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?

Paper Proposals

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
students
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
education
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.”

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

Content

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

A Dansih version

Math Modeling and Models

Content

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 Forecasting
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 Bridge
Project Driving
Project Vine Box
Project Golf
Project Family Firm
Classic Word Problems
References
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/.
Projects from:
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

ABSTRACTof the paper: DE-MODELING NUMBERS, OPERATIONS AND EQUATIONS: FROM INSIDE-INSIDE TO OUTSIDE-INSIDE UNDERSTANDING
Adapting to the outside fact Many, children internalize social number-names, but how do they externalize them when communicating about outside numerosity? Mastering Many, children use bundle-numbers with units; and flexibly use fractions and decimals and negative numbers to account for the unbundled singles. This suggests designing a curriculum that by replacing abstractbased with concrete-based psychology mediates understanding through de-modeling core
mathematics, thus allowing children to expand the number-language they bring to school.

New Curricula for Primary and Middle and High School

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

Material from the Educating Educators Conference ion Freiburg, Germany, October 7-8 2019

Educating Educators PPP

Can Grounded Math and Education and Research Become Relevant to Learners, presentation

Self-explanatory Learning Material to Improve your Mastery of Many, poster

Recounting Before Adding Makes Teachers Course Leaders and Facilitators, poster

Educating Educators, material market

Developing the child’s mastery of many, part 1

Developing the child’s mastery of many, part 2

Developing the child’s mastery of many, part 3

The paper may be found on page 329-334 in the proceedings

ICMT3_proceedings

Finger counting math
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 leaning materials?’.
This motivates another question ‘How much mathematics can be learned as ‘finger-counting math’ using the examples of Many coming from the body as 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.
The text is taken from the paper ‘The Same Mathematics Curriculum for Different Students’ written for the ICMI Study 24, School Mathematics Curriculum Reforms: Challenges, Changes and Opportunities, held in Tsukuba, Japan, 26-30 November 2018. It builds on the article
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.
http://mathecademy.net/journofmathed1101/