Keith DevlinH-STAR Institute, Stanford University

Mathematics Educationfor a Flat World

What should our children learn for life in the twenty-first century?

Sobering (Scary? Exciting?) Thought

Sobering (Scary? Exciting?) Thought

✦ I have a bachelors and a doctorate in mathematics.

Sobering (Scary? Exciting?) Thought

✦ I have a bachelors and a doctorate in mathematics.

✦ Every technique and method I learned at university can now be outsourced to where it can be completed faster and more cheaply.

Sobering (Scary? Exciting?) Thought

✦ I have a bachelors and a doctorate in mathematics.

✦ Every technique and method I learned at university can now be outsourced to where it can be completed faster and more cheaply.

✦ The only things I have left of market value are experience, a human network, and a powerful collection of metacognitive skills.

Sobering (Scary? Exciting?) Thought

✦ I have a bachelors and a doctorate in mathematics.

✦ Every technique and method I learned at university can now be outsourced to where it can be completed faster and more cheaply.

✦ The only things I have left of market value are experience, a human network, and a powerful collection of metacognitive skills.✴ – including mathematical thinking

Reassuring Fact

✦ Since 2001, I have had a steady stream of consulting projects, many for the US DoD.

Reassuring Fact

✦ Since 2001, I have had a steady stream of consulting projects, many for the US DoD.✴ all those connected with intelligence analysis.

Reassuring Fact

✦ Since 2001, I have had a steady stream of consulting projects, many for the US DoD.✴ all those connected with intelligence analysis.

✦ One DoD episode explains what is going on.

Reassuring Fact

✦ Since 2001, I have had a steady stream of consulting projects, many for the US DoD.✴ all those connected with intelligence analysis.

✦ One DoD episode explains what is going on.

Reassuring Fact

NIMD

✦ Since 2001, I have had a steady stream of consulting projects, many for the US DoD.✴ all those connected with intelligence analysis.

✦ One DoD episode explains what is going on.

Reassuring Fact

NIMDMulti-agent context-sensitive reasoning

✦ Since 2001, I have had a steady stream of consulting projects, many for the US DoD.✴ all those connected with intelligence analysis.

✦ One DoD episode explains what is going on.

Reassuring Fact

NIMDMulti-agent context-sensitive reasoning

A feature F is contextual for an action A if F constrains A and may affect the outcome of A, but is not a constituent of A.In particular, a situation, environment, or set of circumstances C is said to be a context for an action A if C constrains A or affects the outcome of A but is not a constituent of A.

Working Definition of Context

Repeatedly:

My instinctive or “too obvious to mention” steps

were key to progress.

Repeatedly:

My instinctive or “too obvious to mention” steps

were key to progress.

It was not the mathematics I knew or could apply that was important; it was the way I approached the

problem and framed the question!

What constitutes a successful mathematics education program?

What constitutes a successful mathematics education program?

World class research

What constitutes a successful mathematics education program?

Fields MedalsUSA 13France 11Russia/USSR 9 UK 6Japan 3Belgium 2Australia, Finland, Germany, Israel, Italy, New Zealand, Norway, Sweden, Vietnam 1

World class research

International ranking

What constitutes a successful mathematics education program?

Successful economy - GDP

What constitutes a successful mathematics education program?

Innovation

Global Innovation Index

What constitutes a successful mathematics education program?

My perspective

YORKSHIRE

I grew up here

I live here

What constitutes a successful mathematics education program

for an innovation economy?

✦ Mathematics has always developed in response to changing needs in society

✦ Changes in math education usually lag behind

What constitutes a successful mathematics education program

for an innovation economy?

The need for mathematicsMedieval times

Medieval timesThe mathematics education provided

Liber abbaci, 1202

Nineteenth centuryThe need for mathematics

The mathematics education providedNineteenth century

Twentieth centuryThe need for mathematics

The mathematics education providedTwentieth century

The mathematics education providedTwentieth century

Late twentieth centuryThe need for mathematics

The mathematics education providedLate twentieth century

Time for a change?

The ions of mathematics

The big picture

Important mathematical ions over the centuries

Important mathematical ions over the centuries

AdditionSubtraction

MultiplicationDivision

Important mathematical ions over the centuries

AdditionSubtraction

MultiplicationDivision

Important mathematical ions over the centuries

AdditionSubtraction

MultiplicationDivision. . .

DifferentiationIntegration

Important mathematical ions over the centuries

AdditionSubtraction

MultiplicationDivision. . .

DifferentiationIntegration

Communication

. ..

Collaboration

Important mathematical ions over the centuries

AdditionSubtraction

MultiplicationDivision. . .

DifferentiationIntegration

Communication

. ..

Collaboration

Innovation

Innovative mathematical thinkers

WANTED

Innovative mathematical thinkers

Must be able to:

WANTED

Innovative mathematical thinkers

✦ Think outside the box

Must be able to:

WANTED

Innovative mathematical thinkers

✦ Think outside the box✦ Find/adapt existing methods/techniques for novel situations

Must be able to:

WANTED

Innovative mathematical thinkers

✦ Think outside the box✦ Find/adapt existing methods/techniques for novel situations✦ Find new approaches/methods/techniques for new problems

Must be able to:

WANTED

Innovative mathematical thinkers

✦ Think outside the box✦ Find/adapt existing methods/techniques for novel situations✦ Find new approaches/methods/techniques for new problems✦ Collaborate with others – work in multidisciplinary teams

Must be able to:

WANTED

Innovative mathematical thinkers

✦ Think outside the box✦ Find/adapt existing methods/techniques for novel situations✦ Find new approaches/methods/techniques for new problems✦ Collaborate with others – work in multidisciplinary teams✦ Communicate well

Must be able to:

WANTED

Innovative mathematical thinkers

✦ Think outside the box✦ Find/adapt existing methods/techniques for novel situations✦ Find new approaches/methods/techniques for new problems✦ Collaborate with others – work in multidisciplinary teams✦ Communicate well

Must be able to:

Must have:

WANTED

Innovative mathematical thinkers

✦ Think outside the box✦ Find/adapt existing methods/techniques for novel situations✦ Find new approaches/methods/techniques for new problems✦ Collaborate with others – work in multidisciplinary teams✦ Communicate well

✦ Broad sense of the scope, power, and limitations of mathematics

Must be able to:

Must have:

WANTED

Innovative mathematical thinkers

✦ Think outside the box✦ Find/adapt existing methods/techniques for novel situations✦ Find new approaches/methods/techniques for new problems✦ Collaborate with others – work in multidisciplinary teams✦ Communicate well

✦ Broad sense of the scope, power, and limitations of mathematics ✦ Good (not necessarily stellar) mathematical ability

Must be able to:

Must have:

WANTED

Innovative mathematical thinkers

✦ Think outside the box✦ Find/adapt existing methods/techniques for novel situations✦ Find new approaches/methods/techniques for new problems✦ Collaborate with others – work in multidisciplinary teams✦ Communicate well

✦ Broad sense of the scope, power, and limitations of mathematics ✦ Good (not necessarily stellar) mathematical ability✦ Ability to quickly master new mathematical techniques

Must be able to:

Must have:

WANTED

Innovative mathematical thinkers

✦ Think outside the box✦ Find/adapt existing methods/techniques for novel situations✦ Find new approaches/methods/techniques for new problems✦ Collaborate with others – work in multidisciplinary teams✦ Communicate well

✦ Broad sense of the scope, power, and limitations of mathematics ✦ Good (not necessarily stellar) mathematical ability✦ Ability to quickly master new mathematical techniques

Must be able to:

Must have:

WANTED

It’s good to have some people who can do this

But innovation nations need a lot of people who can participate in the production of this

TextPhaneesh Murthy, CEO of iGate Patni, a top-ten Indian outsourcing company with 26,000 highly skilled employees, performing tasks such as writing iPhone apps, processing mortgage applications, and redesigning supply chains:

“The U.S. education system is much more geared to innovation and practical application. It’s really good from high school onward. To compete long term, we need more brainstorming, not memorization; more individuality, not standardization.”

Things that cannot easily be outsourced

✦ Should the primary goal be “teaching mathematics” (as commonly understood)?

✦ i.e., a collection of definitions, facts, rules, representations, methods, algorithms, formulas.

✦ Or should we focus on developing mathematical thinking?

✦ i.e., a habit of mind of approaching problems in a particular (but flexible) way

The big question

Mathematical Thinking

Mathematical Thinking

Mathematical Thinking

Mathematical Thinking

Mathematical Thinking

1. Introductory material2. Analysis of language – the logical combinators3. Analysis of language – implication4. Analysis of language – equivalence5. Analysis of language – quantifiers6. Working with quantifiers7. Proofs8. Proofs involving quantifiers9. Elements of number theory10.Beginning real analysis

Course topics

Mathematical Thinking

1. Introductory material2. Analysis of language – the logical combinators3. Analysis of language – implication4. Analysis of language – equivalence5. Analysis of language – quantifiers6. Working with quantifiers7. Proofs8. Proofs involving quantifiers9. Elements of number theory10.Beginning real analysis

Course topics

Mathematical Thinking

The barrier

?

The barrier

The barrier

“Where’s the math?”

Symbols of success?

For over two thousand years, books were the only means to store and disseminate information to society – a technology limitation!

Textbook delivery has shaped our view of what mathematics is and how to do it.

Mathematics is about doing, not knowing. Mathematical thinking is primarily a way of

thinking about entities, issues, and problems in the world. Though much advanced mathematics is linguistically defined, a lot

of mathematical thinking can be done (or learned) without formal notation (symbols), or perhaps with new representations.

What is the best way to learn the piano?

What is the best way to learn the piano?

The piano is a great interface to music

What is the best way to learn the piano?

The piano is a great interface to music

✦Learning by doing (the real thing)✦Same device used throughout, from

initial learning to concert performance✦Sense of direct connection to the music✦ Instant feedback✦The piano tells you when you are wrong✦The piano tells you (in part) how you are

wrong✦Easy to gauge your progress✦The instructor is your guide, not the

arbitrator of right or wrong✦Adaptive learning – the learner adapts,

by adjusting the degree of difficulty

Can music learning point the way?What is the best way to learn mathematical thinking?

WANTED“Instruments” on which to “play” mathematics

Can music learning point the way?What is the best way to learn mathematical thinking?

Can music learning point the way?What is the best way to learn mathematical thinking?

Can music learning point the way?What is the best way to learn mathematical thinking?

98% v 37%

The adoption of this interface to arithmetic in 13th Century Europe, starting in Pisa, ...

?Banking (banca)

Insurance (polizza)

Double-entry bookkeeping (Medicis)

Internationaltrading empires

... facilitated this revolutionThe birth of modern finance

1202

?To be continued . . .

2013