Traditional K-12 Math Education

• Knowledge Transmission
• Why Learn K-12 Math?
• How is K-12 Math Traditionally Learned?
• Understanding The Process of Building A Personal Math Knowledge Base
• How is K-12 Math Traditionally Taught?

---

Knowledge Transmission

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.

Why is K-12 Math Learned?

The traditional reasons are:

1. For the practical math-related needs of daily life.
2. To prepare for occupations that use math.
3. To develop the power of the mind to think logically and abstractly.
4. To experience the step-by-step process of building a remembered knowledge base, relative to a structured knowledge domain.
5. As foundational knowledge for learning more advanced math.
6. 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

• 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:
1. 4 = 3 + 1, by the definition of 4.
2. 3 = 2 + 1, by the definition of 3.
3. 4 = (2 +1) + 1, by the "substitution of equals for equals" axiom.
4. 4 = 2 + (1 + 1), by the "associative property of addition" axiom.
5. 4 = 2 + 2, by the definition of 2 and "substitution of equals for equals" axiom.
6. 2 + 2 = 4, by the "symmetric property of equality" axiom.

1. 2 + 2 = 4, by Theorem 1.
2. (2 + 2) + 1 = 4 + 1, by the "add equals to equals" axiom.
3. 2 + (2 + 1) = 4 + 1, by the "associative property of addition" axiom.
4. 2 + 3 = 5, by the definitions of 3 and 5 and "substitution of equals for equals."

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:
1. y = mx + b
2. y = c
3. x = c
• Self question: Do I remember relevant math skills (Do I remember a procedure for applying the remembered math facts)?
• Self answer:
1. 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.
2. 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.
3. 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.
4. Apply 3 to the point (1,2) to get: 2 = m + b.
5. Apply 3 to the point (-1,4) to get : 4 = -m + b.
6. Add equals to equals:  2 + 4 = 6 = (m +b) + (-m + b) = 2b.
7. 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:
1. Provide an orienting framework.
2. Clearly explain key concepts.
3. Invite the student to fill in gaps (with knowledge remembered from earlier lessons).
4. 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.

• Chapter 2 ..... The NCTM Calls Them "Standards"
• Links to Other Chapters ...... Understanding the Original NCTM Standards
• Understanding the NCTM's Math "Standards" and NCTM "Math Reform" (Home)