In another thread, Lou Talman and Robert Hansen are discussing sequence.
Lou brings up the point that the ancient Greeks invented much of what we call mathematics without being especially proficient in arithmetic in the way our own subcultures were recently trained. We aren't the same way about arithmetic today, since calculators and other devices made everything so easy, although we still school in "the algorithms" in ways the Greeks never did. Were they "unsuccessful in math"?
Quoting Lou: """ 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.) """
Of course once you get through whatever sequence growing up, you're not done with sequences. It's always "one damn thing after another" (Henry Ford, on history). So even if we argue about the "one right sequence" (I'm against the notion) for child-to-adult math learning, we're not done. What else might we try with adults?
I'm probably more into adult guinea pigging than most. For me, it's not all about what we teach to 10 year olds.
Even with kids though, I emphasize V + F == E + 2. I don't worry about what if it has more holes too much. It's a wire frame to begin with probably and a simple convexity, a polyhedron for which we have a name probably, or part of a series.
[ Waterman Polyhedra for example, I worked on them, with Steve, their definer and early sculptor of their form (he used spreadsheets and actual physical models made of little balls) -- our team on Synergetics-L (e.g. Gerald de Jong, myself) did a bunch of the computer graphics, using Qhull. Other yet more expert implementations by other collaborators (with Steve Waterman) came later. Google and ye shall find. ]
Lou again: """ 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"? """