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.
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.
The power of bundle-numbers and per-numbers
THE POWER OF BUNDLE- & PER-NUMBERS UNLEASHED IN PRIMARY SCHOOL: CALCULUS IN GRADE ONE – WHAT ELSE?
In middle school, fraction, percentage, ratio, rate, and proportion create problems to many students. So, why not teach it in primary school instead where they all may be examples of per-numbers coming from double-counting a total in two units. And bundle-numbers with units is what children develop when adapting to Many before school. Here children love counting, recounting, and double-counting before adding totals on-top or next-to as in calculus, also occurring when adding per-numbers. Why not accept, and learn from the mastery of Many that children possess until mathematics takes it away?
Proposals for the 12th Classroom Teaching Research for All Students Conference (CTRAS), as formulated by Shanghai Normal University.
“We are pleased to announce the 2020 Classroom Teaching Research for All Students (CTRAS) Conference, to be held on July 9-11, 2020 in Shanghai, China. The 2020 CTRAS is sponsored this year by the School of Mathematics & Science at Shanghai Normal University.
Background of CTRAS
The Classroom Teaching for All Students Research Working Group was initiated at the U.S.- Sino Workshop in June 2008, which included 11 universities – five from China, and six from the U.S. Since then the CTRAS has been expanded to universities and school districts spanning seven countries and regions from East to West. Each year, the CTRAS has a main research focus area and has an annual conference.
Conference Theme: STEM for All Students in AI Era and the key Competence Education
The 2020 CTRAS will share and discuss the innovative training practices and research initiatives combined with Eastern and Western cultures to support all students’ mathematics learning. The 2020 CTRAS conference will address the following:
1. The research about mathematics classroom teaching based on the key competence development of
2. How mathematics history and culture support the development of key competence of students
3. How digital technologies boost the development of the key competence of students in AI era
4. How Chinese mathematics classroom tradition affects the development of the key competence
5. The research on mathematics curriculum innovation based on the key competence education
6. The strategies to support prospective teachers in STEM teaching design
7. Innovative cases of STEM integration combined with Eastern and Western cultures
8. Engineer-focused STEM education research
9. The most contemporary initiatives of STEM curriculum research
10. The comparison between Eastern and Western mathematics classroom teaching from international perspectives
Submissions and Important Deadlines
We welcome submissions related to the conference theme. Submission can be a project report or a research report.”
- The same Mathematics curriculum for different students
- Comments to a discussion paper
- A Mathematics Teacher Using Communicative Rationality Towards Children
- Addition-free STEM-based Math for Migrants
- Bundle-Counting Prevents & Cures Math Dislike
- Flexible Bundle-Numbers
- Workshop in Addition-free STEM-based Math
- Addition-free STEM-based Math for Migrants, PPP
- Developing the Child’s Own Mastery of Many, outline
- Math Dislike Cured with Inside-Outside Deconstruction
- Learning from The Child’s Own Mathematics
- Five Alternative Ways to Teach Proportionality
- New Textbooks, but for Which of the 3×2 Kinds of Mathematics Education
- Developing the Child’s Own Mastery of Many, paper
- The Child’s Own Mastery of Many, PPP
- Addition-Free Math Make Migrants and Refugees Stem Educators
- Recounting Before Adding Makes Teachers Course Leaders and Facilitators
- Self-explanatory Learning Material to Improve your Mastery of Many
- Can Grounded Math and Education and Research Become Relevant to Learners
- Can Grounded Math and Education and Research Become Relevant to Learners, PPP
- Recounting in Icon-Units and in Tens Before Adding Totals Next-To and On-Top + posters
- What is Math – and Why Learn it?
- Mathematics with Playing Cards
- Mathematics Predicts, PreCalculus
- Sustainable Adaption to Quantity: From Number Sense to Many Sense
- Per-numbers connect Fractions and Proportionality and Calculus and Equations
- Sustainable Adaption to Double-Quantity: From Pre-Calculus to Per-Number Calculations
- A Lyotardian Dissension to the Early Childhood Consensus on Numbers and Operations: Accepting Children’s Own Double-Numbers with Units, and Multiplication Before Addition
- Salon des Refusés, a Way to Assure Quality in the Peer Review Caused Replication Crisis?
- Bundle Counting Table
- Proposals for the 2020 Swedish Math Biennale
- De-Modeling Numbers, Operations and Equations: From Inside-Inside to Outside-Inside Understanding
- De-Model Numbers, Operations and Equations, PPP
- Visit to Ho Chi Minh City University of Education December 7-13 2019
- Review 01 ICMT3
- Review 02 ICMT3
- Comments to ICMT3 Reviewers
- Educating Educators Reviews
Math with playing cards
This booklet contains short articles, most of which have been printed in the LMFK member magazine for Danish upper secondary school math teachers. Thus, (2013.6) indicates that the article has been published in magazine nr. 6 from 2013. The goal is to show how mathematics formulas may be discovered by working with ordinary playing cards. Some formulas are limited by the fact that cards only have positive numbers, so the question if the formulas also apply to negative numbers may be partly answered by testing.
The article on Heron’s formula is the only one not using playing cards.
Allan Tarp, Aarhus, January 2016
01. The little, medium and big Pythagoras with 3, 4 and 5 playing cards (2015, 2) 1
02. PI with three playing cards (2014, 6) 3
03. Proportionality with the 2 playing cards (2015, 1) 4
04. Product rules with 2-4 playing cards (2014, 6) 5
05. The quadratic equation with 2 playing cards (2014, 4) 6
06. Change by adding and multiplying with playing cards 7
07. The saving formula with 9 playing cards (2014, 2) 8
08. The change of a product with 3 playing cards (2013, 6) 9
09. Integral- and differential calculus with 2 playing cards (2015, 1) 10
10. Differentiating sine and cosine with 3 playing cards (2014, 4) 11
11. Topology with 6 playing cards 12
12. Heron’s formula, triangle circles and Pythagoras in factor form (2011, 6) 13
Math Modeling and Models
What is Math – and Why Learn it? iii
A Short History of Mathematics v
Mathematics Before or Through Applications 01
Fact, Fiction, Fiddle – Three Types of Models 09
Applying Mathe-Matics, Mathe-Matism or Meta-Matics (ICME 10) 16
Applying Pastoral Metamatism or Re-Applying Grounded Mathematics (ICME 11) 24
Saving Dropout Ryan with a TI-82 (ICME 12) 35
Sustainable Adaption to Double-Quantity: From Pre-calculus to Per-number Calculations 40
Project Population and Food Growth
Project Saving and Pension
Project Supply, Demand and Market Price
Project Collection and LafferCurve
Project Linear Programming
Project Game Theory
Project Distance to a Far-away Point
Project Vine Box
Project Family Firm
Classic Word Problems
Tarp, A. (2001). Fact, fiction, fiddle – three types of models. in J. F. Matos, W. Blum, K. Houston & S. P. Carreira (Eds.), Modelling and mathematics education. ICTMA 9: Applications in Science and Technology. Proceedings of the 9th International Conference on the Teaching of Mathematical Modelling and Applications (pp. 62-71). Chichester, UK: Horwood Publishing.
Tarp, A. (2001). Mathematics before or through applications, top-down and bottom-up understandings of linear and exponential functions. In J. F. Matos & W. Blum & K. Houston & S. P. Carreira (Eds.), Modelling and Mathematics Education. ICTMA 9: Applications in Science and Technology. Proceedings of the 9th International Conference on the Teaching of Mathematical Modelling and Applications (pp. 119-129). Chichester UK: Horwood Publishing.
Tarp, A. (2019). Sustainable Adaption to Double-Quantity: From Pre-calculus to Per-number Calculations. Paper written for the MADIF 12 Conference in Sweden, rejected. Retrieved at http://mathecademy.net/madif12-2020/.
Tarp, A. (2019). Mathematics Predicts, PreCalculus, How to add constant PerNumbers. Compendium retrieved at http://mathecademy.net/various/us-compendia/
Demodeling mathematics PPP extended
A contribution to an international seminar in Ho Chi Minh City University of Education on Psychology and Mathematics, December 7, 2019