# IconCounting in PreSchool solves the Fraction Paradox From Mathe-matism to MANY-matics MATHeCADEMY.net.

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IconCounting in PreSchool solves the Fraction Paradox From Mathe-matism to MANY-matics Allan.Tarp@ MATHeCADEMY.net

Five Questions In Arabic, the word Algebra means __________ 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

The Fraction Paradox The TeacherThe Students What is 1/2 + 2/3?Well, 1/2 + 2/3 = (1+2)/(2+3) = 3/5 No! 1/2 + 2/3 = 3/6 + 4/6 = 7/6 But 1/2 of 2 cokes + 2/3 of 3 cokes is 3/5 of 5cokes! How can it be 7 cokes out of 6 cokes? Inside this classroom 1/2 + 2/3 IS 7/6 ! MATHeCDMY

Solved by Children Uneducated, kids see numbers as they are: Blocks of Stacked Bundles. Base ten: T = 38 = 3x10 + 8x1, Base five: T = 1x5^2 + 2x5 + 3x1. CountNumbers & BaseNumbers: 2x5 = 2 5s; 2 is Counter and 5 is Base. Bases ad: 1 4s + 1 2s = 1 6s. Counters dont: 1 4s + 1 2s = 3 2s = 1½ 4s. Fractions: Not bases, but counters, so they dont add. Schools teach MatheMatism, true inside but not outside the classroom. GrandMotherGrandChild How old will you be next time?Four (shows four fingers) Four, you said? (shows four fingers 2 by 2) No, that is not four! That is two twos! MATHeCDMY

Teach Numbers or Blocks? Replacing Numbers with Blocks and IconCounting and NextTo Addition: Before counting in tens, (preschool) children learn the Core of Mathematics: Negative and Rational Numbers Proportionality (Linearity) Integrate and Differentiate Solve Equations Testing ‘1Digit Math’ by 8 MicroCurricula M1 – M8 MATHeCDMY

Eight MicroCurricula M1. Create Icons M2. Count in Icons (Rational Numbers) M3. ReCount in the Same Icon (Negative Numbers) M4. ReCount in a Different Icon (Proportionality) M5. Add OnTop (Proportionality) M6. Add NextTo (Integrate) M7. Reverse Adding OnTop (Solve Equations) M8. Reverse Adding NextTo (Differentiate)

M1. Create Icons Counting in ones means naming the different degrees of Many. We stop at nine since when counting by bundling, ten becomes 1 bundle, ten = 1B, needing no icon of its own. 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. MATHeCDMY

M1. Counting Sequences A Natural Number is a decimal number with a unit. Count in tens: no unit & misplaced decimal. A natural number? MATHeCDMY IIIIIIIIIIII 5s1234B1B11B21B31B42B2B12B2 7s123456B1B11B21B31B41B5 tens123456789B1B11B2 in 5s asT = 2B2 = 2x5 + 2 = 2.2 5s 3 4s countedin 7s asT = 1B5 = 1x7 + 5 = 1.5 7s in tens asT = 1B2 = 1xten + 2 = 1.2 tens = 12

M2. Count in Icons A Total of 9 counted in 4s gives T = 9 = 2 4s & 1 IKEA I I I I I I I I I IIII IIII I II)I) 2)1) = 2.1 4s Bundling, stacking, and using cup-writing or decimal-writing with a unit, using decimal points to separate bundles and ones. Shown on a western ABACUS in Geometry (space, base) mode or Algebra (time, counter) mode IIII IIII I MATHeCDMY

M2. IconCounting creates Division, Multiplication & Subtraction - also as 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 = 2x4 + 1 = 2.1 4s 9/4 2.some 9 – 2x4 1 MATHeCDMY

M2. IconCounting creates 2 Counting Formulas IIIIIII = IIIII I I IIIIIII = III III I IIIIIII = II II II I Formulas Predict! ReCounting finds the bundles T = (T/b) x b From T, bs can be taken away and stacked T/b times ReStacking finds the un-bundled T = (T–b) + b From T, b can be taken away and placed next-to T–b Q:T = 7 = ?5sT = 7 = ?3sT = 7 = ?2s 7/5 1.some 7 – 1x5 2 7/3 2.some 7 – 2x3 1 7/2 3.some 7 – 3x2 1 A:T = 7 = 1.2 5sT = 7 = 2.1 3sT = 7 = 3.1 2s MATHeCDMY

M3. ReCount in the Same Unit creates Negative Numbers Unbundle SticksCalculator T = II II II II II II II II II II II II II II II 4x2 – 3x2 4x2 – 2x2 4x2 – 1x2 4x2 – 0x2 4x2 – 5x2 4x2 – 6x2 2 4 6 8 -2 -4 4.0 2s 3.2 2s 2.4 2s 1.6 2s 0.8 2s 5.2 2s 6.4 2s MATHeCDMY ReCounting 4 2s in 2s: (6.4 2s = 6 less 4 2s)

M4. ReCount in a Different Unit Q: A total of 3 4s ReCounted gives ? 5s 3 4s = IIII IIII IIII I I I I I I I I I I I I IIIII IIIII II = 2.2 5s Cups: 2.2 5s 3 4s: II) III)) II) IIII ) CALCULATOR-prediction: 3x4 = 2x5 + 2 = 2.2 5s 3x4/5 2.some 3x4 – 2x5 2 MATHeCDMY

M4. ReCount in a Different Unit Q: A total of 3 4s ReCounted gives ? 5s An ABACUS in G-mode A: 3 4s = 2.2 5s ReCount = Change Unit = Proportionality (Linearity ) MATHeCDMY

M5. OnTop Addition Adding 2 3s and 4 5s gives ? 5s III III IIIII IIIII IIIII IIIII I II III IIIII IIIII IIIII IIIII 2 3s + 4 5s = 1.1 5s + 4 5s = 5.1 5s II) )) IIIIII)) I) IIII) ) I IIII) )I IIII) I) CALCULATOR-prediction: (2x3+4x5) = 5x5 + 1 = 5.1 5s (2x3+4x5)/5 5.some (2x3+4x5) – 5x5 1 MATHeCDMY

M5. OnTop Addition Q: Adding 2 3s and 4 5s gives ? 5s An ABACUS in G-mode: A: 2 3s + 4 5s = 5.1 5s MATHeCDMY

M6. NextTo Addition Q: Adding 2 3s and 4 5s gives ? 8s III III IIIII IIIII IIIII IIIII IIIIIIII IIIIIIII IIIII III II 2 3s + 4 5s = 2 8s + 1.2 8s = 3.2 8s II) ) IIII) )II) IIIIIIII II )I II) II) CALCULATOR-prediction: (2x3+4x5) = 3x8 + 2 = 3.2 8s (2x3+4x5)/8 3.some (2x3+4x5) – 3x8 2 MATHeCDMY

M6. NextTo Addition Q: Adding 2 3s and 4 5s gives ? 8s ABACUS in G-mode: A: 2 3s + 4 5s = 3.2 8s Adding Blocks NextTo = Adding Areas = Integration MATHeCDMY

M7. OnTop Addition Reversed Q: 3 5s is 1 5s added with ? 3s IIIII IIIII IIIII IIIII IIIII IIIII IIIII III III III I 3 5s = 1 5s + 2 5s = 1 5s + 3.1 3s I) ) III) )) IIIIIIIIII )III) I) CALCULATOR-prediction 3x5 = 1x5 + 3x3 + 1 = 1 5s + 3.1 3s (3x5-1x5)/3 3.some (3x5-1x5) – 3x3 1 MATHeCDMY

M7. OnTop Addition Reversed Q: 3 5s is 1 5s added with ? 3s ABACUS in G-mode: A: 3 5s = 1 5s + 3.1 3s ? = T2/3 = (T – T1)/3 =  T/3 = Differentiation MATHeCDMY

M8. NextTo Addition Reversed Q: 4 8s is 2 3s added with ? 5s IIIIIIII IIIIIIII IIIIIIII IIIIIIII III IIIII III IIIII IIIII III II IIIII I 4 8s = 2 3s + 5.1 5s II) ) IIII) ) II) II) )II II) III III)II II I) I) CALCULATOR-prediction 4x8 = 2x3 + 5x5 + 1 = 2 3s + 5.1 5s (4x8-2x3)/5 5.some (4x8-2x3) – 5x5 1 MATHeCDMY

M8. NextTo Addition Reversed Q: 4 8s is 2 3s added with ? 5s ABACUS in G-mode: A: 4 8s = 2 3s + 5.1 5s ? = T2/5 = (T – T1)/5 =  T/5 = Differentiation MATHeCDMY

Five Questions with Answers In Arabic, Algebra means ______?_______ MATHeCDMY This statement is true AlwaysNeverSometimes 2 + 3 = 5x 2 x 3 = 6x 1/2 + 2/3 = 7/6x 1/2 + 2/3 = 3/5x

The Algebra Project: 4 Ways to Unite Algebra means to ReUnite in Arabic: 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

MatheMatism vs. ManyMatics MATHeCDMY MatheMatismManyMatics A number is a symbol A number is a block with a counter-icon & a base-icon 6/3: 6 split by 36/3: 6 split in 3s, 6 counted in 3s 3 x 5 IS 153 x 5 = 3 5s = 2.1 7s = 1.5 tens 2 + 3 IS 5 1/2 + 2/3 IS 7/6 Bases: 1 2s + 1 3s = 1 5s Counters: Depends on the unit Order: + – x / symbolsOrder: / x – + icons Only bundle in tens First bundle in icons to learn ‘1digit Mathematics’ Ten may be a cognitive bomb Only add OnTop Add OnTop and Add NextTo

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

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. To learn MATH: Count&Add MANY 1 Coach 2 Instructors 3 Pairs 2 Teams

MATHeCADEMY.net Material BLOCK-math from the MATHeCADEMY.net may be used in ICONcounting: www.youtube.com/watch?v=R2PQJG3WSQYwww.youtube.com/watch?v=R2PQJG3WSQY PreeSchool Math: www.youtube.com/watch?v=qgCwVZnALXAwww.youtube.com/watch?v=qgCwVZnALXA MATHeCDMY Teacher Training, Pre- & In-Service Distance Education can take place from WestAfrica to Australia 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

BlockMath: Give all Kids a Chance Thank You for Your Time Allan Tarp MATHeCADEMY.net Free Uni Franchise MATHeCDMY