Traditional K-12 Math Education
Knowledge Transmission
This chapter outlines the traditional "knowledge transmission"
philosophy
of K-12 math education. The fundamental assumptions are:
- 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.
Why is K-12 Math Learned?
The traditional reasons are:
- For the practical math-related needs of daily life.
- To prepare for occupations that use math.
- To develop the power of the mind to think logically and
abstractly.
- To experience the step-by-step process of building a remembered
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.
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 facts consist of undefined terms, definitions,
axioms
(fundamental
assumptions), and theorems. For example, the symbol 1 is an undefined
term,
the fact that 2 = 1 + 1 is the definition of 2, the fact that "equals
can
be substituted for equals" is an axiom, and the fact that 2 + 2 = 4 is
a theorem.
- We are assuming the traditional "formalist" philosophy of
mathematics.
From this viewpoint, theorems are "math facts" that are "proven" to
"follow
from" already established math facts by a step-by-step process of
deductive
logic.
- Math skills involve recalling and applying relevant math
facts.
For example, a mathematically correct proof that 2 + 2 = 4
involves remembering and appropriately sequencing the math facts listed
in the following six "by" clauses:
- 4 = 3 + 1, by the definition of 4.
- 3 = 2 + 1, by the definition of 3.
- 4 = (2 +1) + 1, by the "substitution of equals for
equals"
axiom.
- 4 = 2 + (1 + 1), by the "associative property of
addition"
axiom.
- 4 = 2 + 2, by the definition of 2 and "substitution of
equals
for
equals" axiom.
- 2 + 2 = 4, by the "symmetric property of equality"
axiom.
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
- 2 + 2 = 4, by Theorem 1.
- (2 + 2) + 1 = 4 + 1, by the "add equals to equals" axiom.
- 2 + (2 + 1) = 4 + 1, by the "associative property of addition"
axiom.
- 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).
- Self question: What's the math context?
- Self answer: Equations of straight lines in
2-dimensional
space.
- Self question: Do I remember relevant math facts?
- Self answer: Such equations have 3 possible general
forms:
- y = mx + b
- y = c
- x = c
- Self question: Do I remember relevant math skills (Do I
remember
a procedure for applying the remembered math facts)?
- Self answer:
- Rule out form 2: Because the given points have different y
coordinates.
- Skill: The student understands that form 2 applies only
when all points
on the line have the same y coordinate.
- Rule out form 3: Because the given points have different x
coordinates
- Skill: The student understands that form 3 applies only
when all points
on the line have the same x coordinate.
- Remember: If the line y = mx + b passes through (p, q), then
q = mp + b.
- Skill: The student understands the algebraic meaning of the
geometric
fact
that a specific point lies on the line.
- Apply 3 to the point (1,2) to get: 2 = m + b.
- Apply 3 to the point (-1,4) to get : 4 = -m + b.
- Add equals to equals: 2 + 4 = 6 = (m +b) + (-m + b) =
2b.
- Conclude that b = 3, m = -1 and the equation is y = -x
+ 3.
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.
- Example: A first grader needs to memorize 2 + 2 = 4, but a
mathematically
correct proof of this fact (such as given above) must be delayed
until
the child has acquired a richer math knowledge base.
- 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?
- Use a coherent, lesson-by-lesson curriculum based on Genuine
Math 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..
- Present math facts and demonstrate math skills.
- Today's educationists deplore "teaching by telling", but
eliminating
telling
changes the very definition of "to teach".
- How and when to tell may be debatable, but not telling is
malpractice.
- 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.
This
fundamental truth cannot be denied. Although much despised by the
education
establishment, memorization is a powerful and necessary tool.
- 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.
- Let them write, cross-out, revise, and evolve their thinking on
paper.
- 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 NCTM and the "progressive" education establishment.
- Above all, guide your students to correct understanding. Don't
let them
walk away thinking that 9 times 7 equals 97.
Next ?
Copyright
1997-2011
William G. Quirk, Ph.D.