BundleBundle Math on a BundleBundle Board, Math with the Child’s own Numbers

A paradigm-shift from MatheMatism to ManyMath

Existence before Essence means Counting before Adding

BundleBundle Math on a BundleBundle BBBoard

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Grade one Class one, in a Decolonized Future
The teacher: Welcome children, I am your teacher in math, which is about the numbers that you can see on this number line, and that is built upon the fact that one plus one is two as you can see here. So …

Showing a V-sign a child says: “Mister teacher, here is one 1s in space, and here is also one 1s. If we count them in time, we can see how many 1s we have by saying ‘one, two’. So, we have two 1s. But only until we add them as a bundle. Then we have one 2s, so 1s plus 1s become 2s, but one plus one is still one when we count it, and not two as you say. The thumb is also one 1s. They cannot be counted since they are not the same. But they can be added to one 3s. So, again one plus one is one. Here is another 3s on the other hand. They are the same, so we can count them as two 3s. And we can add them as one 6s. Or, we can split the two 3s into six 1s and see that two times three is six. So, the counting numbers two and three can be multiplied, but they cannot be added.

Therefore, please forget adding your line-numbers without units. Instead, help us adding the bundle-numbers with units we bring to school, as 2 3s and 4 5s, that we can add next-to as eights, or on-top as 3s or 5s as we can see on a peg board. If we add them next-to, we add plates, which my uncle calls integral calculus. And if we add them on-top the units must be changed to the same unit, which my uncle calls linearity or proportionality. He says it is taught the first year at college, but we need it here to keep and develop the bundle-numbers with units we bring to school, instead of being colonized with your line-numbers without units.

We know that you want to bundle in tens, and in ten-tens, and in ten-ten tens, but we like to bundle also in 2s, in 3s, in 4s, in half-tens, etc. We know that you have not been taught this and that the textbook doesn’t teach it. But don’t worry, we will teach you what we found out in preschool. Or better, instead of you colonizing our ways let us find out together what math may grow from our bundle-numbers with units. My uncle is a philosopher, and he calls it existentialism if we let existence come before essence.

And, when existence comes before essence, we must count the totals before we can add them. We know you say that 8 divided by 2 is 8 split in 2 parts, but to us 8 divided by 2 is 8 counted in 2s. You cannot split 9 in 2 parts, but we can easily count 9 in 2s as 4 bundles and 1 unbundled that becomes a decimal, 9 = 4B1 2s, or a fraction if we count it in 2s also, 9 = 4 ½ 2s. Or, with negative less-numbers we get 5 bundles less 1, 9 = 5B-1 2s. Now, let us begin with the fingers on a hand. You only see the essence, five, but we see all the ways the five fingers may exist.

F01. A total of fingers many exist as five ones, T = 5 1s, or as one bundle of fives, T = 1B0 5s. Also, the fingers may be bundled in 4s as T = 0B5 = 1B1 4s or as two bundles less 3, T = 2B-3 4s. And the fingers may be bundled in 3s as T = 0B5 = 1B2 = 2B-1 3s. And the fingers may be bundled in 2s as T = 0B5 = 1B3 = 2B1 = 3B-1 2s. But 2 2s is also one bundle of bundles, 1 bundle-bundle, 1 BB, so we also have that T = 1BB0B1 2s. Putting two hands together we see, that eight is one bundle-bundle-bundle, 1 BBB, so that ten is 1BBB0BB1B0 2s. And, if we count ten fingers in 3s, T = 3B1 3s = 1BB0B1 3s. Likewise if we count in tens, twelve is 1B2, and forty-seven is 4B7, and 345 is 3BB4B5.

F02. Here we counted in space, but we also use bundle as the unit when we count in time. If we count our finger in 3s we cannot say ‘1, 2’, and so on since 1 is not 1 3s. Instead, it is 0 bundle 1 3s, so we count ‘0B1, 0B2, 0B3 or 1B0, 1B1, 1B2 or 2B-1’. Or we may count ‘1B-2, 1B-1, 1B0, 2B-2, 2B-1.’

F03. With sticks we see that 5 1s may be bundled as 1 5s that may be rearranged as one icon with 5 sticks. The other digits may also be seen as icons with the number of sticks they repents, where zero is a looking glass finding nothing. We don’t need an icon for ten since here the total is 1B0 if we count in tens.

F04. The calculations are icons also. If we reduce 8 by 2, subtraction is a ‘pull-away icon’ for a rope so that 8-2 means ‘from 8 pull-away 2’. Now a calculator can predict the result, 8 – 2 = 6. And this creates a split formula ‘8 = (8-2) + 2’ telling that 8 remains if the pulled-away is placed next to, or ‘T = (T-B)+B’ with T and B for the total and the bundle.
If we recount 8 in 2s, division is a ‘push-away icon’ for a broom so that 8/2 means ‘from 8 push-away 2s’. Now a calculator can predict the result, 8/2 = 4. If we stack the 4 2s, multiplication becomes a ‘lift icon’ predicting the result, 8 = 4×2. This creates a recount formula ‘8 = (8/2) x 2’ telling that 8 contains 8/2 of 2s, or ‘T = (T/B) x B’ or ‘T = (T/B)*B’ with T and B for the total and the bundle. If we recount 7 in 2s, subtraction is a ‘pull-away icon’ for a rope so 7 – 3*2 means ‘from 7 pull-away 3 2s’.
Finally, addition is a ‘two-ways icon’ showing that two stacks as 2 3s and 4 5s may be added horizontally next-to as areas using integral calculus, or vertically on-top after recounting has made the units like.

F05. A reversed calculation is called an equation using the letter u for the original unknown number. The split and recount formulas may be used to solve equations.
The reverse calculation or equation ‘u+2 = 8’ asks ‘8 is split in 2 and what?’.
The answer, u, is of course if found by the splitting 8 = (8-2) + 2.
So, u+2 = 8 = (8-2) + 2 predicts that u = 8-2, which is also found by simply pulling-away 2 from 8, u = 8-2.
The reverse calculation or equation ‘u*2 = 8’ asks ‘8 is how many 2s?’.
The answer, u, is of course if found by the recounting 8 = (8/2)*2. So, u*2 = 8 = (8/2)*2 predicts that u = 8/2, which is also found by simply recounting 8 in 2s, u = 8/2.
In both cases we see that we find the solution by moving to opposite side with opposite sign. My uncle says that this follows the official definitions. 8-2 is the number u that added to 2 gives 8, so if u = 8-2 then u+2 = 8.
And, 8/2 is the number u that multiplied with to 2 gives 8, so if u = 8/2 then u*2 = 8. And he warned us against a ‘same on both sides’ method you might want to teach us. A combined equation as ‘3*u + 2 = 14’ may be solved by a song:
(3*u) + 2 = 14; 3*u = 14 – 2 = 12; u = 12/3 = 4. TEST: (3*4) + 2 = 12 + 2 = 14.

Equations are the best we know; they’re solved by isolation.
But first the brackets must be placed, around multiplication.
We change the sign and take away, and only u itself will stay.
We just keep on moving, we never give up.
So feed us equations, we don’t want to stop.

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CONTENTS
Abstract
Introduction

● SECTION I, FINDING a New Paradigm, BundleBundleMath

1. Grade one Class one in a Decolonized Future
2. Valid Always or Sometimes,? Mathema-tics or -tism?
3. From Many to Bundle-numbers with Units, for Teachers
4. Micro Curricula, for Learners
5. Many before Math may Decolonize Math, a Video
6. Math Dislike Cured with BundleBundle Math
7. Bundle-counting and Next-to Addition Roots Linearity and Integration
8. Research Project in Bundle-counting and Next-to Addition
9. CATS: Learning Mathematics through Counting & Adding Many in Time & Space
10. The ‘KomMod Report’, a Counter-report to the Ministry’s Competence Report
11. Word Problems
    
    ● SECTION II, REFLECTING on the New Paradigm
    
12. A short History of Mathematics
13. What is Math – and why Learn it?
14. Fifty Years of Research without Improving Mathematics Education, why?
15. Postmodern Enlightenment, Schools, and Learning
16. Can Postmodern Thinking Contribute to Mathematics Education Research
    
    ● SECTION III, SPREADING the New Paradigm
    
17. ICME Conferences 1976, 1996-2024
18. The Swedish MADIF papers 2000-2020
19. The Swedish Mathematics Biennale
20. The MES Conferences
21. CERME Conferences
22. The Catania Trilogy 2015: Diagnosing Poor PISA Performance
23. CTRAS Conferences
24. The 8th ICMI-East Asia Conference on Mathematics Education 2018
25. ICMI Study 24, School Mathematics Curriculum Reforms 2018
26. NORMA 24
27. Curriculum Proposal at a South African teacher college
28. Celebrating the Luther year 1517 with some Theses on Mathematics and Education
29. Invitation to a Dialogue on Mathematics Education and its Research
30. MrAlTarp YouTube videos
    Two-level Table of Contents