it is necessary to understand the philosophy of its predecessor, the 1995 MMCF.

The 1995 MMCF was a "child" of The NCTM Standards, from The National Council of Teachers of Mathematics.  Similar to the parent, the 1995 MMCF failed to identify the specific, grade-by-grade math content that should be taught and learned during the K-12 years. Instead, it promoted constructivist pedagogy.

Constructivists don't believe it's right to pre-specify what kids should learn. This in turn forces them to reject specific content standards, standardized testing,  and traditional teaching methods.  The 1995 MMCF promotes the extended constructivist agenda:

  1. Belief that each child must be allowed to follow their own interests to personally discover the math knowledge that they find interesting and relevant to their own lives.
  2. Belief that knowledge should be acquired as a byproduct of social interaction in real-world settings.
  3. Belief in the primary importance of general, content-independent "process" skills.
  4. Belief that calculators have fundamentally changed the nature of math.
  5. Belief that learning must always be an enjoyable, happy experience, with knowledge emerging naturally from games and group activities.
The constructivist philosophy is not backed up by research in cognitive psychology. Quite the opposite! For more information see the Address to The California State Board of Education by Professor E.D. Hirsch, Jr.  Here you will learn about the difficulty of publishing an important research article, Applications and Misapplications of Cognitive Psychology to Mathematics Education , even though one of the co-authors, Herbert A. Simon, is a winner of the Nobel Prize.

The Traditional Philosophy of Math Education

The traditional "knowledge transmission" philosophy of K-12 math education is based on the following assumptions:

  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.

The traditional reasons for learning math are:

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

Learning Math Means to Build a Remembered Math Knowledge Base

Traditionally, K-12 math is the first man-made knowledge domain where children build a remembered knowledge base of domain-specific content. Conceptually, this knowledge base consists of math facts tightly linked to math skills.

Through a carefully structured learning process, each child gradually comes to understand hundreds of specific ideas that have been developed, organized, and agreed-to by countless contributors over thousands of years. With teachers who know math and sound methods of knowledge transmission, the child remembers more and more math, while moving deeper and deeper into the structured knowledge domain that comprises traditional K-12 math. This first disciplined knowledge-building experience is a key enabler, developing the memorizing and organizing skills of the mind, and thereby helping to prepare the individual to eventually build remembered knowledge bases relative to other knowledge domains in the professions, business, or personal life.

Understanding the Process of Building a Personal Math Knowledge Base

How is K-12 Math Traditionally Taught?

Under constructivist attack, but used with great success in Asia:
  1. Use a coherent, lesson-by-lesson curriculum based on specific math content standards.
  2. Know math yourself. You can't teach math if you don't know math. You should know the underlying "whys" and how to build math knowledge, one concept at a time, in a step-by-step manner.
  3. Present math facts and demonstrate math skills.
  4. Orient the students.
  5. Examples, examples, examples.
  6. Continually ask questions to test understanding.
  7. Give immediate and constant feedback.
  8. Make every effort to help students remember math facts and math skills.
  9. Constantly encourage students to practice their developing math knowledge.
  10. Encourage students to speak and write, using mathematically correct language and symbols.
  11. Expect instant recall for basic math facts only.
  12. Make careful use of "manipulatives".
  13. Make careful use of written instructional materials. Good written materials should be succinct and closely tied to genuine math standards. They should:
  14. Help kids recognize that the challenges of genuine math are exciting and rewarding in ways that go well beyond the "math must be fun and easy" visualization of the constructivist philosophy.
  15. Above all, guide your students to correct understanding. Don't let them walk away thinking that 9 times 7 equals 97.

     Copyright 2000-2011  William G. Quirk, Ph.D.