Date: Nov 29, 2012 5:02 AM
Author: Paul A. Tanner III
Subject: Re: Some important demonstrations on negative numbers

On Wed, Nov 28, 2012 at 5:20 PM, Joe Niederberger <> wrote:
>>Or one can prove it (where "it" could be the more general -(a/b) = a/(-b) for all elements in the set even if the set contains no identity element).
> You no doubt "help" your 4th graders this way. Well, that helps weed 'em out I bet.

That was not intended for them.

> Now, here's a better story -- this is not a 4th grade lesson but a meta-level observation on what and how we teach numbers.
> Negative numbers, and everything that goes with them (sign rules) are useful in some contexts and useless or absurd in other contexts. That's part of learning mathematics -- how to map the concepts from the everyday contexts (e.g., accounting) to the symbols and rules we learn in arithmetic and algebra etc. That "mapping" is crucial to making sense out of the whole enterprise. It is also given short shrift by people like PT III and R.H.

OK. Let's this be about 4th graders - and let's include up through 8th grade, the last year measured by TIMSS.

I'm for including what you say in the mix of what is taught.

But you seem to be against including any notion of proof - even if we allow the notion to be relaxed - for elementary school - and middle school, too?

Consider this:


"Constructing proofs is an important mathematical activity because it provides a reasoned method of verification based on the accepted assumptions and observations of the discipline. Analysis of the videotapes reveals that a greater percentage of the Japanese lessons include proofs than either the German or U.S. lessons. Indeed, 10 percent of the German lessons include proofs while 53 percent of
the Japanese lessons include proofs. None of the U.S. lessons include proofs."

These videos are about 8th grade instruction.

And look at that Liping Ma book "Knowing and Teaching Elementary Mathematics" - and this means really looking at the Chinese teachers justifications (yes, proofs when we allow the term to be more loosely applied o that level) for the mathematics they teach. even though they use actual numerals while also appealing to algebraic properties (associative, distributive etc.), if we were to replace those numerals with letters, we would see that they essentially are same pattern as proofs with variables.

Putting all this above together:

I understand that on international tests, East Asian students blow away all other students in the world on those items that could be considered related to proof.

I do not think that this is an accident.

Example of what I mean: I think that to get to where in Japan in 8th grade, 53 percent of the Japanese lessons include proofs, they have get to that high a percentage smoothly over the years, meaning that proof is used much sooner than 8th grade, even if at a much lower percentage of the time.

I think that all this argues against your seeming position that proof should be not there at all before 8th grade. Or if you're OK with before, then how much before? 7th, 6th, what?

My position is that we Americans should seriously look at what these other countries do who are more successful than the US by some measures, to let these other countries teach us a thing or two.

I think that it;s clear that what they have to teach us is that we should include proof in the teaching of mathematics and that it is can be used in the teaching of mathematics much sooner than so many in the US claim, and much more heavily than they claim, as well. I mean, 53 percent in Japan in 8th grade, I think clearly implying steadily decreasing percentages in prior years. And I think that this steadily decreasing percentages in prior years includes steadily looser uses of the term "proof" in prior years. That is, proofs in 4th grade would not be proofs in 8th grade, even these proofs in 4th grade could still be considered proofs that prepare very well for that 53 percent level in 8th grade. And again recall that by the above, there is 0 percent proof in 8th grade classrooms in the US.

It seems to me that with this anti-proof thinking what we have here is yet another example of how we Americans miss out when we do not look at the rest of the world, to learn from them.