# How Kids Master Many

Outside, addition folds but multiplication holds, since factors are units while addition presupposes like units. This creates two paradigms in mathematics, an outside ‘unit’ paradigm, and an inside ‘no-unit’ paradigm making mathematics a semi-greenhouse. To make mathematics a true science with valid knowledge, we ask what mathematics can grow from bundle-numbers with units, being areas instead of points on a number line. Concretely constructed, digits become number-icons with as many sticks as they represent, and operations become counting-icons for pushing, lifting and pulling away bundles to be added next-to or on-top. Recounting 8 in 2s creates a recount-formula, T = (T/B) x B, saying that T contains T/B Bs. By changing units, it occurs as proportionality formulas in science; it solves equations; and it shows that per-numbers and fractions, T/B, are not numbers, but operators needing numbers to become numbers. Fractions, decimals, and negative numbers describe how to see the unbundled. Recounting sides in a box halved by its diagonal allows trigonometry to precede plane and coordinate geometry. Once counted, total may add on-top after recounting makes the units the same; or next-to addition by adding areas as in integral calculus, which also occurs when adding per-numbers. So, mathematics created outside the ‘no-units’ greenhouse is the same as inside, only the order is different, and all is linked directly to outside things and actions making it easier to be applied. And, with multiplication preceding it, addition only occurs as integral calculus, unless inside brackets with like units.

ICTMT15_Workshop