Post on 14-Dec-2015
IconCountingin PreSchool
solves the
Fraction Paradox
From Mathe-matism to MANY-matics
Allan.Tarp@MATHeCADEMY.net
Five Questions
In Arabic, the word Algebra means __________
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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
The Fraction Paradox
The Teacher The 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 !
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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.
GrandMother GrandChildHow 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!
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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
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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.
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M1. Counting Sequences
A Natural Number is a decimal number with a unit.
Count in tens: no unit & misplaced decimal. A natural number?
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I I I I I I I I I I I I5s 1 2 3 4 B 1B1 1B2 1B3 1B4 2B 2B1 2B2
7s 1 2 3 4 5 6 B 1B1 1B2 1B3 1B4 1B5
tens 1 2 3 4 5 6 7 8 9 B 1B1 1B2
in 5s as T = 2B2 = 2x5 + 2 = 2.2 5s3 4s counted in 7s as T = 1B5 = 1x7 + 5 = 1.5 7s
in tens as T = 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
IIIIIIII I
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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.some9 – 2x4 1
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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 = ?5s T = 7 = ?3s T = 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 5s T = 7 = 2.1 3s T = 7 = 3.1 2s
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M3. ReCount in the Same Unit creates Negative Numbers
Unbundle Sticks Calculator T = II II II II II II II I I II II I I I I II I I I I I I I I I I I I I I II II II II II II II II II II II
4x2 – 3x2 4x2 – 2x2 4x2 – 1x2 4x2 – 0x2 4x2 – 5x2 4x2 – 6x2
2468-2-4
4.0 2s3.2 2s2.4 2s1.6 2s0.8 2s5.2 2s6.4 2s
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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) II)III) ) II) IIII )
CALCULATOR-prediction:3x4 = 2x5 + 2 = 2.2 5s
3x4/5 2.some3x4 – 2x5 2
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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)
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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) ) IIII) ) I IIII) )
I IIII) I)
CALCULATOR-prediction:(2x3+4x5) = 5x5 + 1 = 5.1 5s
(2x3+4x5)/5 5.some(2x3+4x5) – 5x5 1
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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
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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) ) 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
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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
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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) ) I) )III) ) ) IIIIIIIIII ) III) I)
CALCULATOR-prediction3x5 = 1x5 + 3x3 + 1 = 1 5s + 3.1 3s
(3x5-1x5)/3 3.some(3x5-1x5) – 3x3 1
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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 = DT/3 = Differentiation
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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 5sII) ) II) ) II) )
IIII) ) II) II) ) II II) III III) II II I) I)
CALCULATOR-prediction4x8 = 2x3 + 5x5 + 1 = 2 3s + 5.1 5s
(4x8-2x3)/5 5.some(4x8-2x3) – 5x5 1
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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 = DT/5 = Differentiation
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Five Questions with Answers
In Arabic, Algebra means ______?_______
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This statement is true Always Never Sometimes
2 + 3 = 5 x2 x 3 = 6 x1/2 + 2/3 = 7/6 x1/2 + 2/3 = 3/5 x
The Algebra Project: 4 Ways to Unite
Algebra means to ReUnite in Arabic:
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
MatheMatism vs. ManyMatics
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MatheMatism ManyMatics
A number is a symbolA number is a block with a counter-icon & a base-icon
6/3: 6 split by 3 6/3: 6 split in 3s, 6 counted in 3s
3 x 5 IS 15 3 x 5 = 3 5s = 2.1 7s = 1.5 tens
2 + 3 IS 51/2 + 2/3 IS 7/6
Bases: 1 2s + 1 3s = 1 5sCounters: Depends on the unit
Order: + – x / symbols Order: / x – + icons
Only bundle in tensFirst bundle in icons to learn ‘1digit Mathematics’Ten may be a cognitive bomb
Only add OnTop Add OnTop and Add NextTo
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Teach Teachers to Teach MATHEmatics as MANYmatics, a Natural Science about MANY. The CATS method: To learn Math
Count & Add in Time & Space
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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.
To learn MATH:
Count&Add MANY
1 Coach2 Instructors3 Pairs2 Teams
MATHeCADEMY.net Material
BLOCK-math from the MATHeCADEMY.net may be used in
• ICONcounting: www.youtube.com/watch?v=R2PQJG3WSQY• PreeSchool Math: www.youtube.com/watch?v=qgCwVZnALXA
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Teacher Training, Pre- & In-ServiceDistance Education can take placefrom WestAfrica to Australia
C1, A1, T1, S1 : PrimaryC2, A2, T2, S2: Secondary
Preschool & Primary SchoolHome Education M1-M8 Activity Books
Research & PostDoc projectsIconCount & NextToAddThe ICME Trilogy
BlockMath: Give all Kids a Chance
Thank You for Your Time
Allan Tarp
MATHeCADEMY.netFree Uni Franchise
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