[IAEP] reconstructed maths

Albert Cahalan acahalan at gmail.com
Thu Jul 17 05:16:12 EDT 2008

Alan Kay writes:

> Similarly, from the POV of a former guitar teacher and player,
> "playing guitar" has a threshold that excludes "air guitar",
> "Guitar Hero", certain kinds of noise, and "not enough fluency
> to make music yet". We can use "air guitar" as a metaphor for
> (a) taking such a small subset of an activity that only some
> form and essentially no important content is being done, and
> (b) for using form over content to fool ourselves that we are
> "players" and "part of the club".

That works. I also like the term "math appreciation".
Like a music appreciation course, it doesn't get you
to be competent.

> Two of Seymour Papert's most important insights about above
> threshold math-with-computers for children were to (a) find and
> use the real mathematical thinking that children could do at each
> stage of development, and (b) to both pick from the large body of
> existing mathematics and to invent new mathematics that embody the
> most "powerful ideas" that humans have come up with. One of many
> such examples is how to use the children's ability to add (and to
> think additively) and to physically move their bodies to make for
> them a powerful and valid version of Gauss' Differential Geometry
> which covers some of the most important parts of vector calculus
> in a way easily learnable by children.

I do believe that many children can learn vector calculus, and that
this might have some value. However...

When you put the cart before the horse, trying to skip all of the
arithmetic and such, you're teaching math appreciation. It's air math.

An actual bake-off has been conducted. It was the largest educational
study ever done, covering 79000 children in 180 communities. I'm sure
you've heard of it: Project Follow Through.

In that study, Direct Instruction (sage on the stage) trounced all
other programs in multiple ways. More here:


That last article has a lovely graph of the results.

One may question the ethics of such an experiment, since it does in
fact involve experimenting on children in a life-effecting way.
I see no alternative though, so I'm glad it was done. Now it is time
to accept the results, eat some humble pie as required, and teach
children math.

It is unethical to endlessly repeat the experiment, hoping that you
will somehow get results that support a well-loved hypothesis.

Vector calculus is a fine subject for children. Set high expectations
for daily progress, eliminate distractions, and soon enough most kids
will reach vector calculus. (for real, not vector calculus appreciation)

High expectations go something like this:

multi-digit add/subtract with traditional procedure: 1st year
multiplication table memorized: 1st year
long division: 2nd year
2-step word problems: 2nd year
4 basic operations on fractions: 3rd year
4-step word problems: 4th year
order of operations: 4th year
algebra (with proofs): 5th year
geometry (with proofs) and trig: 6th year
regular calculus: 7th year
vector calculus: 8th year

It's doable, but you won't get there if you waste time or if you use
educational methods that are proven to be horrible.

In case anybody wants to look at current curriculum that work:

The two best math programs, unfortunately subject to copyright, are
Saxon Math and Singapore Math. Saxon Math is better for the slower
students, particularly if students are missing school or transferring
in from places that use a different math program. Singapore Math is
better for the faster students. Both are available for purchase.

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