On Nov 9, 2012, at 1:42 AM, Clyde Greeno @ MALEI <email@example.com> wrote:
> You are declaring that children learn arithmetic most naturally via the mathematics of *quantities" which have whole-number numerators. In that much you surely are correct. But much of your rhetoric and logic will be simplified if you rework it all through the context of quantities.
This is false by definition. This would be like saying that children learn to read most naturally by reading prescriptions off medicine bottles. At the heart of arithmetic is number (not quantity) and operation. When we say 3 apples plus 4 apples, this is a physics problem, that involves math. The use of units, while part of a math curriculum, is an application of math to the physical world. Tying math so strongly to the physical world in this manner will retard the development of pure number sense. It is this sense, that includes factoring, LCM, GCD, prime numbers, etc. that a student needs in order to conquer the later stages of arithmetic, namely, fractions and proportional reasoning.
Also note, 3x2 is the multiplication of two numbers. The first factor (3) is labeled the multiplier, the second (2) the multiplicand and the result (6) the product. That is simply a matter of convention. None of these numbers are "scalers" because no context is indicated. Also note, that a young elementary student can easily understand 3 apples plus 6 bananas equals 9 pieces of fruit.
I'll ask again, both of you. When do you proceed past these very strange ideas of teaching arithmetic to actual arithmetic? I mean, when does the student finally escape from all this extraneous litter (vectors and bones and whatnot) to understanding advanced arithmetic with numbers?