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Jun 14, 2008

Teaching the Math Analogy

An analogy is a teaching aid, not knowledge in itself

This is a comment on Makes Me Want To Kill Myself With Occam's Razor at Advice Goddess. The video there presents how multiplication and division is badly taught in some schools.

My aunt Linda was a sweet person, a bit formal, and an elementary school teacher. She had a question for me about teaching math: Can you subtract a negative number? I'm a math/physics college grad.

Her teaching guide used a mailman analogy to teach addition and subtraction. You start with $10. The mailman delivers a check for $5, so you have 10 + 5 = $15.

If the mailman delivers a bill for $8, then 15 - 8 = $7. Also, you can think of this as adding a negative number, so you can also write it as 15 + (-8) = $7.

The problem is that the mailman never takes away a bill, so how can you subtract a negative number? Other teachers at her school didn't have an answer.

After some discussion, she thought it was arbitrary that subtracting a negative is like adding the positive, and it didn't fit the mailman analogy, so she would just not teach it. An inquisitive 10 year old was told that you just can't subtract negative numbers.

I will take the leap to generalize. Schools are teaching mathematical subjects the same way as other subjects. Math is presented as a collection of facts to memorize, with no understanding of the patterns or meanings that the facts are supposed to illustrate. The failure of the textbooks shown in the video (aside from the travelogue approach for filling in 50 pages) is that they don't teach fundamentals (what is it physically like to multiply and divide), and they don't teach compact algorithms derived from the fundamentals.

They teach a middle view that obscures both the fundamentals and the algorithms, leaving math as a large collection of facts that are mostly alike and hard to apply. A mystery best left to the people who magically "get it" from above.

Sorry to say, without a deeper understanding, having a calculator is no help. You punch in a few numbers, make mistakes, get wild answers, and sign for a mortgage with terms that can't make any sense to you.

1 comment :

Micha Elyi said...

"The problem is that the mailman never takes away a bill, so how can you subtract a negative number?"

Oh yes the mailman sure does take away a bill. Say I send in a request for a refund. Or enclose a discount money-off coupon with my payment check when I purchase something by mail order. Or I send a credit voucher off in the mail and the mailman takes that out of my mailbox. Examples abound.

Other teachers at her school didn't have an answer."

Sigh. That's yet another reason we might as well credential K-6 teachers who have no more than a high school diploma themselves.

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