Is Arithmetic Mathematics?

solving problems like X^2 + 2x + 3 = 18 using tactile materials. It *really* shoots down the modern method of teaching math by starting with set theory (from Bertrand Russell, I think).

No, it doesn't, because the 8 year olds of your example *aren't* learning mathematics. They're learning how to calculate. Calculation and mathematics are often confused because calculation is most of what's taught in mathematics courses until the third or fourth year university mathematics courses which are generally only taken by mathematics majors.

I beg your pardon? On what basis do you exclude calculation from mathematics? I'm also not sure where you're drawing the line here. I presume you include calculus and analytical geometry in calculation; fine. How about number theory? Logic? I'm really curious as to what mathematics includes once you've eliminated 97% of the subject matter.

Having a math degree myself, I'm absolutely fascinated by your distinctions. Can you define mathematics so I understand the term also?

Keeping things concrete, while possibly useful for teaching calculation, is a hindrance for teaching mathematics, the primary idea of which is abstract generalization beyond the concrete. Set theory is the best starting point for learning the fundamental abstract concepts of mathematics. You may only be interested in teaching your children calculation and have found a good method for doing so, but that it no way "shoots down" the method of teaching mathematics, not calculation, starting with set theory.

Hmm. Well, I agree with you about set theory, but for an entirely different reason. Sets are less abstract, more concrete-izable than numbers and operators. My husband and I (both with math backgrounds) naturally started speaking about numbers in terms of sets. "You have a set of five blue blocks and a set of three red blocks. How many blocks do you have altogether?" I find that addition is a naturally understood abstraction of set unions and subtraction follows from set intersection nicely.

I agree with you (I think) that the concepts of mathematics are much more important than mere numeric manipulation.

I firmly believe that children need to abstract a step at a time from the concrete, returning to the concrete to illustrate the abstract concepts and to test the abstractions. The concept of "horse-ness" is best illustrated using horses. The concepts of mathematics are best illustrated using real objects.

I have heard other mathematicians argue that math should begin, not with set theory, but with number theory. And I can see the point; one must at some level realize that mathematical abstractions are not equivalent to the real-life applications of mathematics. However, I think these mathematicians are in love with their distilled abstractions (admittedly heady stuff!) and pretty ignorant of child development.

Symbolic systems are wonderful, and I of all people appreciate the beauty and elegance of pure mathematics, but let's remember where it all came from. Mathematics did not arise full-grown in the brain of some erudite mathematician, like Athena from the head of Zeus. Math was built, a step at a time, from concrete reality and the abstractions that had gone before.

One of the areas where I've been fascinated by the divergence between math and reality is with computers. Computers are often viewed as abstract tools that reflect mathematical abstractions very well. Well, they're good within a limited sphere, but, around the edges, computers have quirks that show the overlay of abstract math on the physical machine. The match is far from perfect.

But the topic of how computers distort math is a dissertation, and hardly on topic....