Mathematics as a Language

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On the Versatility of a Math Major

單維彰．2014-10-28

多芸な数学系の学生になるために

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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Think of

Mathematics as a Language

数学を言語として認識すれば

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Math is Built-in

1,2,3,4,…,100,…

余り分

不足分合わせる分

分ける分 ₃

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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

小川洋子《博士の愛した数式》

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Many Layers of

Language Acquisition and Mastery

Math Professors 俳人

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Prepare Yourself

No individual curriculum.

Up to you

what to grasp

how much to grasp to prepare yourself

as versatile as possible

⁸

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Math Education:

Practice and Philosophy

数学教育の実践と理念

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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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Man-Made Universe

• 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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, for ,

a



b a b

 

m  n

, where

m  n m  n 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

職業としての数学教師

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Non-Professions

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

X

• To love and care your students ……

X

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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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時而言

• Not only what to say but also when to say

• Need to know the

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

n n

n

a x a x a

x b x b dx



 



  

 _^

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Cultural Contexts

文化的脈絡

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Greek Paradigm

• 九章算經和算

• Unique Math Style

• Western Culture in

Greek Tradition

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English References

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Japan Emerging

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Life-Long Learning

生涯学習

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Prepare in school

What are about to facilitate the future learning?

• Language

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Natural Languages

• 國語

• English

英文卒業標準

• 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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Math Text Processing

• 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>

其實 HTML 裡面中文也會通。

</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

where list.addressid = address.id and address.zipid = ZIP.id

and status = 'S' and gender = 'M‘

and (ZIP<300 or ZIP>700); ⁴⁰

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Matlab

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Maple

• CAS: Computer Algebra System

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Pedagogical Math Software

• Geogebra: High School Geometry and Algebra

• GSP: Geometers’ Sketch Pad

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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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• Glyphs: Bitmap or Splines

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• Colors: Cylindrical or Cartesian Coordinate System

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• Image: Matrix of Colors

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• Audio: Sequence of Numbers

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Math Curriculum

数学カリキュラム

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Four Corners of Foundation

Basic Concepts of Computers

微積分統計

線形代数

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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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Linear Algebra

•

Some matrices are essentially the same

•

Pure: identify the invariants

•

Applied: conversions

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Epilogue

結語

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The End

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

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Mathematics as a Language

Think of

Mathematics as a Language

Math is Built-in

But why on earth …

We all Speak and Write

秋近き 心の寄るや 四畳半

小川洋子《博士の愛した数式》

Many Layers of

Prepare Yourself

No individual curriculum.

Up to you

what to grasp

how much to grasp to prepare yourself

as versatile as possible

Math Education:

Practice and Philosophy

Native Languages

Math Edu Fundamentals

Man-Made Universe

Roles in a Game

Example: 負数

三人成虎

攻略：Fall Back

, for ,



 

What 負数 are for?

Math Teacher as a Profession

Non-Professions

X

X

知而不言

時而言

• Not only what to say but also when to say

• Need to know the

Context

Context

Go with a Purpose

Cultural Contexts

Greek Paradigm

English References

Japan Emerging

Life-Long Learning

Prepare in school

Natural Languages

A Formal Languages

Mathematics

Languages for Automation

Math Text Processing

Web Publishing

Relational Database

Matlab

Maple

to C or not to C

Not to Waste your Life

ALL are Numbers

Math Curriculum

Basic Concepts of Computers

微積分 統計

線形代数

微積分

Discrete Math Models

Linear Algebra

Some matrices are essentially the same

Pure: identify the invariants

Applied: conversions

Epilogue

The End

秋近き心の寄るや四畳半

微積分統計