Math 2050

Online math may create a communicative turn in number-language education also

Allan Tarp, the,

Summer 2023

Paper presented at the CTRAS 2023 June Conference.



Abstract 1

Background 1

MC01. Digits are icons uniting sticks 3

MC02. Operations are icons created by Bundle-counting and recounting 3

MC03. Bundle-counting in icons 4

MC04. Bundle-counting in tens 5

MC05. Recounting tens in icons gives equations 5

MC06. Recounting icons in tens gives rectangles and multiplication tables 5

MC07. Bundle-Bundles are squares 6

MC08. Recounting rectangles as squares gives square roots to solve quadratics 7

MC09. Recounting in physical units gives per-numbers 8

MC10. Recounting in the same unit gives fractions 8

MC11. Recounting rectangle-sides gives trigonometry before geometry 8

MC12. Adding next-to and on-top gives calculus and proportionality 9

MC13. Adding per-numbers gives calculus 9

MC14. Adding unspecified letter-numbers 10

MC15. The Algebra Square 10

MC16. Algebra and geometry coordinated. 10

MC17. Numbers in time and space: Change and distribution 12

MC18. The Three Tales: Fact, Fiction and Fake 12

MC19. Teacher education in CATS: Count & Add in Time & Space 13

Discussing the difference 13

Testing the difference 17

Conclusion 17

References 18


Teaching mathematics online is different from teaching it offline in a classroom. So, we may ask what else could be different? On a calculator we see the core of mathematics: digits, multidigit numbers, operations, and equations. But they all occur as products in space, not as processes in time. So, maybe teaching mathematics ‘process-based’ instead of ‘product-based’ is different by letting outside Many precede inside Math? And indeed, the math core becomes different when created from tales about Many displayed as rectangles on a concrete ten-by-ten bundle-bundle board, a BBBoard. To see if a ‘process-before-product’ or ‘Many-before-Math’ education will make a difference, micro-curricula are designed to bring outside totals inside by bundle-counting creating flexible bundle-numbers with units: ones, bundles, bundle-of-bundles, etc., that all are included in oral counting sequences. Here digits arise as icons when uniting sticks. Here operations arise as icons as well: 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. Here recounting from tens to icons leads to equations, and when reversed, to tables displayed as the rectangle left when removing the two surplus rectangles 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 rectangle 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 these curriculum scores have been designed, they may be played in online education, as well as in special education to see if a BBBoard may ‘Bring Back Brains’ excluded by the ‘Math-before-Many’ education. But there will be no concert without first designing a score. So actual testing is not addressed here but left to others to perform.