Math with Units & the Child’s own Numbers with Bundle-units to Count & Add in Time & Space.

Numeracy as Math with units where addition folds while multiplication holds.

A paradigm-shift from MatheMatism without units to ManyMath with units.

Existence before Essence means Counting before Adding.

Introduction
Looking at four fingers bundled in twos, an educated person sees four fingers, the essence. But, as uneducated before school, children see what exists, bundles of twos in space, and two of them when counted in time. We thus observe the existence of two kinds of mathematics: ‘half-matics’ that teach only the counting-numbers, 1, 2, 3, etc.; and ‘full-matics’ that connects counting-numbers in time with bundle-numbers in space 1s, 2s, 3s, etc. to teach the child’s own numbers with bundle-units as 2 3s visible and tangible on a ten-by-ten BundleBundleBoard. We may now ask how mathematics may be taught to children if using their own two-dimensional numbers with bundle-units instead of the school’s one-dimensional line-numbers without units. In other words, we may ask how children may learn mathematics by meeting existence before essence to use the two core concepts of Existentialism holding that existence must precede essence.
Counting outside Totals in Bundles then must precede adding them inside. In this ‘Bundle-math’ approach, mathematical concepts are re-rooted in outside existing examples instead of being defined as examples itself inside. Now tens, hundreds and thousands become bundles, bundle-bundles, and bundle-bundle-bundles, as does 2, 4 and 8 when counting in twos instead.
Bundle-counting, one-dimensional lines on a ruler are replaced by two-dimensional rectangles on a BBBoard, containing the outside existing subjects that is linked to inside essence predicates in a number-language sentence as in a word-language sentence. Here units are always included in counting sequences as 0Bundle1, 0B2, …, 1B0. Here digits become icons with as many sticks as they represent. Here also operations become icons created in the counting process. An outside Total may be split by pulling away a Bundle, using a rope as an icon for subtraction. Pulled back, the total is restored in a Reunite-formula, T = (T – B) + B, using a double-rope as an icon for pulling back. Or, the outside Total may be counted in Bundles by pushing away Bundles, using a broom as an icon for division. Pushed back, the total is restored in a Recount-formula, T = (T/B) x B, using a lift as an icon for pushing back. The Recount formula is used all over STEM to change units.
Recounting from tens to icons, the equation, u2 = 8, together with another basic equation, u+2 = 8, are solved by recounting and reuniting, both moving numbers to ‘opposite side with opposite sign’: u2 = 8 = (8/2)x2, so u = 8/2, and, u + 2 = 8 = (8 – 2) + 2, so u = 8 – 2.
Recounting from icons to tens leads to early algebra when including the Bundle-unit. Then 6*7 becomes (B-4) * (B-3) placed on a BBBoard and found by pulling-away the top 4B and side 3B, and adding the 43 pulled away twice. Or by writing 67 = 6* ½B2 = 3B12 = 4B2. And, bundle-bundles squares allow rectangular stacks to be recounted in squares with the square root as the side.
Recounting in physical units creates per-numbers as 4$/5kg bridging the units by recounting: 20kg = (20/5)5kg = (20/5)4$ = 16$. With like units, per-numbers become fractions: 4$/5$ = 4/5.
Recounting the sides in a stack halved by its diagonal leads to trigonometry before geometry.
Recounting now is followed by reuniting (called ‘Algebra’ in Arabic). Stacks may add on-top after recounting has made the units like, or next-to as areas, i.e., as integral calculus (or differential calculus if reversed), also used to add per-numbers and fractions that must be multiplied to unit-numbers to add. Squares add as the square formed by their mutual Bottom-Top line.
Units create an ‘Algebra Square’ to reunite our four number-types. Add and multiply unite unlike and like unit-numbers, where integrate and power unite unlike and like per-numbers. That are split by subtraction, division, differentiation, and the factor-finding root or the factor-counting logarithm.
Counting before adding thus gave us a number-language to tell inside number-tales about outside totals using the same three genres, fact and fiction and fake, as the word-language uses.
Can mathematics be decolonized? Well of course, since mathematics is a socially constructed essence that will always be a colonization of the natural existence it came from and reduces.
Allan.Tarp@gmail.com, Aarhus, Denmark, August 2025

Contents

Contents
A Reaction to a BBM BundleBundle Math Curriculum 5
Introduction i
A Future? From STEM to STeN to make all Youth Numerate by 2030 ii
From STEM to STeN, why? ii
Two different Definitions of ‘Numerate’ exist ii
Economics gives a Fundamental Understanding of Numbers and Calculations ii
Numeracy as Math Counting Totals in Units before Adding them with Units iii
References iv
Micro Curricula in BBM BundleBundleMath, Counting before Adding 1
MC01. Digits as icons in space, IIIII = 5 1
MC02. Tally-counting in time, = IIII I 1
MC03. Bundle-counting in time with units: 0B1, …, 0B5 or 1B0, 3 3s = 1BB 2
MC04. Bundles counted in space with over- and underloads, 5 = 1B3 = 2B1 = 3B-1 2s 3
MC05. Splitting, 8 = (8-2)+2 3
MC06. Recounting, 8 = (8/2)x2 4
MC07. Including the unbundled, 8 = (8/3)3 = 2B2 = 2 2/3 = 3B-1 3s 5 MC08. Recounting in squares, 6 4s = 1 BB ?s 5 MC09. Recounting in another icon, 3 4s = ?5 6 MC10. Recounting from tens to icons, 2 tens = ? 7s 6 MC11. Recounting from icons to tens, 6 7s = ? tens 6 MC12. Recounting in another physical unit creates per-numbers, 3$/5kg 7 MC13. With the same unit, per-numbers become fractions, 3$/5$ = 3/5 8 MC14. Recounting a stack’s sides gives trigonometry, rise = (rise/run)run = tanA*run 8
MC15. Adding next-to or on-top, T = 2 3s + 4 5s = ? 8s; T = ? 3s; T = ? 5s 9
MC16. Subtracting and adding single digit numbers, 8+6 = 1B2 + 1B0 = 2B2 6s 10
MC17. Adding per-numbers and fractions by integral calculus 10
MC18. Adding and subtracting Bundle-Bundle squares 11
MC19. Adding unspecified letter-numbers 12
MC20. Change in time 12
MC21. Bundle-numbers in a coordinate system 13
MC22. Games Theory and damage control 14
MC23. Simple board games 15
The Algebra Square 15
Fact and fiction and fake, the three genres of number-models 16
Modeling and de-modeling 16
Three footnotes 19
Teacher education 19
How different is the difference? 20
Overview of the differences between Essence- math and Existence-math 21
Reactions to a BBM BundleBundle Math Curriculum 22
Grand Theory looks at Mathematics Education 22
Conclusion. In Numeracy education, a Luhmann understanding is essential 25
References 26
BundleBundleMath Wonders 01-24 28
BundleBundleMath Wonder 01. Adding last numbers. 28
BundleBundleMath Wonder 02. Numbers as Pictures, Viking Numbers. 28
BundleBundleMath Wonder 03. Addition 29
BundleBundleMath Wonder 04. Subtraction. 30
BundleBundleMath Wonder 05. Multiplication. 30
BundleBundleMath Wonder 06. Long multiplication. 30
BundleBundleMath Wonder 07. Short division. 31
BundleBundleMath Wonder 08. Long division. 31
BundleBundleMath Wonder 09. Equations Solved: to Opposite Side with Opposite Sign 32
BundleBundleMath Wonder 10. Square Roots. 33
BundleBundleMath Wonder 11. Add and Subtract Squares. 33
BundleBundleMath Wonder 12. Finding the Square Root of a number. 34
BundleBundleMath Wonder 13. Squaring Rectangles. 35
BundleBundleMath Wonder 14. Squaring Triangles. 35
BundleBundleMath Wonder 15. Squaring Circles. 36
BundleBundleMath Wonder 16. Squares in STEM and STeN 37
BundleBundleMath Wonder 17. Pouring Water with Quadratic equations. 37
BundleBundleMath Wonder 18. Multiplication tables on a BundleBundleBoard 39
BundleBundleMath Wonder 19. Adding on-top with proportional ReCounting. 40
BundleBundleMath Wonder 20. Adding next-to with Integral Calculus. 40
BundleBundleMath Wonder 21. ReCounting Goods gives PerNumbers and Fractions 41
BundleBundleMath Wonder 22. The ReCount Formula and per-numbers are the core of STEM and STeN (economics & Numeracy included) 42
BundleBundleMath Wonder 23. Per-numbers add as Areas (Integral Calculus) 42
BundleBundleMath Wonder 24. The Algebra Square ReUnites Unlike & Like, Unit- & Per-numbers 43

Presentation Category at EARCOME9: Special Sharing Groups (SSG). Title of Paper: CAN A DECOLONIZED MATHEMATICS SECURE NUMERACY FOR ALL?
Announcement: This proposal tackles an urgently needed conversation in mathematics education by challenging deeply ingrained assumptions about number systems and arithmetic instruction and proposing a truly decolonized approach that foregrounds learners’ intuitive “bundle‑number” language. Its strength lies in weaving together a compelling theoretical critique—drawing on Habermas’s colonization concept and rich philosophical underpinnings—with concrete instructional innovations like the Algebra Square that reframe operations as intuitive spatial and bundling processes. By aligning this reconceptualization directly with SDG 4’s numeracy targets and illustrating how multiplication‑centered reasoning better reflects real‑world number use, the paper promises to make a bold and impactful contribution to both research and practice. We look forward to seeing how this work can reshape numeracy instruction and foster truly inclusive mathematical literacy for all.

Articles
• Tarp, A. (2001). Fact, Fiction, Fiddle – Three Types of Models, in J. F. Matos & W. Blum & K. Houston & S. P. Carreira (Eds.), Modelling and Mathematics Education: ICTMA 9: Applications in Science and Technology. Proceedings of the 9th International Conference on the Teaching of Mathematical Modelling and Applications (62-71), Horwood Publishing.
• Tarp, A. (2018). Mastering Many by counting, re-counting and double-counting before adding on-top and next-to. Journal of Mathematics Education, 11(1), 103-117.
• Tarp, A. (2018). Good, bad & evil mathematics – tales of totals, numbers & fractions. In Hsieh, F. J. (Ed.), (2018). Proceedings of the ICMI-East Asia Regional Conference on Mathematics Education, Vol2, Taipei, Taiwan: EARCOME8, 163-173.
• Tarp, A. (2020). De-modeling numbers, operations and equations: From inside-inside to outside-inside understanding. Ho Chi Minh City University of Education Journal of Science 17(3), 453-466.
• Tarp, A. (2021). Teaching Mathematics as Communication, Trigonometry Comes Before Geometry, and Probably Makes Every Other Boy an Excited Engineer. Complexity, Informatics and Cybernetics: IMCIC 2021.
• Tarp, A. (2025). Math is fun with bundle-numbers on a bundle-bundle-board. In Kwon, O., Kaur, B., Pang, J., Noh, J., Lee, S., Han, S., Yeo, S., & Lim, M. (Eds.). (2025). Proceedings of the 9th ICMI-East Asia Regional Conference on Mathematics Education (Vol. 1). Seoul National University, Siheung Campus, Korea: EARCOME9, 363-392.