Transcript 5 - MATHeCADEMY.net

IconCounting
in PreSchool
solves the
Fraction
Paradox
From Mathe-matism
to MANY-matics
Allan.Tarp@
MATHeCADEMY.net
MATHeCDMY
Five Questions
This statement is true
Always
Never
Sometimes
2+3=5
2x3=6
1/2 + 2/3 = 7/6
1/2 + 2/3 = 3/5
In Arabic, the word Algebra means __________
MATHeCDMY
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
But 1/2 of 2 cokes + 2/3 of 3 cokes
= 3/6 + 4/6
is 3/5 of 5cokes!
= 7/6
How can it be 7 cokes out of 6 cokes?
Inside this classroom
1/2 + 2/3 IS 7/6 !
MATHeCDMY
Solved by Children
GrandMother
GrandChild
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!
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.
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
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)
MATHeCDMY
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
I
I
I
I
I
I
I
I
I
I
I
I
5s
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
3 4s counted in 7s as
in tens as
T = 2B2 = 2x5 + 2 = 2.2 5s
T = 1B5 = 1x7 + 5 = 1.5 7s
T = 1B2 = 1xten + 2 = 1.2 tens = 12
A Natural Number is a decimal number with a unit.
Count in tens: no unit & misplaced decimal. A natural number?
MATHeCDMY
M2. Count in Icons
A Total of 9 counted in 4s gives T = 9 = 2 4s & 1
IKEA
IIII
IIIIIIIII
IIII IIII I 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
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
9 – 2x4
2.some
1
MATHeCDMY
M2. IconCounting creates
2 Counting Formulas
ReCounting
finds the bundles
ReStacking
finds the un-bundled
IIIIIII = IIIII I I
Q:
T = 7 = ?5s
7/5
7 – 1x5
A:
T = (T/b) x b
From T, bs can be taken away
and stacked T/b times
T = (T–b) + b
From T, b can be taken away
and placed next-to T–b
IIIIIII = III III I
T = 7 = ?3s
1.some 7/3
2 7 – 2x3
T = 7 = 1.2 5s
IIIIIII = II II II I
T = 7 = ?2s
2.some 7/2
1 7 – 3x2
T = 7 = 2.1 3s
Formulas Predict!
3.some
1
T = 7 = 3.1 2s
MATHeCDMY
M3. ReCount in the Same Unit
creates Negative Numbers
ReCounting 4 2s in 2s:
Unbundle Sticks
II II II II
II II II I I
II II I I I I
II I I I I I I
IIIIIIII
II II II II II
II II II II II II
(6.4 2s = 6 less 4 2s)
Calculator
4x2 – 3x2
4x2 – 2x2
4x2 – 1x2
4x2 – 0x2
4x2 – 5x2
4x2 – 6x2
2
4
6
8
-2
-4
T=
4.0 2s
3.2 2s
2.4 2s
1.6 2s
0.8 2s
5.2 2s
6.4 2s
MATHeCDMY
M4. ReCount in a Different Unit
Q: A total of 3 4s ReCounted gives ? 5s
3 4s = IIII IIII IIII
IIII IIII IIII
Cups:
3 4s:
IIIII IIIII II = 2.2 5s
II)
III) )
II)
2.2 5s
II) IIII )
CALCULATOR-prediction:
3x4 = 2x5 + 2 = 2.2 5s
3x4/5
3x4 – 2x5
2.some
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
2 3s +
II) )
IIII) )
4 5s
) IIIIII)
IIII) )
CALCULATOR-prediction:
(2x3+4x5) = 5x5 + 1 = 5.1 5s
I II III
=
IIIII IIIII IIIII IIIII
1.1 5s + 4 5s
) I)
I IIII) )
(2x3+4x5)/5
(2x3+4x5) – 5x5
=
5.1 5s
I IIII) I)
5.some
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
2 3s +
II) )
IIII) )
4 5s
IIIIIIII IIIIIIII IIIII III II
=
2 8s
II)
)
II) IIIIIIII II )
CALCULATOR-prediction:
(2x3+4x5) = 3x8 + 2 = 3.2 8s
+ 1.2 8s = 3.2 8s
I II) II)
(2x3+4x5)/8
(2x3+4x5) – 3x8
3.some
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
3 5s
III) )
IIIII
IIIII IIIII
= 1 5s + 2 5s
I)
)
) IIIIIIIIII )
=
IIIII
III III III I
1 5s
+ 3.1 3s
I) )
III) I)
(3x5-1x5)/3
CALCULATOR-prediction
3x5 = 1x5 + 3x3 + 1 = 1 5s + 3.1 3s (3x5-1x5) – 3x3
3.some
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 = DT/3 = Differentiation
MATHeCDMY
M8. NextTo Addition Reversed
Q: 4 8s is 2 3s added with ? 5s
IIIIIIII IIIIIIII IIIIIIII IIIIIIII
4 8s
IIII) )
III IIIII III IIIII IIIII III II IIIII I
=
II)
)
II) II) )
CALCULATOR-prediction
4x8 = 2x3 + 5x5 + 1 = 2 3s + 5.1 5s
2 3s
+
II)
)
II II) III III)
5.1 5s
II) )
II II I) I)
(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 = DT/5 = Differentiation
MATHeCDMY
Five Questions with Answers
This statement is true
2+3=5
2x3=6
1/2 + 2/3 = 7/6
1/2 + 2/3 = 3/5
Always
Never
Sometimes
x
x
x
x
In Arabic, Algebra means ______?_______
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+n
T–a=n
T=axn
T/n = a
Per-numbers
m/s, $/kg, %
T = ∫ a dn
dT/dn = a
T = a^n
logaT = n, n√T = a
MATHeCDMY
MatheMatism vs. ManyMatics
MatheMatism
ManyMatics
A number is a symbol
A 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
3x5
3 x 5 = 3 5s = 2.1 7s = 1.5 tens
IS 15
2+3
IS 5
1/2 + 2/3 IS 7/6
Order: + – x /
Bases: 1 2s + 1 3s = 1 5s
Counters: Depends on the unit
symbols
Order:
/ x – +
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
icons
MATHeCDMY
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
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
MATHeCDMY
MATHeCADEMY.net Material
BLOCK-math from the MATHeCADEMY.net may be used in
Teacher Training, Pre- & In-Service
C1, A1, T1, S1 : Primary
Distance Education can take place
C2, A2, T2, S2: Secondary
from WestAfrica to Australia
Preschool & Primary School
M1-M8 Activity Books
Home Education
IconCount & NextToAdd
Research & PostDoc projects
The ICME Trilogy
• ICONcounting: www.youtube.com/watch?v=R2PQJG3WSQY
• PreeSchool Math: www.youtube.com/watch?v=qgCwVZnALXA
MATHeCDMY
BlockMath: Give all Kids a Chance
Thank You
for Your Time
Allan Tarp
MATHeCADEMY.net
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