Math Wars in Massachusetts
The Battle Over The Mathematics Curriculum Framework
Bass Attack (Link to document at
- Background Information:
The Constructivist Philosophy of The
Massachusetts Mathematics Curriculum Framework
To understand the battle over the 2000 Massachusetts Mathematics
(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
MMCF failed to identify the specific, grade-by-grade math content that
should be taught and learned during the K-12 years. Instead, it
Constructivists don't believe it's right to pre-specify what kids
learn. This in turn forces them to reject specific content standards,
testing, and traditional teaching methods. The 1995 MMCF
the extended constructivist agenda:
- Belief that each child must be allowed to follow their own
personally discover the math knowledge that they find interesting and
to their own lives.
Belief that knowledge should be acquired as a byproduct of social
in real-world settings.
- Rejection of the concept of a common core of basic math
all children should learn during the K-12 years.
- The 1995 MMCF recommends "moving away from the notion that
be mastered before proceeding to higher level mathematics".
- Rejection of the traditional process of math education whereby
ask questions and present problems that have been carefully chosen to
students to discover teacher-targeted math knowledge.
Belief in the primary importance of general, content-independent
- Devaluation of classroom learning and learning from books.
- Emphasis on knowledge that is needed for everyday living.
Belief that calculators have fundamentally changed the nature of
- Rejection of the need to remember the specific facts and skills
- Belief in the sufficiency of "just-in-time" factual knowledge,
to be readily available through reference materials and computer
Belief that learning must always be an enjoyable, happy
knowledge emerging naturally from games and group activities.
- Rejection of the need to learn the standard algorithms of
- Failure to recognize the foundational nature of the
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.
Jr. Here you will learn about the difficulty of publishing an
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
- 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
is based on the following assumptions:
- Math is a man-made abstraction that only exists in the human mind
- There is an established body of math knowledge that different
understand in the same correct way.
- There is a stable foundational "K-12 math subset" that can be
by different K-12 students in the same correct way.
- K-12 math teachers can lead K-12 students to a correct
The traditional reasons for learning math are:
- For the practical math-related needs of daily life.
- To develop the power of the mind to think logically and
- To experience the step-by-step process of building a remembed
base, relative to a structured knowledge domain.
- As foundational knowledge for learning more advanced math.
- As foundational knowledge for more advanced learning in the many
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
build a remembered knowledge base of domain-specific content.
this knowledge base consists of math facts tightly linked to math
Through a carefully structured learning process, each child
comes to understand hundreds of specific ideas that have been
organized, and agreed-to by countless contributors over thousands of
With teachers who know math and sound methods of knowledge
the child remembers more and more math, while moving deeper and deeper
into the structured knowledge domain that comprises traditional K-12
This first disciplined knowledge-building experience is a key enabler,
developing the memorizing and organizing skills of the mind, and
helping to prepare the individual to eventually build remembered
bases relative to other knowledge domains in the professions, business,
or personal life.
Understanding the Process of Building a Personal
- 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
It becomes increasingly easier to build new knowledge on the already
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
Then the brain can do its magic, leading to what we call
Newly remembered knowledge is integrated with previously remembered
and "understanding" evolves. It may happen instantly, or it may take
- Remembered math is most effective if it is encoded using the
- The language and symbols of math have developed over hundreds
- Math is no place for "expressing it in your own personal way".
How is K-12 Math Traditionally Taught?
Under constructivist attack, but used with great success in Asia:
- Use a coherent, lesson-by-lesson curriculum based on specific
- Know math yourself. You can't teach math if you don't know math.
know the underlying "whys" and how to build math knowledge, one concept
at a time, in a step-by-step manner.
Present math facts and demonstrate math skills.
- The recent (1999) book, Knowing and Teaching Elementary Math
Liping Ma, compared U.S. and Chinese teachers' knowledge of four topics
in elementary math. There was "a striking contrast" with the
teachers demonstrating "algorithmic competence as well as a conceptual
understanding of all four topics". "Considered as a whole, the
of Chinese teachers seemed clearly coherent while that of the U.S.
was clearly fragmented."
Orient the students.
- Today's educationists deplore "teaching by telling", but
changes the very definition of "to teach".
Examples, examples, examples.
Continually ask questions to test understanding.
Give immediate and constant feedback.
Make every effort to help students remember math facts
- Whenever possible, present a new math topic by relating it to a
context of previously learned math knowledge.
Constantly encourage students to practice their
- Math thinking cannot occur without remembered math facts and
Encourage students to speak and write, using mathematically
Expect instant recall for basic math facts only.
- Repetition fixes knowledge in memory. It is the key to
without conscious thought). Instant recall is essential for basic math
- Students can't get bogged down continually "reconstructing" 7
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.
Make careful use of "manipulatives".
- Don't make them think through a multi-step process in their
- Don't make them worry about conserving paper.
Make careful use of written instructional materials. Good written
should be succinct and closely tied to genuine math standards. They
- Concrete materials are teaching aids that can be useful at the
when the child's math knowledge base is initially empty.
- The goal is to discard such "crutches" as soon as possible. Get
to think abstractly and visualize in the mind.
- Manipulatives cannot be used to "prove" math facts. They can be
discussion aids and motivational tools.
- Prolonged reliance on concrete "pacifiers" interferes with the
of genuine math.
Help kids recognize that the challenges of genuine math are
rewarding in ways that go well beyond the "math must be fun and easy"
of the constructivist philosophy.
Above all, guide your students to correct understanding. Don't
walk away thinking that 9 times 7 equals 97.
- Provide an orienting framework.
- Clearly explain key concepts.
- Invite the student to fill in gaps (with knowledge remembered
- Encourage the student to practice.
G. Quirk, Ph.D.