Below is an email from Alan Kay in response to a question I posted to OLPC on teaching Algebra to my kids:

Hi Steve,

If I were trying to do this (and it is just the right thing for a parent to do) I would definitely start off with "a stick and some sand".

Algebra is basically a symbolic form of arithmetic in which the art and skill is to "see/develop" strategies which can be turned into tactics that preserve equality (mostly, and sometimes other relationships).

A lot of it has to do with the idea that "a number is all the ways to make it" and that the basic arithmetic operations have inverses. This last sentence is the one that is missed in most "school math" (which isn't "real math"), and part of the many difficulties which arise is that the children are pushed into trying to deal with "numbers as numerals (that is, their "names" in positional notation)" rather than as ideas which have analytic properties (they can be taken apart and put back together).

And in general one of the difficulties in "school math" is that it winds up concentrating on how ideas are written down using societal conventions, rather on the ideas themselves. (And this is perhaps even more so with regard to science ideas.)

A physical/geometric approach to these ideas is a very good one for all (especially younger) children. There are several good references here, including Hadamard's "Psychology of Invention in the Mathematical Field", a rare and valuable assessment of how mathematicians do their thing by a great mathematician himself. The bottom line in this study is that most real mathematicians do their thinking outside the written down forms, and mostly visually (and configurally) and for about 20% (including Einstein) kinesthetically.

A lot of the real mathematics that is worth learning and understanding was invented to help with science in characterizing observations and results in the real world. This has been perverted in a sense by even some of the well meaning texts which try for "relevance" and to have lots of "story problems" that are situated in the world. But most of these are weak stretches compared to what's interesting about real science.

So I would put a lot of effort into combining "real math" and "real science" as much as possible. A nice observation here is that quite a lot of really good math can be garnered by treating "math as a science" -- that is: where interesting constructions are made and then analyzed (this is like "bridge science", which is studying and modeling the phenomena that bridges manifest).

Back to algebra. The main initial games here are to find ways to "preserve equality by adding 0s and multiplying by 1s). There are scaling games, relationship games, etc. And many of these have nice visual and tactile representations. For example, multiplications can be represented as rectangles, and many of the simple abstract multiplications -- such as (x + y) * (w + z) -- where any or all combinations of lengths can be used for the sides.

A good one to start with is (x+y)*(x+y), and to see how this generates the standard formula from visual inspection alone.

A nice end point is to actually derive "The Quadratic Formula" by "completing the square". I remember being so pissed when I was a kid to be given this formula without any explanation about why it works, when it is a "simple" derivation using algebra(there's an immense irony here).

It's worth trying to understand "simultaneous equations" as simply manifestations of the "main initial games" -- that is, to try to avoid the routinization of method here, but to think what it means to "remove a variable" in algebraic terms. (Another irony here, because much of the second and third steps in algebra are derivable using the first steps. The thinking processes are of the same kind as the way geometry should be taught and learned.)

Functions are often very mysterious for a number of interesting reasons. It is often the case that one part of this mystery can be dispelled by making dynamic interesting relationships using computer graphics. Functional relationships can be learned before formal parameterization. For example, the "Drive a Car" project in Etoys (a favorite early one for 10 year olds) exhibits this in a very memorable form when the heading of the steering wheel is connected to the turn command of the car. For most kids, this is a huge hit of understanding about variables (with the more subtle grasp of the "functional relationship" between the heading of the car and the heading of the steering wheel.

Another good one is to think through what "turn" means and make your own version of "turn 10" from "car's heading <- car's heading + 10" and then make the turn function itself (Etoy scripts can be parametrized). The same kinds of things can be done in Logo (and experiments in parametrization and function making are fun to do in Logo).

Another way to think about a lot of mathematical thinking from numeracy onwards is to use the metaphor of "projection". A mathematical thinker is always *projecting outwards* combinations and dissections of the materials (often numbers) in front of them, and those who haven't gained these skills just let them lie there. (This is also a center of what it means to be a programmer.) One very good route that is often neglected is "mental math". This is hard for most children because they are rarely exposed to it (and I have to cover the screen of their computer or their book when I want them to start thinking internally). So one of the things I would start doing with your son is to pose "in your head" problems. They can start with just simple arithmetic and progress.(You should do them too.) The idea is to get them to internalize their thoughts so they can visualize both images and symbols. (My brother and I had this fun with our Dad (who was a physiologist). A typical example for a 12 or 13 year old -- but after a few years of working up to it -- would be "find the volume of a sphere 35 centimeters in radius". This can take a while, but the results are some great "sticks and sand" in your head that can work on lots more complicated things later on!)

The mantra here is "The music is not in the piano" -- i.e. "The math is not in the computer (or the book or the stick and the sand)"

You are trying to help your son become a kind of musician (where the music is "relationships about relationships" (as von Neumann termed it)).

Cheers,

Alan

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