Spatial thinking – quantity and dimension
I am not exaggerating if I say that most of us grew up learning the geometric progression of numbers without necessarily tying the geometry aspect to the progression of numbers: 2 ,4, 8, 16 and so on. 2, 2-squared, or 2-cubed was not associated with a linear measurement, a surface measurement or a volume measurement at large. The aspect of generalizing 2 raised to the nth power has conveniently dropped geometry from our faculty of understanding (at least in the lower dimensions). Interestingly, keeping geometry tied to numbers in the lower dimensions would actually otherwise help understanding higher dimensions and handling multidimensional data better.
In fact, I try teaching this to my kids; actually starting from a dimensionless point to three dimensional cube. The concept of keeping quantity and dimension together has greater and less understood benefits. Not only in understanding dimensions but the whole concept of 0 becomes so much simpler and clearer when keeping geometry with the counting. During my growing years I struggled much to understand what is to 0 beyond being the starting point of counting. But now, teaching my kids counting alongside of geometry is throwing whole new light on what 0 is.
Zero is the dimensionless point space out of which everything arises and that the periodicity inherent to geometric patterns causes 0 to reestablish itself in the multidimensional world. As for the kids, (my pre-K’ers in context here) it doesn’t really matter if I keep counting and geometry (quantity and dimension) together or separate from the perspective of learning. They have learned them separately or together with the same ease. However, the level of spatial awareness (as I observed in them) that blossoms from keeping geometry associated with counting is outstanding. The idea of front, back, up, down, left, and right come more naturally to us humans anyway and putting math close to this and then going beyond it only keeps the learning faculties grounded in the space/time continuum – simple eh?
Numbers without geometry could help measure linearities but the highly non-linear world we live in starts with the 4-dimensions anyway. I think keeping numbers and geometry together first and then diving deeper into algebraic expression of numbers beyond geometry would make more sense and will help building geospatial awareness from a younger age. I am actually directly noticing this phenomenon blossom in young minds.