Singapour : Les Maths Singapour- Une Methode Miracle

“Singapore Math” was a derivative of ancient Chinese Math, modified and combined with “Polya Problem Solving Method ” by a Singapore Professor Lee Peng Yee (李秉彝) from Nanyang Technological University (NTU)’s NIE (National Institute of Education). Prof Lee was the first batch of Nantah University 南洋大学 (the precursor of NTU) Math undergraduate in late 1950s, who obtained PhD Math in Queen’s University (Belfast).

In 2005 during his public Math Olympiad books launching seminar at NUS bookshop, the 70-year-old Prof Lee started his talk on the genesis of “Singapore Math” idea with this famous ancient Chinese Math “Chicken and Rabbits” (鸡兔问题)。

Comments on Singapore Math :

1. The Pros & Cons

Pros: Concrete Chinese Arithmetic , Polya Problem Solving Pedagogy, Visualisation with Models. PLUS: well trained Math teachers, “educated” parents (revision class for them) or hire private Math home tutor.

Cons: Lack abstract training for Primary 5 & 6 kids (11-12 years old ) in Algebra equations (postponed to Secondary 1 @13 years old). The Chinese (上海) kids start Algebra before Singapore kids. Also French kids start abstract (Set Theory) concepts earlier than Asian kids.

2. UK Textbooks follow Singapore Math

La remise du rapport Villani au Ministre de l’Education nationale Mr. Blanquer préconisant 21 mesures pour l’enseignement des mathématiques en France, a mis à l’honneur, par ricochet et par voix de presse, la méthode de Singapour pour l’apprentissage des maths.

Note: French Mathematician Cédric Villani (Fields Medalist 2010) joins President Macron’s Political Party as a deputé (= Member of Parliament).

https://lepetitjournal.com/singapour/les-maths-singapour-une-methode-miracle-224015

Why are Singapore school kids so strong in Math ? (World #1 PISA Test in 2016)

https://lepetitjournal.com/singapour/actualites/education-pourquoi-les-eleves-singapouriens-sont-ils-si-forts-en-maths-46043

Every (Singapore) school is a good school” – Mr. Heng (Former Minister of Education of Singapore)

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Modelling sans Algebra

Singapore PSLE Modelling Math (for 11 year-old pupils)

Problem:

1) 20% of John saving is $120 more than 30% of Brian savings.

2) After John spent 5/6 of his saving and Brian spent 1/2 of his saving ,
John has $20 more than Brian.

Find the saving of John at first ?

Solution:
(Technique: work backwards from the End.)
Before 2nd condition:
J: 60 units+ $120
[1][2]…..[9][10] [$20]
[1][2]…..[9][10] [$20]
[1][2]…..[9][10] [$20]
[1][2]…..[9][10] [$20]
[1][2]…..[9][10] [$20]
[1][2]…..[9][10] [$20]
B: 20 units
[1][2]…..[9][10]
[1][2]…..[9][10]
After 2nd condition:
J: [1][2]…..[9][10] [$20]
B: [1][2]…..[9][10]
20% of J
= 0.2 x 60 units + 0.2 x $120
= 12 units + $24
30% of B
= 0.3 x 20 units
= 6 units
From 1st condition:
(12 units+$24)- 6 units
->$120
6 units +$24 ->$120
6 units ->$96
1 unit ->$16
J: 60 units+ $120
J= 60x$16+ $120= $1080
[QED]

4-level MathThinking

4 Levels:

L1. S&T (See & Touch) Concrete: 1 apple, 2 oranges…
e.g. Math Modeling: visualise the problem [Primary School]

L2. S~T (See, no Touch but can guess):
e.g. Guess x,y for 2x+3y=8 ? [Secondary School]
e.g. Chimpanzees can guess where you hide the banana.

L3. ~S~T&I (no See, no Touch but Imagine):
e.g. Complex i = [Junior College].

L4. ~S~T~I (no See, no Touch, no imagine)
e.g. Abstract Math: Galois Group, ε-δ Analysis, Ring, Field, etc. [University]

Solution 3 (Modelling): Monkeys & Coconuts

Let x the min number of coconuts initially.

1st monkey took “a” coconuts away, 2nd monkey “b” coconuts….5th monkey took “e” coconuts.

[a][a][a][a][a] + 1 =x

Loan 4 coconuts to the initial pool of x coconuts to divide by 5 evenly at each monkey.

[a][a][a][a][a] + 1 + 4 = [a][a][a][a][a] + 5 = x+4 = X (inflated x by 4 )
1st Monkey: [a’][a’][a’][a’][a’] = X
a’ = \frac {1}{5} X

… Left 4a’= \frac {4}{5} X
[a’][a’][a’][a’] => [b][b][b][b][b]

b= \frac{1}{5} .4a’ = \frac{4}{25}X

… Left 4b= \frac {16}{25}X
[b][b][b][b] => [c][c][c][c][c]
c=\frac {1}{5} .4b= \frac {16}{125}X
… Left 4c= \frac {64}{125}X
[c][c][c][c] => [d][d][d]d][d]
d= \frac {1}{5}.4c= \frac {64}{625}X
… Left 4d= \frac {256}{625}X
[d][d][d]d] => [e][e][e][e][e]
e= \frac {1}{5}.4d= 256.(X/3125)
Since e is integer
=> X = 3125 or multiples of 3125
Minimum X=3125
x+4 = 3125
x= 3121 = minimum Coconut initially.

Note: This solution used the Singapore Modelling Math taught in all Primary Schools for 11-year-old pupils.

Singapore Modelling Math

Modelling Math versus Usual Method

Marie had 20 m of cloth. She used 3/5 of it to make some dresses. How many meters of cloth did she use?
1. Model method:

【】【】【】〖〗〖〗
5 units = 20 m
1 unit = 20 / 5= 4 m
3 units= 3 x 4 = 12 m

2. Usual method:

3/5 x 20 = 12 m

The 2nd method is too abstract for 8 or 9 yr-old kids who will resort to learn fraction “x” or “/” formula by rote.
Models appear ‘silly’ to adults who prefer abstract concept ‘3/5 x 20’;

Note 1:  Singapore Primary School Modelling Math

Adopted since 1990 from Polya’s “How to Solve It” (1945) 4-Step Problem Solving Process

1. Understand the problem in English Word Math.
2. Devise a plan: draw models
3. Carry out the plan: solve
4. Look back: verify

The Concrete-Pictorial-Abstract Math approach is developed by Dr. Kho Tek Hong (retired in 2007) & his MOE team @ 1980s.

Note 2: Models can also help visualize & ‘concretize’ Abstract Algebra (Set, Group, Coset, subgroup, kernel, isomorphism …Galois theory). Feynman invented ‘model’ (Feynman diagram) to visualize quantum theory.