Date: Nov 10, 2012 5:44 PM
Author: kirby urner
Subject: Re: How teaching factors rather than multiplicand & multiplier confuses kids!

> This sounds too linear, as if all math topics were on this timeline in
> a sequence and the job is to hit them in the right order, of
> increasing sophistication.
> I don't care what it sounds like. The preponderance of evidence says it is
> the truth. I am not saying that each and every topic is fixed in concrete,
> but the order is fairly well defined. And this isn't really about math. It
> is about the natural path to formal reasoning. Teachers are so anxious to
> put the cart before the horse and I posit that this is because they don't
> have much experience with the whole race and how it is won. Even I have this
> anxiousness, but I also have sense and the advantage of seeing the whole
> race.

What do you mean by formal reasoning?

Is that with some special notation, or are we being treated to a
display of it here?

Reasoning is often taken up in speech and debate, when coaches go over
the structure of a rational argument.

Is that what you're talking about? Rhetoric?

> In actual fact, it's more like a network of multiple highways and
> byways.
> No, actually it is not. If there were an ounce of truth to this statement
> then it would have shown up in the 2 million exams I studied. I mean, surely
> some percentage of students would have found this hidden network of highways

Like for example the sequence used to have more of the New Math in it,
with more boolean algebra and attention to truth tables.

There was attention to different bases, even in the early grades. I
was a product of this curriculum.

Tom Lehrer sang about it in his song 'New Math' (which is
mathematically correct by the way).

I use this chapter in history to show that the sequence has (and will continue
to be) fiddled with.

Truth tables were considered elite college material until Sputnik, at
which point the University of Chicago swung into action and tried to
turn Americans into little Bertie Russell types.

> and wormholes, even if just by accident. I would have seen kids heading
> straight for the fraction store before they could count to 10. It would show
> up all around us. Yet, it doesn't show up at all. Isn't what you're

You might be alluding to Gattegno's successful experiments.

I shared those videos of his classroom, live demos, Canadian television.

More evidence that there's no reason to see the sequence as fixed in stone.

> suggesting based on a philosophical imagination rather than reality? If all
> of these alternate paths exist, then why don't we see that happening?

We have.

Waldorf is different again and features more of that longitudinal
stuff you were talking about in that kids have the same teacher,
ideally, for more than six years.

Lots of different K-12 math curricula have come and gone, and will
continue to do so. The historical record is clear.

High school didn't always point towards Calculus Mountain so strongly.
There used to be more spherical trig (this was before my time).

> Fragments of group theory fit easily in Algebra 1 and make
> both better, but because Group Theory as a whole is quite
> sophisticated it gets thrown out until after Calculus.
> I am good with the extra for experts inserts, but you have to get the
> algebra first before you can teach the student the why behind it, otherwise
> it is just pretend. Besides, Clyde will tell you that since they are doing
> algebra, and since algebra is controlled by group theory, they understand
> group theory. So, no reason to teach group theory twice.:)

This is reinforcing the stereotype that you need to be especially
gifted to get it in a different sequence.

That's like the MAD magazine parody of what it takes to seem smart:
just seem different.
(excerpt from MAD, 1956)

The segment I'm talking about involves distilling the totatives of a
number N, using the GCD algorithm and then showing how totatives
multiplied modulo N have

(a) Closure
(b) Associativity
(c) Inverse elements
(d) a Neutral element

(CAIN -- plus this group is also Abelian (Biblical pun)).

Explain what each of those properties of a group means. Play around
with more of them. It's simple stuff, easy peasy. Amenable to

This is what's called "spiraling" by the way, where you go into
something a little, from one angle, and then get into it more later,
from another angle. John Saxon was emphatic about "spiraling".

What algorithm for the GCD you ask?

Again, today's USA's K-12 is pretty much devoid of Euclid's Method for
the GCD.

Is that because it's hard? Is that because it's for gifted students
only? Not really, no.

Factoring into primes to find a greatest common divisor is actually
harder to do, stumps even the best computers after a certain size.

Inserting the above sequence (about totatives and groups) into Algebra
would be really smooth and simple to do. Computer languages might be


totatives :: (Integral n) => n -> [n]
totatives n
| n <= 0 = [ ]
| n > 1 = filter (\x -> (gcd n x) == 1) [1..n]

totient :: Integer -> Int
totient = length . totatives

That's enough to get something going at the interactive chat window (GHCI):

GHCi, version 7.4.1: :? for help
Loading package ghc-prim ... linking ... done.
Loading package integer-gmp ... linking ... done.
Loading package base ... linking ... done.
Prelude> :l baby
[1 of 1] Compiling Main ( baby.hs, interpreted )
Ok, modules loaded: Main.
*Main> totatives 50
*Main> totient 50

This isn't for super-intelligent kids. This is for ordinary everyday people.

Sure, it takes some time to build up some Haskell vocabulary. Time we
have. We're learning STEM skills and abilities and that takes some

Why wasn't this done in the 1950s or 1960s?

Which part, the computer languages? They were just being invented.
There's been a lot of water under the bridge since then.

We don't teach slide rule anymore either -- more proof the sequence is
NOT set in stone.

You need more longitudinal awareness I think, to see how it's always changing.

> Junior is saddled with the same
> linear sequence is grandparents had. Is that a good thing? By
> definition?
> Well, Junior is just human, like his grandparents. Naturally, the
> progression would be the same, right?

Is this what you call "formal reasoning" then?

You seem to consider yourself a good example of what your favored
curriculum would turn out.

I assume we're being treated to an example of what "reasoning" means,
am I safe to assume that?


> Bob Hansen

Message was edited by: kirby urner