[Squeakland] the non universals

Alan Kay alan.kay at squeakland.org
Fri Aug 17 06:16:25 PDT 2007

Hi David --

At 02:58 PM 8/16/2007, David Corking wrote:
>Thanks for wrestling with my questioning, Alan  (btw - it seems we
>forgot to share our last two exchanges with the mailing list - my
>fault - I refrained from repeating your responses extensively here in
>case it not your intent to post them.)

I didn't notice this, so just reposted the previous reply to the list.

>On 8/16/07, Alan Kay wrote:
> > Of course, this is far from a scientific survey ....
>You clearly know far more teachers than I do.  I am shocked to hear
>that so few US math and science teachers were math and science majors,
>or were even educated in any college level math and science.

As I said, my little survey this summer wasn't scientific ... it 
would be nice to have a much better assessment of this done in a more 
rigorous fashion.

But I don't think it is an exaggeration to guess that most teachers 
lack the kind of operational mathematical thinking that is the most 
important part of mathematical fluency. And it is very likely that 
"most" is almost total in the elementary grades and means "very 
sparse" in high school.

>I suspect it is normal worldwide to postpone calculus until the
>equivalent of  "Advanced Placement" courses in years 11 and 12 - I
>hope it is mandatory to know calculus before going to college for
>math, science or engineering (and perhaps for social science too.)
>Perhaps by the delay we then rob many kids the chance to (1) see its
>beauty and (2) see that it underpins so much of modern science and

"Knowing calculus" is a tricky phrase. An important idea here (that I 
originally got from Ivan Sutherland) is to ask whether skills are "10 
hour skills", "100 hour skills", "1000 hour skills", etc. Ivan once 
pointed out that e.g. learning to play piano was not a "10 hour 
skill" no matter how much latent talent you might have. And, though 
talent does play a part in time to learn and get fluent at something, 
it takes time for people's brain/minds to build the structures needed 
for doing the thinking in question, and doing it fluently enough, etc.

As I recall, this discussion came up when a bunch of us grad students 
and Ivan at Utah were working out the mathematical transforms for 3D 
graphics (much of which constitutes OpenGL today). We had some 
terrific French grad students, who were better prepared 
mathematically than most of the Americans. I had an undergrad math 
degree, etc. Ivan is quite a bit smarter than most people, etc. Yet, 
this was a real struggle for all of us to "get operational" in a 
theory that we all "kind of knew" really well: the transformations of 
vectors using matrices, with the addition of the homogenous 
coordinates idea from projective geometry that Larry Roberts had suggested.

Ivan's observation was that we had attained the "100 hour skill" 
version of transformations, but not the "1000 hour skill" version. 
And this led to discussions of other skills in other areas. This idea 
is particularly striking and easy to understand in sports and music. 
And also overlaps with some of the findings in cognitive psychology 
about habit formation and habit unlearning. For example, if you put 
in 10 hours a week trying to learn something (2 hours a day, 5 days a 
week) and take a two week vacation, etc., then you will be spending 
about 500 hours a year doing your learning and practicing. Two years 
of this is 1000 hours.

Lots of good things can be learned to the solid mid-intermediate 
level in two years. And 18 months to 2 years is also the time that 
cog psych says it takes to form a solid habit (or unlearn one). Also 
very interesting are the results from many studies of the attempts at 
educational reforms in the 60s which showed that one year of a super 
enriched experience didn't stick, but two or more years did.

So the concept of a "1000 hour skill" is worth contemplating when 
looking at instructional systems.

Back to calculus for a second: I really didn't understand calculus in 
worthwhile ways (that is, to think in terms of what it meant rather 
than just trying to apply the techniques) until I took Advanced 
Calculus with a very good prof (who was a fabulous mathematical 
thinker and teacher). This was quite shocking to me (because I didn't 
really have a sense that I didn't understand calculus until I 
understood it so much better). And all of us working with Ivan a few 
years later had a similar sense about transformations. We only 
thought we understood them until we understood them deeply enough to 
think in terms of them, not just try to use them.

I think there are also real analogies here to stages of learning a 
foreign language. Seymour and I have talked a lot about this, and he 
thinks so also. The differences between being able to use another 
language a little and being able to think in "its perfume" are profound.

This is where more longitudinal approaches and immersion are 
critical. One of the reasons I loved Seymour's ideas and approach was 
that it would be possible to have children immersed in "CalculusLand" 
in ways meaningful to them for years so they could gradually build up 
real "CalculusThink".

>As you point out, the algebraic model of calculus is not interesting
>to many people, but the difference model would, I imagine, be useful
>to every aspiring mechanic, lab technician or customer service

Its not that the algebraic model of calculus is not interesting (it 
really is) but it is much further removed from most people's 
fluencies. The difference model is just simple accumulation by 
addition, and the equivalent of higher order differential equations 
is just more accumulations by addition lined up. I have written about 
how Julia Nishijima (the first grade teacher who had a real 
mathematical sense) could set up projects that would induce the 
children to discover and derive second order discrete DEs (first 
order is steady growth, second order is quadratic, etc.), These are 
the very  same progressions that can be used for velocity and 
acceleration, F = ma, Galilean gravity, etc. so it is terrific to get 
started with these as tools one has derived in first grade.

We have a nice way to (later, perhaps in 7th or 8th grade) reconcile 
the easier incremental approach to the algebraic formulas by deriving 
the latter from the former. This is not just pro forma but is a very 
useful way to start thinking about what it is that is being said (and 
how universally) using quantification. It's also very illuminating to 
start thinking about integration and what it means in the universally 
quantified world (leading to the fundamental theorem of calculus).

But having quite a few years of calculus thinking and doing under 
one's belt is a much better way (in my opinion) to approach some of 
the deep and initially non-intuitive properties of calculus.

> > But what if the
> > secondary math teachers complained loudly? I don't think they are in
> > any decision process that I can find.
>I don't know the US systems very well.  I would like to think that
>school boards and education departments consult professionals first.
>Are there countries where that does happen?

It's very tricky in the US -- in part because there are 25,000 or 
more individual school districts. There are state and national 
standards. Professionals are consulted. Etc. I only have speculations 
on how the system has not managed to do better with mathematics curricula.

One thing that seems to be almost universal around the world, is that 
the notion of children learning some subject (like mathematics) is 
almost always posed as "how can children be taught the adult version 
of this subject?", rather than, as Montessori, Piaget, Bruner and 
Papert have shown "how can we find an honest children's version of 
this subject?".

Another important idea here is that there are likely to be other 
approaches that are also better than the standard ones. Seymour (and 
I and others in his footsteps) simply have worked out one set of 
insights that can allow children to actually be real mathematicians 
starting at an early age. This is not a religion, nor is it exclusive 
to "the Seymour way".

Over the last 30+ years of my own experience I have been greatly 
surprised at some of the things children have shown they can do (the 
4 year olds at Reggio Emilia, the 6 year olds of Julia Nishijima, 
that 5th graders could do the Galilean gravity project I designed for 
9th graders, etc.). Basically, we still don't really know what 
children can learn at different ages if the subject matter is 
properly formed. The experiments are very difficult to do, and lots 
of them need to be done (in part because there are so many things 
that can prevent a good reading of the children).



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