Fresh start precalculus
Using Sociological Imagination to Create a Paradigm Shift
As a social institution, mathematics education might be inspired by sociological imagination, seen by Mills and Baumann as the core of sociology.
Thus, we might have s Kuhnian paradigm shift if, as a number-language, mathematics would follow the communicative turn that took place in language education in the 1970s lead by  Halliday in 1973 nad Widdowson in 1978 by prioritizing its connection to the outside world higher than its inside connection to its grammar.
So why not try designing a fresh-start precalculus curriculum that begins from scratch to allow students gain a new and fresh understanding of basic mathematics, and of the real power and beauty of mathematics, its ability as a number-language for modeling to provide an inside prediction for an outside situation? Therefore, let us try to design a precalculus curriculum through, and not before its outside use.
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.
Teaching material may be found in a compendium called ‘Mathematics Predicts’ (Tarp, 2009).
http://mathecademy.net/various/us-compendia/

What is Math and Why Learn it

”What is math – and why learn it?” Two questions you want me to answer, my dear nephew.

0. What does the word mathematics mean?
In Greek, ‘mathematics’ means ’knowledge’. The Pythagoreans used it as a common label for their four knowledge areas: Stars, music, forms and numbers. Later stars and music left, so today it only includes the study of forms, in Greek called geometry meaning earth-measuring; and the study of numbers, in Arabic called algebra, meaning to reunite. With a coordinate-system coordinating the two, algebra is now the important part giving us a number-language, which together with our word-language allows us to assign numbers and words to things and actions by using sentences with a subject, a verb and a predicate or object:
“The table is green” and “The total is 3 4s” or “T = 3*4”. Our number-language thus describes Many by numbers and operations.

1. Numbers and operations are icons picturing how we transform Many into symbols
2. Placeholders
3. Calculation formula predict
4. Reverse calculations may also be predicted
5. Double-counting creates per-numbers and fractions
6. Change formulas
7. Use
8. Conclusion

Same math for different students

To offer mathematics to all students, parallel tracks often occur from the middle of secondary school. The main track continues with a full curriculum, while parallel tracks might use a reduced curriculum leaving out e.g. calculus; or they might contain a different kind of mathematics meant to be more relevant to students by including more applications. Alternatively, a single curriculum may be designed for all students no matter which track they may choose if mathematics as a number-language follows the communicative turn that took place in language education in the 1970s by prioritizing its connection to the outside world higher than its inside connection to its grammar. We will consider examples of all three curricula options.

Written as a proposal for a chapter for the ICMI STUDY 24: School Mathematics Curriculum Reforms: Challenges, Changes and Opportunities, key question B2: How are mathematics content and pedagogical approaches in reforms determined for different groups of students.

Reference: ICMI study 24 (2018). School Mathematics Curriculum Reforms: Challenges, Changes and Opportunities. Pre-conference proceedings. Editors: Yoshinori Shimizu and Renuka Vithal.

CTRAS 2019 contributions

Paper & poster & workshop

Addition-free STEM-based math for migrants……………………………………… 01

Decreased PISA performance despite increased research……………………… 01

Social theory looking at mathematics education…………………………………… 01

Different kinds of education………………………………………………………………… 02

The tradition of mathematics education……………………………………………….. 03

Theorizing the success of mathematics education research…………………… 04

Difference research looks at mathematics education…………………………….. 07

Meeting many creates a ‘count-before-add’ curriculum………………………… 07

A short curriculum in addition-free mathematics………………………………….. 09

Meeting Many in a STEM context……………………………………………………….. 10

Adding addition to the curriculum……………………………………………………….. 13

Conclusion and recommendation…………………………………………………………. 15

Poster in outside-inside mathematics…………………………………………………… 17

Workshop in addition-free STEM-based mathematics………………………….. 21

Paper PowerPointPresentation……………………………………………………………..25

Defining, as Habermas, communicative rationality as ‘wanting to reach understanding to secure the participant speakers an intersubjectively shared lifeworld, thereby securing the horizon within which everyone can refer to one and the same objective world’; and defining the objective world as ‘the totality of entities concerning which true propositions are possible’ (thus, to avoid self-reference, not seeing propositions as part of the objective world); and seeing a speech act as ‘a speaker pursuing the aim of reaching understanding with a hearer about something’, we might ask:

How can a math teacher use communicative rationality to establish a non-patronizing power-free rational dialogue with grade one children about the objective fact Many, present in both the children and the teacher’s life-world; thus accepting four fingers held together two by two being rationalized as (as do the children) ‘the total I two twos’ and not just as ‘four’?

A Habermas note

A NEW CURRICULUM BUT FOR WHICH OF THE 3×2 KINDS OF MATHEMATICS EDUCATION
An essay on observations and reflections at the ICMI study 24 curriculum conference

As part of institutionalized education, mathematics needs a curriculum describing goals and means. There are however three kinds of mathematics: pre-, present and post-‘setcentric’ mathematics; and there are two kinds of education: multi-year lines and half-year blocks. Thus, there are six kinds of mathematics education to choose from before deciding on a specific curriculum; and if changing, shall the curriculum stay within the actual kind or change to a different kind? The absence of federal states from the conference suggests that curricula should change from national multi-year macro-curricula to local half-year micro-curricula; and maybe change to post-setcentric mathematics.

Core Papers 2017-2018
This selection contains four papers written in 2017 and 2018
01. The Simplicity of Mathematics Designing a STEM-based Core Math Curriculum for Outsiders and Migrants.
This article is due to be published in the next number 34 of Philosophy of Mathematics Education Journal.
The abstract says that Swedish educational shortages challenge traditional mathematics education offered to migrants. Mathematics could be taught in its simplicity instead of as ‘meta-matsim’, a mixture of ‘meta-matics’ defining concepts as examples of inside abstractions instead of as abstractions from outside examples; and ‘mathe-matism’ true inside classrooms but seldom outside as when adding numbers without units. Rebuilt as ‘many-matics’ from its outside root, Many, mathematics unveils its simplicity to be taught in a STEM context at a 2year course providing a background as pre-teacher or pre-engineer for young migrants wanting to help rebuilding their original countries.

02. Addition-free migrant-math rooted in STEM re-counting formulas

A short version of the article above was sent to the Topic Working Group 26 on STEM mathematics at the CERME 11 conference. It was rejected as a paper, so it was redrawn.

The abstract says that a curriculum architect is asked to avoid traditional mistakes when designing a curriculum for young migrants that will allow them to quickly become STEM pre-teachers and pre-engineers. Typical multiplication formulas expressing re-counting in different units suggest an addition-free curriculum. To answer the question ‘How many in total?’ we count and re-count totals in the same or in a different unit, as well as to and from tens; also, we double-count in two units to create per-numbers, becoming fractions with like units. To predict, we use a re-count formula as a core formula in all STEM subjects.

03. Mastering Many by Counting, Re-counting and Double-counting before Adding On-top and Next-to

This article was published in the Journal of Mathematics Education, March 2018, 11(1), 103-117.

The abstract says that observing the quantitative competence children bring to school, and by using difference-research searching for differences making a difference, we discover a different ‘Many-matics’. Here digits are icons with as many sticks as they represent. Operations are icons also, used when bundle-counting produces two-dimensional block-numbers, ready to be re-counted in the same unit to remove or create overloads to make operations easier; or in a new unit, later called proportionality; or to and from tens rooting multiplication tables and solving equations. Here double-counting in two units creates per-numbers becoming fractions with like units; both being, not numbers, but operators needing numbers to become numbers. Addition here occurs both on-top rooting proportionality, and next-to rooting integral calculus by adding areas; and here trigonometry precedes geometry.

04. A Twin Curriculum Since Contemporary Mathematics May Block the Road to its Educational Goal, Mastery of Many

This article was accepted at the conference ICMI Study 24, School Mathematics Curriculum Reforms: Challenges, Changes And Opportunities, in Tsukuba Japan, 26-30 November 2018. The abstract says that mathematics education research still leaves many issues unsolved after half a century. Since it refers primarily to local theory, we may ask if grand theory may be helpful. Here philosophy suggests respecting and developing the epistemological mastery of Many children bring to school instead of forcing ontological university mathematics upon them. And sociology warns against the goal displacement created by seeing contemporary institutionalized mathematics as the goal needing eight competences to be learned, instead of aiming at its outside root, mastery of Many, needing only two competences, to count and to unite, described and implemented through a guiding twin curriculum.

**Counting before Adding, The Child’s Own Twin Curriculum, Count & ReCount & DoubleCount before Adding NextTo & OnTop

This is a Power Point Presentation made from the article above.

Addition-free migrant-math rooted in STEM re-counting formulas
Paper for the STEM working group at the CERME 11 conference
A curriculum architect is asked to avoid traditional mistakes when designing a curriculum for young migrants that will allow them to quickly become STEM pre-teachers and pre-engineers. Typical multiplication formulas expressing re-counting in different units suggest an addition-free curriculum. To answer the question ‘How many in total?’ we count and re-count totals in the same or in a different unit, as well as to and from tens; also, we double-count in two units to create per-numbers, becoming fractions with like units. To predict, we use a re-count formula as a core formula in all STEM subjects.
AdditionFree Math

A TWIN CURRICULUM SINCE CONTEMPORARY MATHEMATICS MAY BLOCK THE ROAD TO ITS EDUCATIONAL GOAL, MASTERY OF MANY
Count & ReCount & DoubleCount before Adding NextTo & OnTop
Mathematics education research still leaves many issues unsolved after half a century. Since it refers primarily to local theory, we may ask if grand theory may be helpful. Here philosophy suggests respecting and developing the epistemological mastery of Many children bring to school instead of forcing ontological university mathematics upon them. And sociology warns against the goal displacement created by seeing contemporary institutionalized mathematics as the goal needing eight competences to be learned, instead of aiming at its outside root, mastery of Many, needing only two competences, to count and to unite, described and implemented through a guiding twin curriculum.

A Twin Curriculum in Counting & Adding PPP
A Twin Curriculum in Counting & Adding Paper

MathEdResearch_2018

A collection of articles about Mathematics Education written in 2018

1. Posters at the 2018 Swedish Biennial stand
2. Migrants Master Many by Re-counting in Block- & Per-numbers – Short
3. Migrants Master Many by Re-counting in Block- & Per-numbers – Long
4. Math: Useful Abstractions or an Undiagnosed Compulsory Cure? Conflicting Theories in Mathematics Education – Short
5. Math: Useful Abstractions or an Undiagnosed Compulsory Cure? Conflicting Theories in Mathematics Education – Long
6. A STEM-based Mathematics without Addition – Short
7. A STEM-based Mathematics without Addition – Long
8. Cure Math Dislike with 1 Cup & 5 Sticks: New Meanings to Numbers, Counting, Operations & Fractions – Short
9. Curing Math Dislike with 1 Cup & 5 Sticks: New Meanings to Numbers, Counting, Operations & Fractions – Long
10. Good Math & Goofy Math, which is Truer? – Short
11. Good Math & Goofy Math, which is Truer? – Long
12. A Core STEM Curriculum for Young Migrants – Short
13. A Core STEM Curriculum for Young Migrants – Long
14. Fifty Years of Ineffective Math Education Research, Why? Oops, Wrong Numbers, Sorry
15. Rethinking Line-Number Arithmetic as Block-Number Algebra
16. A Count-before-Adding Curriculum for Preschool and Migrants
17. Difference-research Saving Dropout Ryan with a TI-82 Calculator
18. Conflicting Theories Help Teachers Improve Mathematics Education
19. Addition-free Core STEM Curriculum for Late Learners along the Silk Road
20. Good, Bad & Evil Mathematics – Sociological Imagination in Math Education
21. Remedial Math MicroCurricula – When Stuck in a Traditional Curriculum
22. Mastering Many by Counting, Re-counting and Double-counting before Adding On-top and Next-to
23. Good, Bad & Evil Mathematics – Tales of Totals, Numbers & Fractions
24. Good, Bad & Evil Mathematics – Tales of Totals, Numbers & Fractions, PPP
25. The Simplicity of Math reveals a Core Curriculum
26. The Simplicity of Math reveals a Core Curriculum, PPP
27. Addition Free Migrant-Math Rooted in STEM Re-Counting Formulas
28. A Twin Curriculum Since Contemporary Mathematics May Block the Road to its Educational Goal, Mastery of Many
29. Counting Before Adding, a PPP for the Article on a Twin Curriculum
30. A New Curriculum – But for Which of the 3×2 Kinds of Mathematics Education