Math Education Evolution: From Function to Set to Category

Interesting Math education evolves since 19th century.

“Elementary Math from An Advanced Standpoint” (3 volumes) was proposed by German Göttingen School Felix Klein (19th century) :
1)  Math teaching based on Function (graph) which is visible to students. This has influenced  Secondary school Math worldwide.

2) Geometry = Group

After WW1, French felt being  behind  German school, “Bourbaki” Ecole Normale Supérieure students rewrote all Math teaching – aka “Abstract Math” – based on structure “Set” as foundation to build further algebraic structures (group, ring, field, vector space…) and all Math.

After WW2, the American prof MacLane & Eilenburg summarised all these Bourbaki structures into one super-structure: “Category” (范畴) with a “morphism” (aka ‘relation’) between them.

Grothendiek proposed rewriting Bourbaki Abstract Math from ‘Set’ to ‘Category’, but was rejected by the jealous Bourbaki founder Andre Weil.

Category is still a graduate syllabus Math,  also called “Abstract Nonsense”! It is very useful in IT  Functional Programming for “Artificial Intelligence” – the next revolution in “Our Human Brain” !

Elementary Math from an Advanced Standpoint

Guesstimate Question  (no calculator allowed) –

[Hint]: use Abstract Math “Homomorphism”  {f * +}

What is 35823 x 23412 ?

A) 845203402
B) 838688076
C) 812343296

The mapping f defined by
\forall  a_1, a_2, a_3 \in Z
f(a_1a_2a_3) = a_1+ a_2+ a_3
such that,

\forall a, b \in Z,
f (a * b) = f(a) * f(b)

=> f is an homomorphism.

Examples:

f (35823) = 3+5+8+2+3 = 21
f (21)= 2+1= 3

f (23412) = 2+3+4+1+2 =12
f (12) =1+2 = 3

f (35823 x 23412) = 3 x 3 = 9

Verification  (B):
f (838688076)= 8+3+8+6+8+8+0+7+6=54 ..5+4 =9 ☆

Wrong :
f(845203402)= 28…=10 = 1

f (812343296) = 38… =11 =2

Java Family

Year Ver Code Name Description
1995 1.0 Java Applets
1977 1.1 Java Event, Beans, Internationalization
Dec 1978 1.2 Java 2 J2SE, J2EE, J2ME, Java Card
2000 1.3 Java 2 J2SE 1.3
2002 1.4 Java 2 J2SE 1.4
2004 1.5 Java 5 J2SE 1.5
Nov 2006 6 Java 6 Open-Source Java SE 6. “Multithreading” by Doug Lea
May 2007 OpenJDK free software
2010 Oracle acquired Sun
Jul 2011 7 Java 7 “Dolphin”
Mar 2014 8 Java 8 Lambda Function

Javac: Java Compiler

Java Distributions:

1. JDK (Java Developer Kit )
◇JRE & Javac & tools

2. JRE (Java Runtime Environment)
◇ JVM & core class libraries
◇ Windows / Mac / Linux

Java is Object-Oriented Programming (OOP):
1. Class
public class Employee {

public int age;
public double salary;

public Employee () { [<– constructor with no arg]
}

public Employee (int ageValue, double salaryValue) { [<- constructor with args]

age = ageValue;
salary = salaryValue;

}
}

2. Construct = create / instantiate objects from the Class template.

3. Constructor : used to construct an object
◇ Every class must have at least 1 constructor (default add one by compiler)
◇ Same name as the Class
◇ It is like a Method without return value.

3. Method
returnType methodName (args)

returnType : primitive | object | void

void : (method returns nothing)

Examples:
♡ Primitive : int

int add (int a, int b)

♡ Object : Address

Address getAddress ()

◇ Void
(Special Method)

public static void main (String [] args)

static members are not tied to class instances. The method main acts as the entry point to a class, is static because it must be called before any object is created (instantiated).

Naming Conventions:
◇ Class: Employee , Address
◇ Fields: firstName, maxAge, blockNumber…
◇ Method: getAddress ()

The Success of French New Math Education in Secondary Schools

New Math  education is successful in France but failed miserably in the U. S. (1960s) because of ONE simple reason: Math teacher’s quality and quantity.

In France, each year the best of the undergraduate Math students are selected via the CONCOURS for Ecole Normale  Supérieure – the Top university in France, and the World’s BEST incubator for Fields Medalists.

Below is a report by a Chinese Math educator studying the French Math Education success:

[Source:]

image

image

image

image

法国中学数学教材的特色 (The unique characteristics of French Secondary School New Math Syllabus)

Singapore Math (PSLE)

image

This is the typical Singapore PSLE Math for 12-year-old school students sitting for Primary School Leaving Exams (PSLE).

The Primary 6 kids have not learnt Algebra, which will be taught only 1 year later in Secondary schools.

Singapore is proud of her unique “Singapore Math“, characterised by Polya-style Problem-Solving Process , aided with visual Modelling and Guesstimation techniques.

It is Singapore’s secret of being the World’s 2nd position in PISA Test (Math), after China (Shanghai) which beats us with Algebra teaching at this age.

image

See if you can solve this problem without Algebra.

If stuck … (Answer below) …
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
..
.
.
.

.
..