1. Allan.Tarp@ MATHeCADeMY.net How to teach Proportionality & Integration in PreSchool through IconCounting & NextTo Addition

2. First: 6 Questions to be Answered How many fingers here: ___ ___ MATHeCDMY This statement is true AlwaysNeverSometimes 2 + 3 = 5 2 x 3 = 6 1/2 + 2/3 = 7/6 1/2 + 2/3 = 3/5

3. Mathematics in PreSchool ? A great idea – but is Mathematics well defined? Freudenthal: Among Pythagoras’ adepts there was a group that called themselves mathematicians, since they cultivated the four “mathemata”, that is geometry, arithmetic, musical theory and astronomy. OOPS! Mathematics is not an activity, it is a label! Rebuild MATHEmatics as MANYmath: to deal with MANY we need 4 COUNT & 4 ADD Activities. MATHeCDMY :::::::::::

4. 8 Activities Teach Preschoolers to C1. COUNT in 1s ( Rearrange Many as Icons) C2. COUNT in Icons (Decimal numbers, Operations as Icons with a natural order: Division first, then Multiplication, then Subtraction) C3. ReCOUNT in the Same Icon (Negative Numbers) C4. ReCOUNT in a Different Icon (Proportionality) A1. ADD OnTop (Abacus in Geometry- & Algebra-mode) A2. ADD NextTo (Integrate) A3. ADDing OnTop Reversed (Backwards calculation) A4. ADDing NextTo Reversed (Differentiate) MATHeCDMY

5. C1. Create Icons: I I I I → IIII → Counting in ones means naming the different degrees of Many. Counting in icons means changing four 1s, I I I I, to 1 fours, IIII, rearranged as a 4-icon with four sticks or strokes. A numeral is an ICON with as many sticks as it represents. MATHeCDMY

6. C2. Count in Icons: 9 = ? 4s 9 = I I I I I I I I I IIII IIII I II)I) 2)1) = 2.1 4s Bundle, stack, 2 cups, cup-writing or decimal-writing with a unit. Decimal points separate the bundles and the un-bundled singles. Shown on a western ABACUS in Geometry/space mode or Algebra/time mode IIII IIII I MATHeCDMY

7. C2. IconCounting makes Operations Icons ‘From 9 take away 4s’ we write 9/4 iconizing the sweeping away by a broom, called Division. ‘2 times stack 4s’ we write 2x4 iconizing the lifting up by a jack called Multiplication. ‘From 9 take away 2 4s’ to look for leftovers we write 9 – 2x4 iconizing the dragging away by a stroke called Subtraction. CALCULATOR-prediction: 9/4 2.some 9 – 2x4 1 MATHeCDMY

8. ReCounting the Total T = 3 2s in 2s: (4.2 2s = 4 less 2 2s) And 2digit Numbers if using Bundles of Bundles: I I I I I I = II II II = II II II 6 = 3 B = 1BB 1B 6 = 3) ) = 1) 1) ) = 11 2s C3. ReCount in the Same Unit creates Negative Numbers SticksCalculator T = II II II II II II II 3x2 – 2x2 3x2 – 4x2 2 -2 3.0 2s 2.2 2s 4.2 2s MATHeCDMY T = 3.0 2s = 2.2 2s = 4.2 2s

9. C4. ReCount in a Different Unit: 3 4s = ? 5s 3 4s = IIII IIII IIII I I I I I I I I I I I I IIIII IIIII II = 2.2 5s Change Unit = Proportionality CALCULATOR-prediction: 3x4/5 2.some 3x4 – 2x5 2 MATHeCDMY

10. A1. OnTop Addition III III + IIIII IIIII IIIII IIIII I II III IIIII IIIII IIIII IIIII CALCULATOR-prediction: (2x3 + 4x5)/5 5.some (2x3 + 4x5) – 5x5 1 MATHeCDMY T = 2 3s + 4 5s ? 5s

11. A2. NextTo Addition III III + IIIII IIIII IIIII IIIII IIIIIIII IIIIIIII IIIII III II Adding Blocks NextTo = Adding Areas = Integration CALCULATOR-prediction: (2x3 + 4x5)/8 3.some (2x3 + 4x5) – 3x8 2 MATHeCDMY T = 2 3s + 4 5s ? 8s

12. A3. OnTop Addition Reversed IIIII IIIII IIIII IIIII IIIII IIIII IIIII III III III I CALCULATOR-prediction: (3x5 - 1x5)/3 3.some (3x5 - 1x5) – 3x3 1 MATHeCDMY T = ? 3s + 1 5s 3 5s

13. A4. NextTo Addition Reversed III IIIII III IIIII III IIIII III IIIII IIIIIIII IIIIIIII IIIIIIII IIIIIIII ABACUS in G-mode: ? = (T – T1)/5 =  T/5 = Differentiation CALCULATOR-prediction: (4x8 - 2x3)/5 5.some (4x8 - 2x3) – 5x5 1 MATHeCDMY T = 2 3s + ? 5s 4 8s

14. Counting in Tens: ByeBye to Golden Math C1. No ten-icon, so no more calculator prediction C2. Count in tens: T = 2.3 tens = 23, oops, no unit & misplaced decimal C3. ReCount in the Same Icon: Impossible since 23 is not 17 C4. ReCount in a Different Icon: 3 5s = 3x5 = 15, 24 = ? 3s = (24/3) x 3 Postponed to multiplication, to division & to proportionality A1. Add OnTop: Cup writing or Carry, 24 + 58 = 7)12) or 7 1 2 = 82 Subtract: Cup writing or Carry, 74 – 39 = 4) -5) = 35 or 74 = 6 10 4 A2. Add NextTo (Integrate): Impossible before Calculus A3. Reversed Adding OnTop: Postponed to Solving Equations A4. Reversed Adding NextTo: Impossible before Calculus, turned upside down by presenting differentiation before integration. MATHeCDMY

15. Conclusion Algebra means to ReUnite in Arabic. Writing 345 = 3 x B^2 + 4 x B + 5x1 shows the 4 ways to Unite IconCounting and NextTo Addtion also teach preschoolers the core of Algebra Operations Unite/ split into VariableConstant Unit-numbers m, s, $, kg T = a + n T – a = n T = a x n T/n = a Per-numbers m/s, $/kg, % T = ∫ a dn dT/dn = a T = a^n log a T = n, n √T = a

16. Myth vs. Reality: Before or Through Myth Mathematics must be taught and learned before it can be applied to outside activities! But we teach MATHEMATISM, true inside, but not outside the classroom: 2+3 IS 5, but 2w+3d = 17d? Reality MATHEMATICS is learned through the 2 activities dealing with the outside fact MANY: COUNT & ADD MATHeCDMY

17. 6 Questions – 2 correct How many fingers here: 4 v 4 MATHeCDMY This statement is true AlwaysNeverSometimes 2 + 3 = 5x 2 x 3 = 6 x v 1/2 + 2/3 = 7/6x 1/2 + 2/3 = 3/5x

18. 6 Questions: 6 correct How many fingers here: 4 2 2s MATHeCDMY This statement is true AlwaysNeverSometimes 2 + 3 = 5, 2w+3w=5w but 2w+3d=17d x 2 x 3 = 6, 2x3 = 2 3s = 6 1s x 1/2 + 2/3 = 7/6 1/2 of my p + 2/3 of my p = 7/6 of my p x 1/2 of 2 p + 2/3 of 3 p = 3/5 of 5 p 1/2 of 4 p + 2/3 of 3 p = 4/7 of 7 p 1/2 + 2/3 = 3/5 x

19. MATHeCADEMY.net Teach Teachers to Teach MATHEmatics as MANYmath, a Natural Science about MANY. The CATS method: To learn Math Count & Add in Time & Space MATHeCDMY

20. MATHeCADEMY.net Material MANYmath from the MATHeCADEMY.net may be used in ICONcounting: www.youtube.com/watch?v=R2PQJG3WSQYwww.youtube.com/watch?v=R2PQJG3WSQY PreSchool Math: www.youtube.com/watch?v=qgCwVZnALXAwww.youtube.com/watch?v=qgCwVZnALXA MATHeCDMY Teacher Training, Pre- & In-Service World-wide Distance Education C1, A1, T1, S1 : Primary C2, A2, T2, S2: Secondary Preschool & Primary School Home Education M1-M8 Activity Books Research & PostDoc Projects IconCount & NextToAdd The ICME Trilogy

21. PYRAMIDeDUCATION MATHeCDMY In PYRAMIDeDUCATION, 8 learners are organized in 2 teams of 4 choosing 2 instructors and 3 pairs by turn. Each pair works together to solve Count&Add problems. The coach assists the instructors when instructing their team and when correcting the Count&Add assignments. Each learner pays by coaching a new group of 8 learners. 1 Coach 2 Instructors 3 Pairs 2 Teams To learn MATH: Count&Add MANY

22. MANYmath: Give all Kids a Chance So Please COUNT first in Icons then in tens before you ADD first NextTo then OnTop Allan.Tarp travels the World to give cost free talks on MANYmath MATHeCADEMY.net Free Uni Franchise MATHeCDMY