Many before Math, Math decolonized by the child’s own BundleBundle-Numbers with units, a YouTube video, and a PDF version.

Flexible Bundle-numbers Develops Children’s Innate Mastery of Many workshop, YouTube video

Flexible Bundle Numbers Workshop Web, PDF-version

Bundle-Bundle-numbers with Units may make Children stay Numerate, booklet

Many-Math, sheet

Many-Math, folder

Apparently, we have 2 Mathematics Paradigms, one without Units, and one With Units

• an inside ‘no-unit-math’ paradigm, where 1 plus 2 is 3 always despite 1week + 2days is 9days, and
• an outside ‘unit-math’ paradigm, where 1 plus 2 depends on the units

The ‘unit-math’ paradigm builds on the philosophy, EXISTENTIALISM, where EXISTENCE precedes ESSENCE
So, unit-math describes real existence, and neglects institutionalized essence

The outside, ‘unit-math’ paradigm, provides the same mathematics, as the inside, ‘no-unit-math’ paradigm, only in a different order. And, the ‘unit-math’ paradigm, avoids the inside paradigm’s ‘mathema-tism’, with its falsifiable, addition-claims.

So, to become a full science, mathematics should leave, its 1 plus 2 is 3, ‘no-unit-math’ greenhouse, and accept that, of course, numbers cannot add, without units.

It should teach the outside ‘counting-before-adding’, ‘unit-math’, paradigm, where Numbers and operations are icons, linked directly to existing things, and actions

The Canceled Curriculum Chapter in the ICMI Study 24

A curriculum for a class is like a score for an orchestra. Follow it, and the result will be a perfect performance. In music perhaps, but not always in a class.
It begins so well. Textbooks follow curricula, and teachers follow the textbooks supposed to mediate perfect learning. But, as shown in international tests, this does not always take place for all learners. But then, other scores may be more successful? Well, with few variations, scores seem to teach the same in the same way: numbers, operations, calculations, formulas, and forms. Why is there so little room for improvisation as in jazz?
So, with the transformation of modern society into a postmodern version, the time has come to ask: How about jazzing up the curricula to allow children’s quantitative competences and talents to blossom?
As a curriculum architect using difference research to uncover hidden differences that may make a difference, I warmly greeted the announcement of ICMI study 24 with the title ‘School Mathematics Curriculum Reforms: Challenges, Changes and Opportunities’. I was especially excited about including opportunities, which would allow hidden differences to be noticed and perhaps tested. And I jumped for joy with the acceptance of my paper ‘A Twin Curriculum Since Contemporary Mathematics May Block the Road to its Educational Goal, Mastery of Many.’
At the conference I was asked to contribute writing a report on part B2 asking ‘How are mathematics content and pedagogical approaches in reforms determined for different groups of students (for e.g. in different curriculum levels or tracks) and by whom?’ The deadline was end June 2019, but shortly before I was told that this part would be canceled and not appear in the report. Still, I finished my contribution and sent it in. But as expected, it has not been included. Consequently, I have chosen to publish it as an appendix to the ICMI 24 study.

Introduced at the beginning of the century, eight competencies should solve poor math performance.

Adopted in Sweden together with increased math education research mediated through a well-funded centre, the decreasing Swedish PISA result came as a surprise, as did the critical 2015 OECD-report ‘Improving Schools in Sweden’. But why did math competencies not work?

A sociological view looking for a goal displacement gives an answer: Math competencies sees mathematics as a goal and not as one of many means, to be replaced by other means if not leading to the outside goal.

Only the set-based university version is accepted as mathematics to be mediated by teachers through eight competencies, where only two competences are needed to master the outside goal of mathematics education, Many.

The Danish KOM-report Kompetencer og matematiklæring

The Danish version of the KOMMOD counter report

The English version of the KOMMOD counter report

Math Competenc(i)es – Catholic or Protestant? a paper for the MADIF 11 2018 conference

An ICME Trilogy from ICME 10, 11, and 12

ICME 13 papers

ICME 14 2020 and 2021 papers

• The power of bundle- & per-numbers unleashed in primary school: calculus in grade one – what else?
• From loser to user, from special to general education, learning inside mathematics through outside actions

ICME 15 papers:

• Modeling Eased by DeModeling and ReRooting

• Against Alien AI: A TEXT-FREE Math Education

The MADIF papers 2000-2020

Decolonizing 1D Mathema-tism into 2D Many-math

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?

The goal of math education, is that to learn to master math to later master Many, or the other way around?

● 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 the math core is different when created as tales about Many existing as rectangular stacks of bundles on a plastic ten-by-ten bundle-bundle board, a BBBoard. To see if a ‘process-based’ ‘Many-first’ education will make a difference to the traditional ‘product-based’ ‘Math-first’ education, micro-curricula are designed using bundle-counting to bring outside totals inside as flexible bundle-numbers with units, that are rectangular where the bundle-bundles are squares.
● Here both digits and operations arise as icons. Digits when uniting sticks. And operations with division to push-away bundles that multiplication lifts into a stack.
Now subtraction pulls-away stacks so unbundled are included as decimals, fractions, or negatives. The addition cross shows the two ways to add, next-to & on-top.
● Once counted, changing unit may be predicted on a calculator by the recount formula T = (T/B) x B, saying that the total T contains T/B Bundles.
Here recounting from tens to icons and vice versa leads to equations, and to multiplication tables displayed as the stack left when removing the two surplus stacks from the full bundle-bundle on a BBBoard. And here recounting from rectangles to squares introduces its side as the square root, and a way to solve quadratics.
● Here recounting in two physical units leads to per-numbers bridging the two units and becoming fractions with like units.
● Here mutual recounting the sides and the diagonal in a stack leads to trigonometry before geometry.
● Once counted, totals may add on-top after recounting has provided like units, or next-to as areas as in integral calculus becoming differential calculus if reversed.
● As operators needing numbers to become numbers, per-numbers and fractions also add by their areas after being 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 from the ‘Math-first’ education.

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

Papers presented at the CTRAS 2023 June Conference, Classroom Teaching Research for All Students

Artificial Intelligence makes Difference Research more relevant
Allan Tarp, the MATHeCADEMY.net
Research typically is seen as an example of a top-down or bottom-up lab-lib cooperation where laboratory observations are deduced from or are inducing library concepts. In the top-down version, library theory generates a hypothesis that, validated or falsified in the laboratory, leads to a strengthened or adapted theory. In the bottom-up version, laboratory observations lead to categories, that additional observations may split into subcategories.
Artificial Intelligence has access to the library, but laboratory data will be input. Top-down research thus may be generated very quickly with a quality depending on the reliability of the input, which may be difficult to check.
In contrast, AI is of less help to bottom-up research typically generating new categories not yet present in the library.
Also Difference Research searching for differences making a difference (Tarp, 2018) may now be more relevant since although AI may locate existing differences, it cannot invent new differences. Nor can it examine the difference they make.
Examples of difference research are bundle-numbers with units, operations as icons for counting, re-counting to change unit, per-numbers coming from recounting in two units, integration as addition of locally constant per-numbers, trigonometry before geometry, and mathematism adding numbers without units.
References:
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.

Online math opens for a communicative turn in number language education
Allan Tarp, the MATHeCADEMY.net
“Of course, the goal of mathematics education is to master mathematics before it can be applied to later master Many.” Seeing this as a ‘goal displacement’ making a means a goal, sociology (Bauman, 1990) and difference research (Tarp, 2018) suggests a communicative turn as in foreign language education in the 1970s (Widdowson, 1978): Maybe mastering Many is a more accessible way to later master Mathematics. Likewise, existentialism (Sartre, 2007) holds that existence should precede essence.
Seeing 4 fingers 2 by 2 as ‘2 2s’ shows that preschool children master Many with 2D bundle-numbers with units. In this ‘BundleNumber-math’ adding is preceded by counting, which de-models (Tarp, 2020) division and multiplication as icons for a broom and a lift to push-away the unit-bundles to be lifted as a stack. They combine in a ‘recount-formula’ (Tarp, 2018), T = (T/B)xB, predicting that T contains T/B Bs. Subtraction iconizes a rope to pull-away the stack to find unbundled that are placed on-top as decimals, fractions, or negatives. Addition iconizes adding on-top and next-to.
This allows both school and education students to be guided by the concrete subject on their desktop instead of by an instructor on a screen, as exemplified by the MATHeCADEMY.net.
References:
Bauman, Z. (1990). Thinking sociologically. Blackwell.
Sartre, J.P. (2007). Existentialism is a humanism. Yale University Press.
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. (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.
Widdowson, H. G. (1978). Teaching language as communication. Oxford University Press.

Matematik er let, utrolig let, næsten for let. Så hvorfor er det så så svært i skolen?

Se videoen “Matematik er bare så let, hvis Mange-mestring får forrang. Se også dens PDF version her.

Der findes to slags tal, styk-tal og per-tal, der kan være uens eller ens, og som vi skal genforene (arabisk: algebra). Matematiks ærinde er derfor ikke at ’matematikke’, for det kan man jo ikke, men at handle:

”at genforene uens & ens styk-tal & per-tal”.

• 3 kroner og 2 kroner er uens styktal, og her forudsiger regnestykket 3+2 = 5 resultatet af at forene dem.

• 3 gange 2 kroner er ens styktal, og her forudsiger regnestykket 3*2 = 6 resultatet af at forene dem.

• 3 gange 2% er ens per-tal, og her forudsiger regnestykket 102%^3 = 106,12% resultatet af at forene dem til 6% samt 0,12% ekstra i ’rentes-rente’.

• Uens per-tal er fx blandinger som 2kg á 3kr/kg og 4 kg á 5kr/kg. Her kan styk-tallene 2 og 4 forenes direkte, medens per-tallene 3 og 5 først skal opganges til styk-tal, før de kan forenes som arealer, også kaldet at integrere, hvor gange kommer før plus: T = (2+4) kg til (2*3+4*5) kr, altså 6 kg á 26/6 kr/kg.

Fortsæt her

Folder

English versions: Folder and Sheet

Woke-Math respects flexible bundle-numbers for a total instead of colonizing it by forcing a linear number upon it.

So, Woke-Math wants to decolonize by warning that imposing line-numbers without units as five upon a total of fingers will disrespect the fact that the actual total may exist in different forms, all with units: as 5 1s, as 1 5s, as 1Bundle3 2s, as 2Bundle1 2s, as 3Bundle-1 2s, etc.    

Woke-Math builds on a basic observation asking a 3year old “How many years next time?” The answer typically is four showing four fingers. Holding them together 2 by 2, that child objects “That is not 4, that is 2 2s.”

The child thus sees what exist in space and time, bundles of 2s in space, and 2 of them in time when counted. So, what exist are totals to be counted and added in time and space, as T = 2 2s.

Woke-Math builds on the philosophy called existentialism holding that existence precedes essence to prevent the latter from colonizing the former, i.e., that what exists outside precedes what we say about it inside.

By de-modeling mathematics instead of modeling reality, Woke-Math offers “Master Many to master Math” as a decolonized alternative to the traditional approach, “Master Math to master Many”.

Woke-Math respects, that outside totals inside may be counted and recounted in various two-dimensional bundle-numbers with units; and rejects one-dimensional line-numbers without units since they lead to ‘mathematism’ true inside, but seldom outside, where 2+3 = 5 is falsified by 2weeks + 3days = 17 days.

WokeMath intro

Folder

Sheet

Flexible Bundle-numbers Develops Children’s Innate Mastery of Many workshop, YouTube video

Flexible Bundle Numbers Workshop Web, PDF-version

Apparently, we have 2 Mathematics Paradigms, one without Units, and one With Units
• an inside ‘no-unit-math’ paradigm, where 1 plus 2 is 3 always, and
• an outside ‘unit-math’ paradigm, where 1 plus 2 depends on the units
The ‘unit-math’ paradigm builds on the philosophy, EXISTENTIALISM, where EXISTENCE precedes ESSENCE
So, unit-math describes real existence, and neglects institutionalized essence
The outside, ‘unit-math’ paradigm, provides the same mathematics, as the inside, ‘no-unit-math’ paradigm, only in a different order. And, the ‘unit-math’ paradigm, avoids the inside paradigm’s ‘mathema-tism’, with its falsifiable, addition-claims.
So, to become a full science, mathematics should leave, its 1 plus 2 is 3, ‘no-unit-math’ greenhouse, and accept that, of course, numbers cannot add, without units.
It should teach the outside ‘counting-before-adding’, ‘unit-math’, paradigm,
where Numbers and operations are icons, linked directly to existing things, and actions

PP1 Allan Tarp(MATHeCADEMY.net)/ Ethical Math Welcomes Primary and Middle School Calculus (https://youtu.be/PedfLp1XiQU)

PP2 Allan Tarp(MATHeCADEMY.net)/ In Ethical Math, Mastery of Many Precedes Mastery of Math (https://youtu.be/xELn2-6vefc)

PP3 Allan Tarp(MATHeCADEMY.net)/ Demodeling Math Outside its ‘no-unit-math’ Greenhouse (https://youtu.be/K4RhJtll1W8)

PP4 Allan Tarp(MATHeCADEMY.net)/ Bring Back Brains from Special Ed: Recount, don’t Add (https://youtu.be/yg5994TVH0Y)

PP5 Allan Tarp(MATHeCADEMY.net)/ 8 Competencies? No, only 2 Competences: Count & Add (https://youtu.be/wTIbOvVZCZs)

PP6 Allan Tarp(MATHeCADEMY.net)/ Including All in PreCalculus with a PerNumber Curriculum (https://youtu.be/l-BHSh8Kq94)

PP7 Allan Tarp(MATHeCADEMY.net)/ From STEM to STEAM – or to STEEM with Economics Instead? (https://youtu.be/W-njyKtiBWY)

Outside, addition folds but multiplication holds, since factors are units while addition presupposes like units. This creates two paradigms in mathematics, an outside ‘unit’ paradigm, and an inside ‘no-unit’ paradigm making mathematics a semi-greenhouse. To make mathematics a true science with valid knowledge, we ask what mathematics can grow from bundle-numbers with units, being areas instead of points on a number line. Concretely constructed, digits become number-icons with as many sticks as they represent, and operations become counting-icons for pushing, lifting and pulling away bundles to be added next-to or on-top. Recounting 8 in 2s creates a recount-formula, T = (T/B) x B, saying that T contains T/B Bs. By changing units, it occurs as proportionality formulas in science; it solves equations; and it shows that per-numbers and fractions, T/B, are not numbers, but operators needing numbers to become numbers. Fractions, decimals, and negative numbers describe how to see the unbundled. Recounting sides in a box halved by its diagonal allows trigonometry to precede plane and coordinate geometry. Once counted, total may add on-top after recounting makes the units the same; or next-to addition by adding areas as in integral calculus, which also occurs when adding per-numbers. So, mathematics created outside the ‘no-units’ greenhouse is the same as inside, only the order is different, and all is linked directly to outside things and actions making it easier to be applied. And, with multiplication preceding it, addition only occurs as integral calculus, unless inside brackets with like units.

YouTube Video: Children’s innate Mastery of Many developed by flexible bundle-numbers

ICTMT15_Workshop