Author Archives: Allan Tarp

A DeColonized Curriculum

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

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?

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.

Korea KSME 2020

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
Copyright (c) 2003, The Korean Society of Mathematical Education. All rights reserved. Mail to Webmaster
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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

‘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.

Corona Infection Model

Calculation Models Fact or Fiction, a letter to the editor

The 3FactorFormula says balance meetings, a letter to the editor

On Use and Misuse of Corona Mathematics, shows how mathematics is used and misused

Comparing the Danish and the American Infection curves, shows how the Danish curve was copied directly from the American without giving the source


YouTube video The two Infection formulas part 1

YouTube video The two Infection formulas part 2

Video pictures and text

Wrong Numbers and Formulas Locked Denmark Down, a letter to the editor


Population Based Logistic Hospitalization Model, a report

Population Based Logistic Hospitalization Model, a spreadsheet model

newly-admitted-over-time data


Debatindlæg om corona situationen

 


Direct links

The two Infection formulas part 1:

 

The two Infection formulas part 2:

 

Learn Math trough Kid’s Tile-Math

Learn Core Math Through Kid’s Tile-Math
with worksheets here Math Dislike Cured with flexible bundle numbers

Asked ‘How old next time?’, a 3-year-old says ‘Four’ showing four fingers; but objects when seeing them held together two by two: ‘That is not four, that is two twos!’ A child thus sees what exists in the world, bundles of 2s, and 2 of them. So, adapting to Many, children develop bundle-numbers with units as 2 2s having 1 1s as the unit, i.e. a tile, also occurring as bundle-of-bundles, e.g. 3 3s, 5 5s or ten tens.
Recounting 8 in 2s as 8 = (8/2)x2 gives a recount-formula T = (T/B)xB saying ‘From the total T, T/B times, B can be pushed away’ occurring all over mathematics and science. It solves equations: ux2 = 8 = (8/2)x2, so u = 8/2. And it changes units when adding on-top, or when adding next-to as areas as in calculus, also occurring when adding per-numbers or fractions coming from double-counting in two units. Finally, double-counting sides in a tile halved by its diagonal leads to trigonometry.

The following papers present close to 50 micro-curricula in Mastering Many inspired by the bundle-numbers children bring to school.
Learn Core Mathematics Through Your Kid’s Tile-Math:
Recounting Bundle-Numbers and Early Trigonometry
This first paper is written for the conference ‘The Research on Outdoor STEM Education in the digiTal Age (ROSETA) Conference’ planned to take place between 16th and 19th June 2020 at Instituto Superior de Engenharia do Porto in Portugal.
The Power of Bundle- & Per-Numbers Unleashed in Primary School:
Calculus in Grade One – What Else?
This second paper is written for the International Congress for Mathematical Education, ICME 14, planned to be held in Shanghai from July 12th to 19th, 2020, but postponed one year.