Category Archives: CupCount

Matematik er let

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

Der findes jo kun to slags tal i verden, styk-tal og per-tal, som kan være uens eller ens., og som skal opsamles eller opdeles.

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

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

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

Uens per-tal findes fx i blandinger som 2kg a 3kr/kg og 4 kg a 5kr/kg. Her kan styktallene 2 og 4 samles direkte, medens per-tallene 3 og 5 først skal opganges til styktal, før de kan samles som arealer, også kaldet at integrere, hvor gange kommer før plus:

2 kg til 2*3 kr + 4 kg til 4*5 kr = 6 kg til 26 kr, altså 6 kg a 26/6 kr/kg.

Fortsæt her



Woke-Math never Forces Fixed Forms upon Flexible Totals.

Woke-math respects flexible bundle-numbers for a total instead of imposing a linear number upon it.

So, woke-math warns 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 old 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 thus builds on the philosophy called existentialism holding that existence precedes essence, 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 an 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

Korean Math Ed 2021

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

The Korean Society of Mathematical Education has their yearly conference i mid December.
I had the presentation above, a workshop, and 7 YouTube posters accepted.

PP1 Allan Tarp( Ethical Math Welcomes Primary and Middle School Calculus (

PP2 Allan Tarp( In Ethical Math, Mastery of Many Precedes Mastery of Math (

PP3 Allan Tarp( Demodeling Math Outside its ‘no-unit-math’ Greenhouse (

PP4 Allan Tarp( Bring Back Brains from Special Ed: Recount, don’t Add (

PP5 Allan Tarp( 8 Competencies? No, only 2 Competences: Count & Add (

PP6 Allan Tarp( Including All in PreCalculus with a PerNumber Curriculum (

PP7 Allan Tarp( From STEM to STEAM – or to STEEM with Economics Instead? (

How Kids Master Many

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


Master Many, ReCount before Adding

To explain 50 years of low-performing mathematics education research, this paper asks: Can mathematics and education and research be different? Difference-research searching traditions for hidden differences provides an answer: Traditional mathematics, defining concepts from above as examples of abstractions, can be different by instead defining concepts from below as abstractions from examples. Also, traditional line-organized office-directed education can be different by uncovering and developing the individual talent through daily lessons in self-chosen half-year blocks. And traditional research extending its volume of references can be different, either as grounded theory abstracting categories from observations or as difference-research uncovering hidden differences to see if they make a difference. One such difference is: To improve PISA performance, Count and ReCount before you Add.

To master Many Recount before Adding paper

To master Many Recount before Adding Power Point Presentation

To master Many Recount before Adding video

Canceled Curriculum Chapter in ICMI 24

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.



  • Yes, core mathematics may be learned through its historic root, economics, describing how humans share what they produce
  • Asking “How many did I produce?” roots counting, predicted by division iconizing a broom pushing away bundles, to be stacked by a multiplication-lift, to be pulled away by a subtraction-rope to look for unbundled singles, to be added on-top or next-to, thus rooting decimal and negative numbers
  • Recounting in a new unit creates a recount formula, used to solve equations, and to change units as in most STEM formulas
  • Uniting stacks on-top or next-to roots proportionality or calculus
  • So why make mathematics hard when it may also be easy & meaningful?

Calculus adds PerNumbers

The key to core mathematics is one simple question: “5 4s and 3 2s add to what?”
Adding next-to as areas brigs you directly to integral calculus. And adding on-top after shifting units brings you directly to proportionality that leads on to per-numbers as 2$/5kg when including physical units.
Which again leads to the root of calculus, adding per-numbers as in mixture problems: Adding 2kg at 3$/kg and 4kg at 5$/kg, the unit-numbers 2 and 4 add directly to 6, but the per-numbers 3 and 5 must be multiplied to unit-numbers before adding, thus, adding as areas as integral calculus, becoming differential calculus when reversing the question: “2kg at 3$/kg and 4kg at how many $/kg add to 6kg at 5$/kg”, or “5 4s and how many 2s add to 5 6s?”.
Calculus thus occurs in three versions.
Primary school: adding bundle-numbers by their areas.
Middle school: adding piece-wise constant per-numbers as the area under the per-number graph.
High school: adding locally constant per-numbers as the area under the per-number graph.

Calculus papers

Calculus Reified extended abstract

Calculus and Linearity in Grade One Dramatically Improve College Performance abstract

As operators, per-numbers are multiplied before adding as areas abstract

From evil to good calculus that adds locally constant per-numbers abstract

Refugee Camp Math

A Refugee Camp Curriculum

The name ‘refugee camp curriculum’ is a metaphor for a situation where mathematics is taught from the beginning and with simple manipulatives. Thus, it is also a proposal for a curriculum for early childhood education, for adult education, for educating immigrants, and for learning mathematics outside institutionalized education.
It considers mathematics a number-language parallel to our word-language, both describing the outside world in full sentences, typically containing a subject and a verb and a predicate.
The task of the number-language is to describe the natural fact Many in space and time, first by counting and recounting and double-counting to transform outside examples of Many to inside sentences about the total; then by adding to unite (or split) inside totals in different ways depending on their units and on them being constant or changing.
This allows designing a curriculum for all students inspired by Tarp (2018) that focuses on proportionality, solving equations and calculus from the beginning, since proportionality occurs when recounting in a different unit, equations occur when recounting from tens to icons, and calculus occurs when adding block-numbers next-to and when adding per-numbers coming from double-counting in two units.
Talking about ‘refugee camp mathematics’ thus allows locating a setting where children do not have access to normal education, thus raising the question ‘What kind and how much mathematics can children learn outside normal education especially when residing outside normal housing conditions and without access to traditional learning materials?’.
This motivates another question ‘How much mathematics can be learned as ‘finger-math’ using the examples of Many coming from the body like fingers, arms, toes and legs?’
So, the goal of ‘refugee camp mathematics’ is to learn core mathematics through ‘Finger-math’ disclosing how much math comes from counting the fingers.