Math Wars in Massachusetts

• The Bass Attack   (Link to document at mathematicallycorrect.com)
• Massachusetts Curriculum Frameworks
• NCTM Letter to the Massachusetts Commissioner of Education and the Board of Education
• Background Information:
• The Constructivist Philosophy of The 1995 Massachusetts Mathematics Curriculum Framework    Versus
• The Traditional Philosophy of Math Education

The Constructivist Philosophy of The 1995 Massachusetts Mathematics Curriculum Framework

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.
• Rejection of the concept of a common core of basic math knowledge that all children should learn during the K-12 years.
• The 1995 MMCF recommends "moving away from the notion that basics must be mastered before proceeding to higher level mathematics".
• Rejection of the traditional process of math education whereby teachers ask questions and present problems that have been carefully chosen to lead students to discover teacher-targeted math knowledge.
2. Belief that knowledge should be acquired as a byproduct of social interaction in real-world settings.
• Devaluation of classroom learning and learning from books.
• Emphasis on knowledge that is needed for everyday living.
3. Belief in the primary importance of general, content-independent "process" skills.
• Rejection of the need to remember the specific facts and skills of elementary mathematics.
• Belief in the sufficiency of "just-in-time" factual knowledge, believed to be readily available through reference materials and computer information-accessing tools.
4. Belief that calculators have fundamentally changed the nature of math.
• Rejection of the need to learn the standard algorithms of arithmetic.
• Failure to recognize the foundational nature of the standard algorithms of arithmetic.
5. Belief that learning must always be an enjoyable, happy experience, with knowledge emerging naturally from games and group activities.
• Rejection of the need for memorization and practice.
• Rejection of attempts to challenge a child to work harder.

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

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

• Progress is slow at the beginning:
• Each of us begins with no remembered math knowledge.
• Initially, the mind has no orientation information.
• Progress is  faster and faster as the knowledge base grows:
• As knowledge grows, the mind has an increasingly richer frame of reference. It becomes increasingly easier to build new knowledge on the already existing and ever expanding remembered math knowledge base.
• "Understanding" grows as the knowledge base grows
• Newly acquired knowledge helps to clarify old knowledge.
• Frequently we just need to memorize, to get the knowledge in our brain. Then the brain can do its magic, leading to what we call "understanding". Newly remembered knowledge is integrated with previously remembered knowledge and "understanding" evolves. It may happen instantly, or it may take years.
• Remembered math is most effective if it is encoded using the precise language of math:
• The language and symbols of math have developed over hundreds of years.
• Math is no place for "expressing it in your own personal way".

How is K-12 Math Traditionally Taught?

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.
• The recent (1999) book, Knowing and Teaching Elementary Math by Liping Ma, compared U.S. and Chinese teachers' knowledge of four topics in elementary math.  There was "a striking contrast" with the Chinese teachers demonstrating "algorithmic competence as well as a conceptual understanding of all four topics".  "Considered as a whole, the knowledge of Chinese teachers seemed clearly coherent while that of the U.S. teachers was clearly fragmented."
3. Present math facts and demonstrate math skills.
• Today's educationists deplore "teaching by telling", but eliminating telling changes the very definition of "to teach".
4. Orient the students.
• Whenever possible, present a new math topic by relating it to a familiar context of previously learned math knowledge.
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.
• Math thinking cannot occur without remembered math facts and skills.
9. Constantly encourage students to practice their developing math knowledge.
• Repetition fixes knowledge in memory. It is the key to reflexive use (use without conscious thought). Instant recall is essential for basic math facts.
• Students can't get bogged down continually "reconstructing" 7 times 9 equals 63. They have to achieve automatic use of such facts. Then their minds will be free to focus on the next level of math knowledge.
10. Encourage students to speak and write, using mathematically correct language and symbols.
11. Expect instant recall for basic math facts only.
• Don't make them think through a multi-step process in their heads.
• Don't make them worry about conserving paper.
12. Make careful use of "manipulatives".
• Concrete materials are teaching aids that can be useful at the beginning, when the child's math knowledge base is initially empty.
• The goal is to discard such "crutches" as soon as possible. Get the child to think abstractly and visualize in the mind.
• Manipulatives cannot be used to "prove" math facts. They can be used as discussion aids and motivational tools.
• Prolonged reliance on concrete "pacifiers" interferes with the learning of genuine math.
13. Make careful use of written instructional materials. Good written materials should be succinct and closely tied to genuine math standards. They should:
• Provide an orienting framework.
• Clearly explain key concepts.
• Invite the student to fill in gaps (with knowledge remembered from earlier lessons).
• Encourage the student to practice.
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.

Next?

• TERC Hands-On Math: The Truth is in the Details
• The Truth About The Revised NCTM Standards
• NCTM Math in the NCEE America's Choice Performance Standards
• The Anti-Content Mindset: The Root Cause of the "Math Wars"
• Memorize Multiplication Facts?  Cheney, Yes.  Romberg, Abstain.
• The Truth About the NCTM Standards
• Searching For The Truth About The TIMSS 4th Grade Math Test
• The Truth About Math Standards and Information-Age Math Reform (Home)

 Copyright 2000-2002  William G. Quirk, Ph.D.