Math Talk Like TED

Math Talk Like TEDe.g. How to Teach Calculus

Qun LinChinese Academy of Sciences

Aijun ZhangHKBU Center for Big Data in Education

Asian Technology Conference in Mathematics (ATCM) 2015

1

Agenda for Math Talk like TED

2

1. Calculus in a number : Less is Better

2. Math Elite ( 数学英才 ) via WeChat

3. Math-Micro-MOOC M3-Initiative ( 数学微慕课倡议 )

3数学英才慕课系列微信公众号： shuxueyingcai

Lin Qun

[email protected]

Working Principle: Think Simple

To count 2 after 9

BlitzkriegSpeedway

Rather than ask Better think

Calculus in a number, 80 hours 18 minutes

To count 9 after 2

数学英才慕课系列微信公众号： shuxueyingcai 4

1. Introduction

• Nowadays, numbers are frequently adopted in many TV programs, such as

“News in numbers” ( 数说新闻 )

“Daily Talk in numbers” ( 每日数说 )

Often limited by 8 minutes.

• Example in art: Leonardo da Vinci’s “Painting in a number, 0.618”.

• In mathematics: we propose “Calculus in a number, ”

Less is Better


Good news: Calculus might be regarded as a theorem that is called the “Common

Theorem” or “Common Platform”

∫𝑎

𝑏

𝑓 ′ (𝑥 )𝑑𝑥= 𝑓 (𝑏)− 𝑓 (𝑎 ) (Newton-Leibniz formula)

The rest of calculus plays only the trick on such a “common theorem”. One

theorem wins the Calculus world! (One Course, One Theorem.)

Bad news: Traditional way of deriving Newton-Leibniz formula costs too high!

Normally it requires at least 80 hours’ lessons to complete the course, which to

many people is a high threshold.


Circumference

DifferentiationSub−heigh 𝑡 =0.999⋯T angent length

¿−curve length=0.999⋯Arc lengthTangent length=0.999⋯

Curve length Curve height

Good news: In “Calculus in a number, ”, we use the comics

It only takes about 18 minutes to explain and demonstrate the

number involved, which is much easily understood.

Less is Better!


With focus on the last picture that solves for the hill-height, we only need 5 minutes

to quickly demonstrate the “common theorem”.

When the numerator (Differentiation) is so close to the denominator (Sub-height)

that the ratio reaches , we can deduct that the ratio:

By = 1, Differential Summation = Full Height. This is we call the "common

theorem", or Newton-Leibniz formula.


This is also what Tolstoy used in his “War and Peace”:

Only by taking infinitesimally small units for

observation (the differential of history, that is,

the individual tendencies of men) and

attaining to the art of integrating them (that is,

finding the sum of these infinitesimals) can

we hope to arrive at the laws of history.

Now, we go to the complete demonstration of our “common theorem”. Let’s start

with Zhuangzi’s Saying.


Great Oaks from Little Acorns Grow – Calculus based on the Concept of Infinite

Zhuangzi (~300 B.C.):Take half from a foot long stick each day, you will never exhaust it in million years.

Tear paper into 1/2, 1/4, 1/8, 1/16, pieces. Put

together all the pieces,

This process brings about the infinite concept and

infinite arithmetic.

Stepping from finite to infinite is a giant leap! https://en.wikipedia.org/wiki/1/2_%2B_1/4_%2B_1/8_%2B_1/16_%2B_⋯

2. Zhuangzi’s Saying in numbers

庄子：一尺之棰，日取其半，万世不竭。


The process in numbers:

#Terms Sum

4 0.9375

7 0.9921875

10 0.9990234375

14 0.99993896484375

17 0.99999237060546875

…… ……

34 0.99999999994179233...

PS: Gold’s purity is also measured the percentage, such as 9, 99, 999,

9999. They are incredibly similar!

(one 9)

(two 9’s)

(three 9’s)

(ten 9’s)

(four 9’s)

(five 9’s)


Concern: Will it stop for such a process of adding 9?

Answer： No! It must have infinite numbers of 9’s, because otherwise it could not

reach 1.

The Zhuangzi’s principle can be a prerequisite for learning calculus.

Next, we enter the core part of calculus, the trilogy of calculus.

9.0161

81

41

21


3. The 1st Play: Circle and Polygon• Calculus originates from measuring the circumference.

• Length could be measured by a ruler. How about the curve?

• The circle drawn by computer, if magnified, becomes a regular polygon.

This is the cyclotomic method of Archimedes and Liu Hui.

Observation: the chord length and the arc length get closer and closer. What is the

meaning of “close”? It is also explained in numbers …

Secant line approximation(Lower approximate)


N-Polygon Perimeter Percentage6 6 0.95492965867 6.0743723476 0.96676638538 6.1229349178 0.974495358412 6.2116570824 0.988615929514 6.2305861508 0.991628584316 6.2428903046 0.9935868512

Percentage:

chord lengtharc length (¿ sin θ2

θ2

)=0.999⋯⟶∑ of chord lengthsC ircumference

=0.999⋯

Infinite number of 9’s will be added, as or denominator numerator. So define:Integral of chord length = Circumference


Integral of tangent length = Circumference

An important step: simplify the secant line to tangent line (only related to a single tangent point)

Observation: the tangent length and the arc length get closer and closer.

By similar process, collect numbers, then observe the percentage,

Bad news: The above definitions cannot obtain the exact value. They are always

approximate values. This has been a historical puzzle.

Tangent line approximation(Upper approximate)

Arc lengthTangent length (¿ θ

tan θ )=0.999⋯→ CircumferenceSumof tagent lengths=0.999⋯

Infinite number of 9’s will be added, as or numerator ＝ denominator. So,


4. The 2nd Play: Curved Triangle and Tangent• Let us extend from the circle to the general curve (e.g. hillside).

• The curve length is unmeasurable. We may change it to straight line.

• The curved triangle can be changed to straight triangle, e.g. the tangent triangle with hypotenuse tangent to the curve.

Similar as last play about circle’s tangent (i.e. collect numbers, observe percentage)

Tangent point

Curve length(unmeasurable)

Tangent length(measurable)

= 0.999 by contradiction

(The latter is the average version of the former.)


Integral of tangent length = Curve length

PS: Why is the curve length calculated based on tangent line rather than secant line? Our daily experience suggests to use pace to measure, i.e. the secant length step by step. When the pace becomes shorter and shorter, the secant length would become identical to the tangent length. The tangent length can be measured from the tangent point, without need to consider the other end point.

Infinite number of 9’s will be added, as or denominator =numerator. Define

This is an extension of circle’s circumference.

Good news: The tangent can be defined as the differential in the next section.


5. The 3rd Play: Calculate Height of Hillside

No need to start over again. Simply take the isomorphism and change one to the other.

Percentage:

DifferentiationS ub − height

=0.999⋯ (Definition ) quick proof →

∑ of differentiation 𝑠Height of hillside

= 0.999 ⋯ ( Integrability Theorem)by contradiction(The latter is the average version of the former.)


When the hillside is a parabolic curve:Data Collection

#Subintervals 2 4 10 20 100

0.5 0.75 0.90 0.95 0.99

The percentages also have one 9, two 9’s, three 9’s, four 9’s …

Infinite number of 9’s will be added, as or numerator = denominator. Define

Integral of differentiations = Height of hillside

Here the integral is deterministic and finite.

Good news: The integral could be exact value (rather than approximate), thus

solved the historical puzzle (e.g. circumstance). It is called the fundamental

theorem in calculus textbooks ,and here we call it the common theorem.


6. Common Theorem of Calculus from Humanities to Sciences

Humanities Calculus:The "common theorem" built above doesn’t need more mathematical symbols but only the humanities languages, and so it might be served as the Humanities Calculus, including the principle of "War and Peace". Different majors may have different acception or rejection. Indeed, it’s an elastic teaching material.

Sciences CalculusIn sciences, we need not only the "common theorem" itself, but also play the "common theorem". Then, we need mathematical symbols and more calculation. Let us omit the details but put them in the Appendix, as the discussion class or exercises course.

Document


20





Math Elite Program – Quick Introduction

21

Elite Secondary Students ProgramOrganized by CAST( 中国科协 )including 5 subjects: Mathematics, Physics, Chemistry, Biology, Computer Science

Math Elite ( 中学生数学英才计划 )Director: Tian Gang (PKU)Involving top universities in China to open mathematics resources to elite secondary students.

http://www.ycjh.org/


Math Elite via WeChat


Trend within 4 months


24




25

Recommendation to China GovernmentProposal for Microlecture development: a) Better production tools; b) MicroMOOC

Full-text (in Chinese though) available upon request.

26

Microlecture/MOOC Tools

来源：教育部科技委《专家建议》总第 217期

27

Quick Demo of “iWeike” App

28

Math-Micro-MOOC M3-InitativeMOOC MicroMOOC

Providers Coursera, edX, Udacity, xuetangX, Study163 … Our Proposal

Methods creating videos

Classroom recording, Green screen keying, etc Micro-video apps

Fit Mobile Internet? No Yes

Video duration ≥ 40min (mostly) ≤ 10min

Storage Volume ≥ 100MB ≤ 1MB

Production Complexity Rather complicated Extremely Simple

Production Cost Very High Very Low

PlatformSupport

None has social-networking;Some lack LCMS

Strong-tie social-networkingE-Learning platform

Marketing Approach

Top-down approach:Partner w/ prestigious universities

Bottom-up approach:Praise by teachers and students

Target Market Higher education (mostly) Both K12 and Higher Education

Typical example of Math-Micro-MOOC:

29

Math-Micro-MOOC M3-TeamMMOOC R&D and Operation by: HKBU Center for Big Data in Education

30

Thank you!

Contact us:

Prof. Lin, Qun: [email protected]

Dr. Zhang, Aijun: [email protected]