Allan.Tarp@MATHeCADeMY.net

How to teach

Proportionality & Integration

in PreSchool

through

IconCounting &NextTo Addition

First: 6 Questions to be Answered

How many fingers here: ___ ___

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This statement is true Always Never Sometimes

2 + 3 = 5 2 x 3 = 6 1/2 + 2/3 = 7/6 1/2 + 2/3 = 3/5

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.

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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)

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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.

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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

IIIIIIII I

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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.some9 – 2x4 1

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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

Sticks Calculator T = II II II II II I I II II II II

3x2 – 2x2 3x2 – 4x2

2-2

3.0 2s2.2 2s4.2 2s

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T = 3.0 2s = 2.2 2s = 4.2 2s

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.some3x4 – 2x5 2

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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

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T = 2 3s + 4 5s ? 5s

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

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T = 2 3s + 4 5s ? 8s

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

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T = ? 3s + 1 5s 3 5s

A4. NextTo Addition Reversed

III IIIII III IIIII III IIIII III IIIII IIIIIIII IIIIIIII IIIIIIII IIIIIIII

ABACUS in G-mode:

? = (T – T1)/5 = DT/5 = Differentiation

CALCULATOR-prediction: (4x8 - 2x3)/5 5.some(4x8 - 2x3) – 5x5 1

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T = 2 3s + ? 5s 4 8s

Counting in Tens: ByeBye to Golden Math

C1. No ten-icon, so no more calculator predictionC2. Count in tens: T = 2.3 tens = 23, oops, no unit & misplaced decimal C3. ReCount in the Same Icon: Impossible since 23 is not 17C4. ReCount in a Different Icon: 3 5s = 3x5 = 15, 24 = ? 3s = (24/3) x 3

Postponed to multiplication, to division & to proportionalityA1. Add OnTop: Cup writing or Carry, 24 + 58 = 7)12) or 712 = 82 Subtract: Cup writing or Carry, 74 – 39 = 4) -5) = 35 or 74 = 6104

A2. Add NextTo (Integrate): Impossible before CalculusA3. Reversed Adding OnTop: Postponed to Solving EquationsA4. Reversed Adding NextTo: Impossible before Calculus, turned upside down by presenting differentiation before integration.

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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 Variable Constant

Unit-numbers m, s, $, kg

T = a + nT – a = n

T = a x nT/n = a

Per-numbers m/s, $/kg, %

T = ∫ a dndT/dn = a

T = a^nlogaT = n, n√T = a

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?

RealityMATHEMATICS is learned through the 2 activities dealing with the outside fact MANY:

COUNT & ADD

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6 Questions – 2 correct

How many fingers here: 4 v 4

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This statement is true Always Never Sometimes

2 + 3 = 5 x2 x 3 = 6 x v1/2 + 2/3 = 7/6 x1/2 + 2/3 = 3/5 x

6 Questions: 6 correct

How many fingers here: 4 2 2s

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This statement is true Always Never Sometimes

2 + 3 = 5, 2w+3w=5w but 2w+3d=17d x2 x 3 = 6, 2x3 = 2 3s = 6 1s x1/2 + 2/3 = 7/6 1/2 of my p + 2/3 of my p = 7/6 of my p x1/2 of 2 p + 2/3 of 3 p = 3/5 of 5 p1/2 of 4 p + 2/3 of 3 p = 4/7 of 7 p1/2 + 2/3 = 3/5

x

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

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MATHeCADEMY.net Material

MANYmath from the MATHeCADEMY.net may be used in

• ICONcounting: www.youtube.com/watch?v=R2PQJG3WSQY• PreSchool Math: www.youtube.com/watch?v=qgCwVZnALXA

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Teacher Training, Pre- & In-ServiceWorld-wide Distance Education

C1, A1, T1, S1 : PrimaryC2, A2, T2, S2: Secondary

Preschool & Primary SchoolHome Education M1-M8 Activity Books

Research & PostDoc ProjectsIconCount & NextToAddThe ICME Trilogy

PYRAMIDeDUCATION

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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 Coach2 Instructors3 Pairs2 Teams

To learn MATH:Count&Add MANY

MANYmath: Give all Kids a ChanceSo Please COUNT first in Icons then in tensbefore you ADD first NextTo then OnTop

Allan.Tarptravels the World to give cost free talks on MANYmath

MATHeCADEMY.netFree Uni Franchise

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