Math Wars in Massachusetts
The Battle Over The Mathematics Curriculum Framework
- The
Bass Attack (Link to document at
mathematicallycorrect.com)
- Background Information:
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:
- 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.
- 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.
- 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.
- 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.
- 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 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:
- Math is a man-made abstraction that only exists in the human mind
or in
written form.
- There is an established body of math knowledge that different
people
can
understand in the same correct way.
- There is a stable foundational "K-12 math subset" that can be
understood
by different K-12 students in the same correct way.
- K-12 math teachers can lead K-12 students to a correct
understanding of
K-12 math.
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
abstractly.
- To experience the step-by-step process of building a remembed
knowledge
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
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
- 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?
Under constructivist attack, but used with great success in Asia:
- Use a coherent, lesson-by-lesson curriculum based on specific
math
content
standards.
- 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."
- Present math facts and demonstrate math skills.
- Today's educationists deplore "teaching by telling", but
eliminating
telling
changes the very definition of "to teach".
- Orient the students.
- Whenever possible, present a new math topic by relating it to a
familiar
context of previously learned math knowledge.
- Examples, examples, examples.
- Continually ask questions to test understanding.
- Give immediate and constant feedback.
- Make every effort to help students remember math facts
and math
skills.
- Math thinking cannot occur without remembered math facts and
skills.
- 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.
- Encourage students to speak and write, using mathematically
correct
language
and symbols.
- 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.
- 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.
- 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.
- 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.
- 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.