Traditional K-12 Math Education
This chapter outlines the traditional "knowledge transmission"
of K-12 math education. The fundamental assumptions are:
- Math is a man-made abstraction that only exists in the human mind
- There is an established body of math knowledge that
can understand in the same correct way.
- There is a stable foundational "K-12 math subset" that can
by different K-12 students in the same correct way.
- K-12 math teachers can lead K-12 students to a correct
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
- To experience the step-by-step process of building a remembered
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.
How is K12 Math Traditionally
Learning Math Means to Build a Remembered Math
Traditionalists believe that learning math is a process of building a
math knowledge base that is stored in the brain. Conceptually, this
base consists of math facts tightly linked to math skills.
- Math facts consist of undefined terms, definitions,
assumptions), and theorems. For example, the symbol 1 is an undefined
the fact that 2 = 1 + 1 is the definition of 2, the fact that "equals
be substituted for equals" is an axiom, and the fact that 2 + 2 = 4 is
- We are assuming the traditional "formalist" philosophy of
From this viewpoint, theorems are "math facts" that are "proven" to
from" already established math facts by a step-by-step process of
- Math skills involve recalling and applying relevant math
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:
Math is a structured domain. "New" math facts are "built on"
math facts. For example, now we can use Theorem 1: 2 + 2 = 4 to
establish the "new" Theorem 2: 2 + 3 = 5
- 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
- 4 = 2 + (1 + 1), by the "associative property of
- 4 = 2 + 2, by the definition of 2 and "substitution of
- 2 + 2 = 4, by the "symmetric property of equality"
Math thinking involves the mind in a question and answer process. This
process depends on remembered math facts and remembered math skills.
is not a simple matter of remembering a few facts. As the following
illustrates, it requires remembering and understanding a whole series
linked facts. If there are gaps in knowledge, the whole process breaks
- 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"
- 2 + 3 = 5, by the definitions of 3 and 5 and "substitution of
Example: Find the equation of a
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
- Self question: Do I remember relevant math facts?
- Self answer: Such equations have 3 possible general
Self question: Do I remember relevant math skills (Do I
a procedure for applying the remembered math facts)?
- y = mx + b
- y = c
- x = c
- Rule out form 2: Because the given points have different y
Rule out form 3: Because the given points have different x
- Skill: The student understands that form 2 applies only
when all points
on the line have the same y coordinate.
Remember: If the line y = mx + b passes through (p, q), then
q = mp + b.
- Skill: The student understands that form 3 applies only
when all points
on the line have the same x coordinate.
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) =
Conclude that b = 3, m = -1 and the equation is y = -x
- Skill: The student understands the algebraic meaning of the
that a specific point lies on the line.
Understanding the Process of
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
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.
- Example: A first grader needs to memorize 2 + 2 = 4, but a
correct proof of this fact (such as given above) must be delayed
the child has acquired a richer math knowledge base.
- 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
- Use a coherent, lesson-by-lesson curriculum based on Genuine
- 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.
- Today's educationists deplore "teaching by telling", but
changes the very definition of "to teach".
- How and when to tell may be debatable, but not telling is
- Orient the students.
- Whenever possible, present a new math topic by relating it to a
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
- Math thinking cannot occur without remembered math facts and
fundamental truth cannot be denied. Although much despised by the
establishment, memorization is a powerful and necessary tool.
- Constantly encourage students to practice their
- 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.
- Encourage students to speak and write, using mathematically
- Let them write, cross-out, revise, and evolve their thinking on
- Expect instant recall for basic math facts only. Don't make
through a multi-step process in their heads. Don't make them worry
- Make careful use of "manipulatives".
- 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.
- Make careful use of written instructional materials. Good written
should be succinct and closely tied to genuine math standards. They
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 NCTM and the "progressive" education establishment.
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.
William G. Quirk, Ph.D.