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.

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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.

Should the U.S. Be Talking to Iran, Syria, Hamas, and Al-Quaeda?

We should be talking, and we are talking.

Mr. Obama has said that he would talk personally, as President, without preconditions, to many countries and organizations that have declared hatred for the U.S. and its allies. He didn't specifically leave out any group.

The wags say that he didn't specifically include the leadership of Iraq, and the military officers and generals serving in Iraq.

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This is a bold position. It implies that President Bush and his administration are just too timid, stupid, or war-loving to take this obvious step toward communicating. It implies that Mr. Obama is unusually courageous, intelligent, and peace-loving, willing to go the extra step to eliminate misunderstandings and find a truce with our enemies. One smart man, with bold ideas, who can talk well, can step in and make the difference.

There are problems with this position. The U.S. already talks with all of these groups, and negotiations are vastly complicated.

The State Department has ambassadors and diplomats who devote themselves to talking. The communication goes through foreign embassies or "informal" contacts when there is no direct contact. The leaders of these groups often give speeches and issue statements. These are translated and discussed within the State Department and in universities, at least.

All this talk gives results that need analysis. For example, "Syria" is not a single entity; there are various power groups within Syria having different views. Who has real power, what are they saying, and what are they doing? The U.S. is also not a single entity from this point of view. If one President says something, will the Congress or the next President say or do the same thing?

Sometimes there is general agreement after working out the details at lower levels. The Presidents of the two (or more) groups meet to confirm the agreement as the last step.

Mr. Obama's idea does not work, because it is impossible for him to work out an agreement in that way. He can't do all the negotiation needed for the groups on his end, and neither can his adversary. He would be in the position of a diplomat, except that his political prestige would be at risk. The most he could do is offer concessions.

Consider this possible exchange:
US Pres: Hi.
Antagonist: What does the Western dog want?
US Pres: I demand that you stop attacking us.
Antagonist: Attacking you is in our blood, a service to God, and a requirement of keeping my power.
US Pres: (a) We will squash you with cluster bombs, or (b) You can have $5 Billion in aid each year.

A threat like (a) has to be delivered in a public speech to be believed. A concession like (b) is a mistake early in a negotiation.

I think Mr. Obama's idea is posturing. It sounds good but is not thought through, and he can't believe in it. He would be ignorant and arrogant if he did believe in it.

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Negotiating Treaties in the Global Environment by the Australian Government: Department of Foreign Affairs and Trade. A quick scan of this paper gives a feeling for the complexity of a negotiation between nations or powerful groups.