Let Kids teach teachers teach MatheMatics as ManyMath, a natural science about MANY using flexible bundle-numbers with units to count and recount before adding on-top and next-to; the CATS approach: Count & Add in Time & Space
Lessons in BBM Bundle-Bundle Math Fairy-told by Bo, a self-educated pre-teen child still living in an enchanted world with Bundle-numbers as 2 3s and 4B2 5s existing on a BundleBundleBoard.
Allan.tarp@gmail.com published on LinkedIn April-May 2025, https://www.linkedin.com/in/allantarp
Lesson 01. I count my fingers in 2s 5 = 0B5 = 1B3 = 2B1 = 3B-1 2s. Also 5 = 1BB 0B1 2s, and ten = 2BB0B2 = 1BBB 0BB 1B 0 2s. Likewise 37 = 3B7 = 2B17 = 4B-3. Otherwise, ten = 3B1 = 1BB 0B1 3s.
Lesson 02. I build squares with BundleBundles I see that 3 3s = 1BB 2B 1 2s = 1BB -2B 1 4s. So now I can learn the square-numbers 1, 4, 9, 16, 25 from the bottom and 81, 64, 49, and 25 from the top, of course sharing the same last digits.
Lesson 3. I square 6 4s to find its square-root I square 6 4s to find its square-root by moving half the excess form the top to the side to give 5 5s. So my first guess is that the square root of 6 4s is 5, which is too much since both 5s must give away a slice to fill the empty corner. Lesson 04. I will re-count 8 1s in 2s to change the units
“No, 1+1 is not 2, but 1 as shown by a Collapsing V-Sign” said the Children in their Declaration of Independence.
The 4th UN Sustainable Development Goal wants to ensure that all youth and most adults achieve literacy and numeracy.
This makes education a core institution meant to ‘teach learners something’.
But an institution must choose between different views on teaching and learning, and on what to master first, the outside goal or an inside means, Many or math.
These choices are discussed by the three grand theories, philosophy and sociology and psychology.
When adapting to Many in time and space before school children use their innate number-sense to develop bundle-numbers with units like two 3s.
The educational choice then is: shall existence precede essence, or shall essence be allowed to colonize existence with a ‘no-unit regime’?
Will children stay numerate if their own 2D bundle-numbers with units are left uncolonized instead of being colonized by the institutionalized 1D line-numbers without units?
The children may think so and formulate their own declaration of independence inspired by the parallel American one.
The paper ‘The 12 Math-Blunders of Killer-Mathematics’ was written for the fifth Swedish Mathematics Education Research Seminar, MADIF 5, in Malmoe in Sweden in 2006. After its rejection it was presented at the 41. Tagung für Didaktik der Mathematik in Berlin Germany. The paper defines ‘killer-mathematics’ as the authorized routines that threaten to kill the enrolment to mathematics based education by creating enrolment problems; and that threatens to kill the relevance of mathematics. Using the principles of natural learning and natural research concepts and theory are based upon laboratory observations and validations. In this way a natural mathematics can be recreated revealing 12 math-blunders in mathematics education. The blunders are treating both numbers and letters as symbols, 2digit numbers before decimal numbers, fractions before decimals, forgetting the units, addition before division, fractions before pernumbers and integration, proportionality before doublecounting, balancing instead of backward calculation, killer equations instead of grounded equations, geometry before trigonometry, postponing calculus, and finally the 5 metablunders of mathematics education. The paper was rejected and only accepted as a short presentation. The way it was rejected led to the production of the following mini paper.
The mini-paper called ‘Moo Review and Tabloid Review’ was written in response to the rejection of the paper above. The paper defines ‘Moo-review’ as a review containing at most a single sound; and ‘Tabloid-review’ as a review containing only a single sentence. Three reviewers reviewed the paper above. The review contained 14 examples of moo-review, and 11 examples of tabloid-review. Only 2 statements contained two sentences. The paper uses this observation to raises some questions and to give some recommendations as how to use other forms of review questions. Also it proposes that moo-review and tabloid-review should not be accepted since statements as ‘yes’ and ‘no’ are verdicts, but in any democratic society a verdict always rests upon testimonies and cross-examination. The paper was sent to the conference editors, but it was not published.
I wrote a 20 pages 10K words chapter “Demodeling Calculus from Deriving Functions to Adding Per-numbers” to a coming Springer Nature book “New trends in teaching and learning of calculus” published in the book series “Research in Mathematics Education”.
The chapter points out that calculus is needed in grade one to add the bundle-numbers with units that children bring to school, e.g., ‘2 3s + 4 5s = ? 8s’, i.e., if you want to teach mathematics instead of ‘mathematism’ claiming that 1+1 = 2 despite 1week+1day = 8 days.
However, it was rejected based on two peer reviews. I gladly accept quality reviews, but in this case I think they may deserve the label ‘Poor Peer Review’.
And frankly, I am a little tired of Poor Peer reviews, so I have prosed to Springer to write a book with examples of Poor Peer Reviews and a Journal called ‘Preventing Poor Peer Review’.
You find the two reviews below as well as an extract from the paper.
The full paper may be seen here until the end of February 2025.
Review 01.
The proposed work is new and original, and it also has interesting historical and philosophical aspects. However, its connection with the teaching and learning of Calculus is minimal , so it does not seem to be the appropriate book for this work. On the other hand, the proposal is not truly a research work , since it lacks a methodological section, field work, results, etc. In summary, I do not recommend the publication of this work for the book “New Trends in Teaching and Learning of Calculus” and I would suggest the author to send it to a publication more appropriate to his subject matter.
Review 02.
The proposed chapter presents a number of difficulties that, given my expertise, I believe require a detailed examination, particularly in relation to the intended audience. One of my first observations is that throughout the text several shortcomings in its structuring are identified. In several sections , the relevance of sustainable development is mentioned (e.g. the goal By 2030, ensure that all youth and a substantial proportion of adults, both men and women, achieve literacy and numeracy), but this idea is presented in a superficial way , without a clear link to school reality and the description of the objectives of the 2030 Agenda. In my opinion, the author takes only the title of this agenda and develops his own interpretation , without going deeper into the subject. During the reading, doubt arises about the target audience of the proposal , and although it is mentioned in a general way that it should be implemented in primary education, it is not made explicit in such a clear way. Another important aspect is the lack of empirical studies to support the proposal, which would demonstrate its relevance for teaching and learning of calculus. Instead, the approach seems more like a personal intention of the author . While this approach is not necessarily negative, in the context of a book aimed at the teaching and learning of calculus , it is not appropriate, or at least not at this stage of development. As a specialist in mathematics didactics, I am concerned about the implementation of this proposal in primary school students. In a relatively recent history (around 1960), the ‘New Math’ trend generated significant problems in teaching by prioritising mathematical formality over students’ cognitive readiness to understand these concepts. This chapter reminds me, to some extent, of that trend, so I would pay attention to it. Now, in mathematical terms, the proposal is interesting, but I think it would be more suitable for students training to be teachers or mathematicians. This approach would allow them to visualise and explore mathematics from another perspective, as well as to work on demodeling processes, an aspect that I also find valuable on the part of the author. In conclusion, I consider that the proposal needs a thorough restructuring . This includes clearly defining the target audience , conducting a more detailed analysis of the school curriculum (including a review of educational reforms in mathematics teaching), and supporting the proposal with empirical studies that demonstrate its relevance .
Math decolonized by the child’s own BundleBundle-Numbers with units.
The Mathematics Education for the Future Project A Symposium on New Ways of Teaching & Learning The Historic City of Bologna, Italy, August 6-10, 2024
Project Home Pages: https://sites.google.com/view/alan-rogerson-home/home
And, https://sites.unipa.it/grim/21project.htm Organising Committee:Alan Rogerson,Jasia Morska and Simone Brasili Local Organisers: Chiara Chiarini and Alan Rogerson In Cooperation with: Budapest Semesters in Mathematics Education, Hong Kong Institute of Education, MUED, DQME3, MAV, AWM, ATM, AAMT, Wholemovement, MACAS, AIMSSEC, Mathematics Education Centre, Institute for Mathematics, Faculty of Sciences, Eötvös Lóránd University, Budapest, International Symmetry Association and WTM-Verlag.
OUR PROJECT and OUR CONFERENCES
The programme for the Symposium is 5 nights B&B, Aug 6-10, 2024 Aug 6: Arrival ONLY & Registration: NO meals, NO sessions (B&B only) Aug 7-10: 4 work Days with presentations and lunch Aug 7: Welcome Reception at 6.15pm – cocktails/snacks Aug 10: Gala Dinner 7pm in Hotel Restaurant Aug 11: Departure ONLY with check-out after breakfast
Bologna is a city full of history. Its medieval center is among the largest in Europe and its forty-two kilometers of porticoes, and the medieval towers, give Bologna its unique character. Bologna University was founded in 1088 and is the oldest university in the world! (yes.. older even than “the other two places” in the UK!) Bologna has an international airport and is also well served by High Speed Trains that run the whole length of Italy with frequent connections to Milan, Firenze, Rome, Naples etc.
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Video: Many before Math, Math decolonized by the child’s BundleBundle-Numbers with units. Workshop: Flexible Bundle Numbers Develop the Childs Innate Mastery of Many. A BBBoard shows that 67 = (B-4)*(B-3) = (10 – 4 – 3)*B + 4*3 = 3B12 = 4B2 = 42
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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
Grade one Class one in a Decolonized Future
Valid Always or Sometimes,? Mathema-tics or -tism?
From Many to Bundle-numbers with Units, for Teachers
Micro Curricula, for Learners
Many before Math may Decolonize Math, a Video
Math Dislike Cured with BundleBundle Math
Bundle-counting and Next-to Addition Roots Linearity and Integration
Research Project in Bundle-counting and Next-to Addition
CATS: Learning Mathematics through Counting & Adding Many in Time & Space
The ‘KomMod Report’, a Counter-report to the Ministry’s Competence Report
Word Problems
● SECTION II, REFLECTING on the New Paradigm
A short History of Mathematics
What is Math – and why Learn it?
Fifty Years of Research without Improving Mathematics Education, why?
Postmodern Enlightenment, Schools, and Learning
Can Postmodern Thinking Contribute to Mathematics Education Research
● SECTION III, SPREADING the New Paradigm
ICME Conferences 1976, 1996-2024
The Swedish MADIF papers 2000-2020
The Swedish Mathematics Biennale
The MES Conferences
CERME Conferences
The Catania Trilogy 2015: Diagnosing Poor PISA Performance
CTRAS Conferences
The 8th ICMI-East Asia Conference on Mathematics Education 2018
ICMI Study 24, School Mathematics Curriculum Reforms 2018
NORMA 24
Curriculum Proposal at a South African teacher college
Celebrating the Luther year 1517 with some Theses on Mathematics and Education
Invitation to a Dialogue on Mathematics Education and its Research
MrAlTarp YouTube videos Two-level Table of Contents
Et BBBræt, der viser, at 67 = (B-4)*(B-3) = (10 – 4 – 3)*B + 4*3 = 3B12 = 4B2 = 42
Matematik: MatemaTisme, eller MangeMatik med barnets 2D BundtBundtTal på etBBBræt?
Matematik blir så let hvis Mange mestres først med MangeMatik. MATERIALE, se senere.
MangeMatik respekterer, at MANGE beskrives med barnets egne BundtBundt-tal med enheder – i stedet for at få påtvunget falsk WOKE-identitet som linjetal uden enheder, der blir matematisme ved at påstå, at 2+1 er 3 altid, til trods for at 2dage+1uge er 9dage.
MangeMatik ses ved at spørge en 3-årig “Hvor mange år næste gang?” Svaret er 4, med 4 fingre vist. Men holdt sammen 2 og 2, indvender barnet “Det er ikke 4, det er 2 2ere.” Barnet ser således, hvad der findes i rum og tid, bundter af 2ere i rummet, og 2 af dem i tid, når de tælles. Så det, der eksisterer, er totaler, der kan optælles til (gen)forening (algebra på arabisk) i rum og tid, som fx 2B1 2ere.
MangeMatik bygger på filosofien eksistentialisme, der anbefaler, at eksistens går forud for essens, der ellers vil kolonisere eksistensen. Det eksternt eksisterende går altså forud for interne ’essens-regimer’ som bør dekonstrueres & demodeleres så eksistensen afkoloniseres
BundtTal med enheder: 8, 0B8, 1B-2, eller 1B0 8ere. Og 87, 8B7, 9B-3.
Many before Math, Math decolonized by the child’s own BundleBundle-Numbers with units, en YouTube video.
INDHOLD Sammendrag Oversigt Fra Mange til bundt-tal med enheder, for lærere Mikro-læseplaner for lærende ML01. Cifre som ikoner i rummet, IIIII = 5 ML02. Styk-tælling i tid, = IIII I ML03. Bundt-tælling i tid med enheder: 0B1, …, 0B5 eller 1B0, 3 3ere = 1BB ML04. Fleksibel bundt-tælling i rum med over- og under-læs, 5 = 1B3 = 2B1 = 3B-1 2ere ML05. Opdeling, 8 = (8-2) + 2 ML06. Omtælling, 8 = (8/2)x2 ML07. Omtælling af de ubundtede, 8 = (8/3)3 = 2B2 = 2 2/3 = 3B-1 3ere ML08. Omtælling i kvadrater, 6 4ere = 1 BB ?ere ML09. Omtælling til et andet ikon, 3 4ere = ? 5ere ML10. Omtælling fra ti’ere til ikoner, 2 ti’ere = ? 7ere ML11. Omtælling fra ikoner til ti’ere, 6 7ere = ? ti’ere ML12. Omtælling til en anden fysisk enhed skaber per-tal, 3kr/5kg ML13. Med samme enhed bliver per-tal til brøker, 3kr/5kr = 3/5 ML14. Omtælles en staks sider giver trigonometri, højde = (højde/bund)bund = tanA*bund ML15. Plusning vandret eller lodret, T = 2 3ere + 4 5ere = ? 8ere; T = ? 3ere; T = ? 5ere ML16. Minus og plus med etcifrede tal, 8 + 6 = 1B2 + 1B0 = 2B2 6ere ML17. Plusning af per-tal og brøker er integralregning ML18. Plusning af Bundt-Bundt kvadrater ML19. Plusning af ukendte bogstavtal ML20. Ændring i tid ML21. Bundt-tal i et koordinatsystem ML22. Spil-teori og skade-kontrol ML23. Enkle brætspil Algebra-tavlen Fakta og fiktion og fup, de tre genrer i tal-modeller Modellering og de-modellering Tre fodnoter Læreruddannelse Hvor forskellig er forskellen? Oversigt over forskellen mellem Essens- and Eksistens-matematik Konklusion Referencer Kronikker og læserbreve om matematik 2023-2024 Matematik-skandalen: Skolens matematisme berøver barnet dets tal-sprog og talsans Fra katolsk til protestantisk matematik, fra bevis til beregning Foredrag for skolebørn om min matematikbog på BogForum 2023 Coronatiden, krise eller skandale, hvad med en høring? Ny matematik og ny skole nu, ellers uddør vi Afkoloniser tal-sproget nu Lær dit barn at matematikke før skolen gør det Kan matematikken afkoloniseres ved at ombytte essens med eksistens?
