Robert Hansen says: >One last thing Joe. If you agree with Clyde then you must understand Clyde, correct?
Please - in this thread I meant that one specific observation. I often find points of agreement with him though, in general, but other times he loses me as well with his full-blown theories and terminology.
I will note that he does know the difference between "vector space" as commonly used in abstract algebra and his particular appropriation of that word. I don't have the URL handy, but he says so right in the first paragraph or so.
Robert Hansen says: >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.
Absolutely true for counting and adding. However, when you first start to learn about "functions" perhaps you have a very concrete idea of functions of a real variable. To an advanced mathematician these days, that's like counting on your fingers - their are so many more uses for mappings in general.
That's why I like "meta" better than "symbolic". There are meta levels upon meta levels, not just two levels. The distinctions are always relevant no matter how far one gets. "Meta" also captures better the idea of studying "expressions with variables" as objects in their own right, rather than simply viewing them as ways of expressing things *about* integers or rationals or what-have-you. I agree with you fourth graders are usually not at that level. Nor, sadly, are a lot of 10th garders.