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

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.

From STEM over STEAM to STEEM

• 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?

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

Here are examples of online mathematics for primary and secondary school

Refugee Camp Math

Learn Math trough Kid’s Tile-Math

Math with Playing Cards

Geometry from Below

Math Modeling & Models

Counting and Recounting before Adding

Math Dislike Cured with flexible bundle numbers

Teaching Mathematics as Communication journal

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.

Mathematics language or grammar    Video

Trigonometry Before Geometry Probably Makes Every Other Boy an Excited Engineer    Video

Introducing the MATHeCADEMY dot net     Video

Teaching Mathematics as Communication journal

All presented at the 12th International Multi-Conference on
Complexity, Informatics and Cybernetics: IMCIC 2021©
March 9 – 12, 2021 ~ Virtual Conference

Thank you for your meaningful presentation at the 2020 International Conference of the Korean Society of Mathematical Education.
There are more than 380 registrations and 180 presentations at the conference.
Please share your research ideas that can be applied in the field for mathematics education.
Thank you again for your contribution to the conference.
Mangoo Park
President of the Korean Society of Mathematical Education
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Tarp_poster paper

Tarp_poster_handout

Tarp poster video

Tarp_workshop

Tarp_workshop-handout

Workshop in developing the childs own mastery of many

Tarp_oral_presentation

Tarp_ video

Geometry from below

‘Geometry from below’ means geometry as tales about a social practice, in this case about ‘earth-measurement’, to which the Greek word ’geo-metry’ can be directly translated.

The earth is what we live on and what we live on. We divide the earth between us by drawing dividing lines.

If these boundaries disappear, it is important to be able to re-establish them, and this restoration of lines and corners requires that these can be measured.

In ancient Egypt, the Nile thus crossed its banks once a year and brought manure to the fields. After retiring, the divisions had to be re-established.

Geometry from below can be understood as the opposite of geometry from above, deducing geometry from metaphysical truths, axioms.

The following material is not a traditional textbook, but rather an activity guide with suggestions for a range of activities that the reader can perform and report.

So, the idea is that the reader builds his own textbook.