On Nov 9, 2012, at 12:50 PM, Joe Niederberger <firstname.lastname@example.org> wrote:
> I agree with Clyde - children learn to count, and even to count perhaps in pure numbers. Its a game, with its own rhythm, like skipping rope. But expressions like "how many" beg for an object: "how many apples". Beginning arithmetic such as addition has always been greatly aided by grounding in real world objects. That's how children > begin to understand expressions like 3 + 5; they relate it to a real-world situation, like "3 apples + 5 apples".
One last thing Joe. If you agree with Clyde then you must understand Clyde, correct? Then explain to us, if you will, Clyde's theory of vector algebra for toddlers. Maybe you jumped into the middle of a long and continuing debate, but I never said that I didn't use apples to teach counting or to start addition. Who doesn't? The difference is that after that relatively short phase, I then went on to numbers, and we haven't looked back since. Clyde on the other hand is doing something with vector algebra. You agree with him, so tell us, WHAT IS IT?
The counting/adding/apple phase ends pretty quickly in the second grade. I shed a tear every time I remember my son adding on his fingers. They grow up so fast.