Establishing what those who succeeded have in common is substantially different from establishing "the elements of success". Cancer victims have in common that they drank milk as children. But that doesn't identify milk as an element that causes cancer.
Those who succeed in mathematics today generally did well at arithmetic as kids. But when I grew up, great numbers of children did well at arithmetic. They had to, because calculators didn't exist. Very few of those people succeeded at algebra, let alone mathematics. There is a serious disconnect here.
To mention just one problem, you can't say much about individuals from your study of aggregates.
And the ancient Greeks---who invented modern mathematics---are certainly a counterexample to your "natural progression". They accomplished a great deal without beginning with the algorithms we ask kids to study today. Indeed, it's likely that they weren't very good at arithmetic at all. So their "progression", if there was such a thing, was entirely different from the one you think you've identified.
This last example suggests very strongly that arithmetic, while it may be *an* entry into mathematics, is not the *only* entry. Your "natural progression" completely ignores a significant possibility: The primacy of arithmetic is simply an artifact of a curriculum that denies entry to those who haven't acquired proficiency at arithmetic. (A curriculum, moreover, that's now strongly distorted by the effects of fifty years of standardized, multiple-guess, truth-or-consequences, mis-matching tests.)
And consider the popularity of puzzles like sudoku---which are based on very mathematical, but non-arithmetic, reasoning---in a nation that despises mathematics. Where do such phenomena fit in your "natural progression"?
On Thu, Nov 1, 2012 at 11:16 PM, Robert Hansen <email@example.com> wrote:
> > On Nov 2, 2012, at 12:55 AM, Louis Talman <firstname.lastname@example.org> wrote: > > But you haven't explained what the unsuccessful students do wrong. Nor, > perhaps even more importantly, why they miss the boat. And what do their > teachers do wrong? > > > No, I didn't. I only said that I established the elements of success. That > there is a natural order, a progression. That is at least a start. It > establishes that whatever the pedagogy is, one of it's requirements is that > it advance authentically through that progression. So I guess that does > mean that I prefer pedagogies that favor authentic success at each level > over social promotion. > > Bob Hansen >
--Louis A. Talman Department of Mathematical and Computer Sciences Metropolitan State College of Denver