Calculus Reified extended abstract

Calculus and Linearity in Grade One Dramatically Improve College Performance abstract

The key to core mathematics is one simple question: “5 4s and 3 2s add to what?”

Adding next-to as areas brigs you directly to integral calculus. And adding on-top after shifting units brings you directly to proportionality that leads on to per-numbers as 2$/5kg when including physical units.

Which again leads to the root of calculus, adding per-numbers as in mixture problems: Adding 2kg at 3$/kg and 4kg at 5$/kg, the unit-numbers 2 and 4 add directly to 6, but the per-numbers 3 and 5 must be multiplied to unit-numbers before adding, thus, adding as areas as integral calculus, becoming differential calculus when reversing the question: “2kg at 3$/kg and 4kg at how many $/kg add to 6kg at 5$/kg”, or “5 4s and how many 2s add to 5 6s?”.

Calculus thus occurs in three versions.

Primary school: adding bundle-numbers by their areas.

Middle school: adding piece-wise constant per-numbers as the area under the per-number graph.

High school: adding locally constant per-numbers as the area under the per-number graph.