With units, mathematism becomes manymath counting and adding in time and space

Teaching mathematics online is different from teaching it offline in a classroom. So, we may ask what else could be different, its goal, its teaching, its learning, and math itself? Traditionally, the goal of math education is seen as learning to master math to later master Many. So, a difference could be to see the goal of math education as learning to master Many directly to indirectly learning math on the way, at least the core math as displayed on a calculator: digits, operations, and equations. Traditionally these all occur as products in space, so a difference could be to see them as processes in time by letting outside Many precede inside math. And indeed, the math core becomes different when created as tales about Many displayed as rectangular stacks of bundles on a plastic ten-by-ten bundle-bundle board, a BBBoard. To see if a ‘process-based’ or ‘Many-first’ or ‘Many-math’ education will make a difference to the traditional ‘product-based’ ‘Math-first’ ‘’set-Math’, micro-curricula are designed using bundle-counting to bring outside totals inside as flexible bundle-numbers with units. Here both digits and operations arise as icons: digits when uniting sticks, and operations when division pushes away bundles that multiplication lifts onto a stack that subtraction pulls away so the unbundled may be included as decimals, fractions, or negatives. Once counted, a unit may be changed by recounting predicted by a calculator. Here recounting from tens to icons and vice versa leads to equations, and to tables displayed as the stack left when removing the two surplus stacks from the full bundle-bundle on a BBBoard. Here recounting in two physical units leads to per-numbers bridging the two units and becoming fractions with like units. Here recounting the sides and the diagonal in a stack leads to trigonometry before geometry. Finally, once counted, and recounted, totals may add on-top after recounting has provided like units, or next-to as areas as in integral calculus that becomes differential calculus when reversed. As operators needing numbers to become numbers, per-numbers and fractions also add by their areas since they need to be multiplied to unit-numbers before adding. So, outside totals inside appear in an ‘Algebra Square’ where unlike and like unit-numbers and per-numbers are united by addition and multiplication, and by integration and power. And later again split by the reverse operations, subtraction, and division, and by differentiation and root or logarithm. Once process-based Many-first Many-math micro curricula have been designed, they may be tested in online education, as well as in special education to see if a BBBoard may ‘Bring Back Brains’ excluded by the ‘Math-first’ education. But there will be no concert without first designing a score. So detailed testing is not addressed here but left to others to perform.

Contents

Abstract 1

Background 1

MC01. Digits are icons uniting sticks 4

MC02. Operations are icons created by Bundle-counting and re-counting 4

MC03. Bundle-counting in icons 5

MC04. Bundle-counting in tens 6

MC05. Recounting in another unit 6

MC06. Recounting tens in icons gives equations 6

MC07. Recounting icons in tens gives rectangles and multiplication tables 7

MC08. Bundle-Bundles are squares 8

MC09. Three square formulas 9

MC10. Recounting stacks as squares gives square roots to solve quadratics 9

MC11. Recounting in physical units gives per-numbers 11

MC12. Recounting in the same unit gives fractions 11

MC13. Recounting the stack sides gives trigonometry before geometry 11

MC14. Adding next-to and on-top gives calculus and proportionality 12

MC15. Adding and subtracting one-digit numbers 13

MC16. Adding per-numbers gives calculus 13

MC17. Adding unspecified letter-numbers 13

MC18. The Algebra Square 13

MC19. A coordinate system coordinates algebra and geometry. 14

MC20. Change in time: Growth and decay 16

MC21. Distributions in time, probability 19

MC22. Distributions in space, statistics 19

MC23. Simple board games 21

MC24. Modeling and de-modeling 21

MC25. The Three Tales: Fact, Fiction and Fake 22

MC26. Game theory 22

MC27. The Three footnotes 23

MC28. Math with playing cards 23

Teacher education in CATS: Count & Add in Time & Space 23

Discussing the difference 24

Testing the difference 27

Conclusion 28

References 29

Appendix 31

Unit-number tasks 31

Per-number tasks 32

Mechanics 33

The Economic Flow Diagram 34

Meeting many in a STEM context 36

The Twelve Math-Blunders 39