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# Mathematics as a Language

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On the Versatility of a Math Major

Associate Professor of Department of Mathematics and Center of Teacher Education at National Central University

http://libai.math.ncu.edu.tw/~shann/

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## Mathematics as a Language

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### Math is Built-in

1,2,3,4,…,100,…

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God Creates Natural Numbers

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### But why on earth …

foreign language 外語

alien language 宇宙人語

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### We all Speak and Write

How many can write 俳句

### 秋近き 心の寄るや 四畳半

How many can create stories

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### Many Layers of

Language Acquisition and Mastery

Math Professors 俳人

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## Practice and Philosophy

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### Native Languages

• Pick up in an immersion of a realistic environment

• Learn by examples

• Postpone the divorce of

Math and Native Language 10

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### Math Edu Fundamentals

• Learn in an immersion environment

• Teach by examples

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• Mathematics

• Computer Games

• Board Games

• Card Games

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### Roles in a Game

• Compatibility

• Inherit old rules

• Interact with old roles

• New rules and Exceptions

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### Example: 負数

Whole Numbers Negatives

Integers

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Man Creates Other Numbers We create negative numbers

just like we create card games

4  ( 6) 15

3 ( 1)  5 ( 3)  ( 2)  ( 1)

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## 三人成虎

No matter how absurd an idea is, three examples are enough for you to believe it.

People can usually derive a general rule by very few

examples. 16

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### 攻略：Fall Back

There are 8 cases of

Fall Back to only Two:

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a

b a b

###  

mn

, where

mn mn for ,m n  

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### What 負数 are for?

• Directional measurement

with a reference center, e.g.

temperature and financing.

• Eliminate subtractions.

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Learn Math as a Language

• Chances are

ノーベル文学賞 フィールド賞

• To read, to speak, and to reason with the common

sense. 19

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## Math Teacher as a Profession

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### Non-Professions

• To work well in 大学入試 test problems ……

### X

• To love and care your students ……

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## 知而不言

• Don’t say it just because you know it

• Don’t do it simply because it is doable

• Don’t prove when there is

no doubt 22

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## Context

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### Context

• What comes before and what goes after

• What students already know before the lecture

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### Go with a Purpose

• High school math is not the purpose in itself

• To make decision with Purpose in mind

• Must:

OK: 25

(15 ( 6))   ( 3)

| |x x | |x

 

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From a Higher Point of View

• Must:

No: remainder of

• Need: solve

Must: 平方完成 Ok:

• 部分分数 26

7 ( 2) 7 ( 1)

     2

7  ( 2)

2 0

ax bx  c

6x2 5x  6 (3x 2)(2 x 3)

1

1 0

1

1 0

m m

m m

n n

n

a x a x a

x b x b dx

 

 

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## Cultural Contexts

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• 九章算經 和算

• Unique Math Style

• Western Culture in

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## Life-Long Learning

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### Prepare in school

What are about to facilitate the future learning?

• Language

• Language

• Language

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### Natural Languages

• 國語

• English

• 華語

• Korean、Thai …

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Chinese Program at NCU

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### A Formal Languages

for relations and changes,

forms and shapes, quantities and the measurement systems and their manipulations, and the description of data and

uncertainties. 35

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### Mathematics

Learn to be a fluent and

efficient reader and speaker, not necessarily as a writer

(俳人), though you may become one.

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## Languages for Automation

コンピューター言語

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• and

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### Web Publishing

HTML: HyperText Markup Language

<HTML>

<BODY>

<H1>My First HTML 文件</H1>

<P>

Hello, world.  How are you doing today?

I am learning HTML now.

</P>

<H2>第一節</H2>

<P>

</P>

</BODY>

</HTML>

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### Relational Database

SQL /sequel/ Structured Query Language

JOIN: 集合の直積

select name, zip, zone, area, addressno from list join address join ZIP

and status = 'S' and gender = 'M‘

and (ZIP<300 or ZIP>700); 40

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### Maple

• CAS: Computer Algebra System

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Pedagogical Math Software

• Geogebra: High School Geometry and Algebra

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### to C or not to C

• FORTRAN for scientific computing

• C is the common core

• Python … maybe

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### Not to Waste your Life

We mathematicians have a trained talent to distinguish the hype from the core.

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## ALL are Numbers

すべてはナンバー

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Basic Concepts of Computers

Characters: CJK (中日韓), Unicode

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Basic Concepts of Computers

Glyphs: Bitmap or Splines

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Basic Concepts of Computers

Colors: Cylindrical or Cartesian Coordinate System

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Basic Concepts of Computers

Image: Matrix of Colors

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Basic Concepts of Computers

Audio: Sequence of Numbers

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## Math Curriculum

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Four Corners of Foundation

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### 微積分

Math for the Continuum 連続体

Super calculator for the past 300 years

Many realistic objects are Discrete

Calculus helped the invention of Computers

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### Discrete Math Models

• Math Tools for Discrete Models

• 離散数学

• 有限数学

• Algorithms アルゴリズム

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## Epilogue

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### The End

Thank you for your attention http://www.ncu.edu.tw

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Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung