the symbol barrier - stanford universitykdevlin/presentations/symbol... · 2012-12-10 · Ñbill...

Keith DevlinH-STAR Institute, Stanford University

THE SYMBOL BARRIERUsing Video Games to Overcome the Greatest

Obstacle to Good Mathematics Learning

EVERYDAY MATH

Counting

The focus

Arithmetic

Proportional Reasoning

Numerical Estimation

Elementary Geometry

Elementary Trigonometry

Elementary Algebra

Basic Probability and Statistics

Logical Thinking

Algorithm Use

Problem Formation

Problem Solving

Modeling

Sound Calculator Use

Devlin makes the case with care, repeatedly drawing on documented studies and educational principles.—Bill Wood, MAA Reviews, September 2011

Keith Devlin’s highly readable book sets the foundation for a new approach to learning mathematics ... The book is based on empirically well supported and lucidly explicated theories of learning, teaching, and gaming. It will become a classic.—James Paul Gee, Arizona State University and author of What Video Games Have to Teach Us About Learning and Literacy

Mathematics Education for a New Era connects Devlin’s deep understanding of mathematics education to the new research in digital-games-based learning to pave a path for re-energizing mathematics education.—Kurt Squire, University of Wisconsin, author of Video Games & Learning: Teaching and Participatory Culture in the Digital AgeAK Peters, 2011. Based on 7 years of

research at Stanford and in a large commercial video game company.

GENERAL BACKGROUND

THE THREE SOURCES

THE GOAL: The NRCʼs Five StrandsConceptual

understanding the comprehension of mathematical concepts, operations, and relations

Procedural fluency

skill in carrying out arithmetical procedures accurately, efficiently, flexibly, and appropriately

Strategic competence

the ability to formulate, represent, and solve mathematical problems arising in real-world situations

Adaptive reasoning

the capacity for logical thought, reflection, explanation, and justification

Productive disposition

a habitual inclination to see mathematics as sensible, useful, and worthwhile, combined with a confidence in one’s own ability to master the material

FOUR OBSTACLESto good (everyday-) math learning that any

successful pedagogy has to overcome

FOUR OBSTACLESto good (everyday-) math learning that any

successful pedagogy has to overcome

FOUR OBSTACLESto good (everyday-) math learning that any

successful pedagogy has to overcome

ymbol barrier

ransfer problem

ngagement

eaning and motivation

FOUR OBSTACLESto good (everyday-) math learning that any

successful pedagogy has to overcome

Traditional approaches can fail on all four counts

ymbol barrier

ransfer problem

ngagement

eaning and motivation

FOUR OBSTACLESto good (everyday-) math learning that any

successful pedagogy has to overcome

Traditional approaches can fail on all four counts

ymbol barrier

ransfer problem

ngagement

eaning and motivationVideo games can eliminate these

THE SYMBOL BARRIERThe lesson from Street Mathematics

Street Mathematics and School Mathematics, by Terezinha Nunes, Analucia Dias Schliemann, and David William Carraher, C.U.P. (1993)

THE SYMBOL BARRIERThe lesson from Street Mathematics

98%Street Mathematics and School Mathematics, by Terezinha Nunes, Analucia Dias Schliemann, and David William Carraher, C.U.P. (1993)

THE SYMBOL BARRIERThe lesson from Street Mathematics

98% 37%Street Mathematics and School Mathematics, by Terezinha Nunes, Analucia Dias Schliemann, and David William Carraher, C.U.P. (1993)

THE SYMBOL BARRIERThe lesson from Street Mathematics

98% 37%Street Mathematics and School Mathematics, by Terezinha Nunes, Analucia Dias Schliemann, and David William Carraher, C.U.P. (1993)

Meaningful, motivating,

real-life contextAbstract and

decontextualized

THE SYMBOL BARRIERThe lesson from Street Mathematics

98% 37%Street Mathematics and School Mathematics, by Terezinha Nunes, Analucia Dias Schliemann, and David William Carraher, C.U.P. (1993)

Meaningful, motivating,

real-life contextAbstract and

decontextualized

EVERYONE CAN DO EVERYDAY MATH

THE PROBLEM IS SYMBOLIC REPRESENTATION

More later on the symbol barrier.meanwhile

How do we overcome the Problem?

More later on the symbol barrier.meanwhile

How do we overcome the Problem?

98% 37%

ONE APPROACHPresent the mathematics symbolically

Tabula Digitaʼs DimensionM Imagine Educationʼs Koʼs Journey

ONE APPROACHPresent the mathematics symbolically

IMPORTANT TO: Develop the game (e.g. the backstory) around the mathematics so it arises naturally, and doing

the mathematics is a natural part of the gameplay

Tabula Digitaʼs DimensionM Imagine Educationʼs Koʼs Journey

Doing math in a meaningful, natural context

Adopting an identity of being a practitioner

IMPORTANT TO: Develop the game (e.g. the backstory) around the mathematics so it arises naturally, and doing

the mathematics is a natural part of the gameplay

ONE APPROACHPresent the mathematics symbolically

Teacher  assigns  homework:“Get  to  the  next  level.”

Student  plays  game  at  home;  makes  progress  but  fails  to  level  up.

In  class,  teacher  explains  themath  required  to  advance

further  in  the  game.

NEW  TOPIC

HOMEWORK-­‐FIRST

LEARNING

Transfer: eUCLID’s approach

Many resources, including a teacher

Student  consults  various  sources  (inc.  family,  

friends,  Internet,  books)  to  try  to  make  progress.

Let the teacher and other resources take care oftransfer, and focus on the really big problem,where video games can have a huge impact:

Transfer: eUCLID’s approach

Let the teacher and other resources take care oftransfer, and focus on the really big problem,where video games can have a huge impact:

Transfer: eUCLID’s approach

98% 37%

EVERYONE CAN DO EVERYDAY MATH

THE PROBLEM IS SYMBOLIC REPRESENTATION

How should we teach everyday math?

98% 37%?

EVERYONE CAN DO EVERYDAY MATH

THE PROBLEM IS SYMBOLIC REPRESENTATION

How should we teach everyday math?

98% 37%?Symbol barrier

EVERYONE CAN DO EVERYDAY MATH

THE PROBLEM IS SYMBOLIC REPRESENTATION

How should we teach everyday math?

98% 37%?Symbol barrier

Grounding

EVERYONE CAN DO EVERYDAY MATH

THE PROBLEM IS SYMBOLIC REPRESENTATION

How should we teach everyday math?

98% 37%?Symbol barrier

Grounding

EVERYONE CAN DO EVERYDAY MATH

THE PROBLEM IS SYMBOLIC REPRESENTATION

We should start on the left and move right

How should we teach everyday math?

98% 37%?Symbol barrier

Grounding

EVERYONE CAN DO EVERYDAY MATH

THE PROBLEM IS SYMBOLIC REPRESENTATION

We should start on the left and move right

& hitherto

LACK OF A SUITABLE MEDIUM

The Symbol Barrier

HABITUATION

How do we think of “doing math”?The Symbol Barrier

The centrality of the textThe Symbol Barrier

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

The centrality of the textThe Symbol Barrier

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

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

The centrality of the textThe Symbol Barrier

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

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

Mathematics is about doing, not knowing.

The centrality of the textThe Symbol Barrier

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

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

Mathematics is about doing, not knowing. Everyday math is primarily a way of thinking about entities,

issues, and problems in the world.

The centrality of the textThe Symbol Barrier

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

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

Mathematics is about doing, not knowing. Everyday math is primarily a way of thinking about entities,

issues, and problems in the world. Though much advanced mathematics is linguistically defined,

everyday math is directly abstracted from the world, and can be done without formal notation (symbols).

The centrality of the textThe Symbol Barrier

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

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

Mathematics is about doing, not knowing. Everyday math is primarily a way of thinking about entities,

issues, and problems in the world. Though much advanced mathematics is linguistically defined,

everyday math is directly abstracted from the world, and can be done without formal notation (symbols).

But are symbolic representations the best way to teach (everyday) mathematics today ?

The centrality of the textThe Symbol Barrier

✦ 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 way

Even bigger question

The Symbol Barrier

✦ 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 way

Even bigger question

The Symbol Barrier

✦ 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 way

Even bigger question

The Symbol Barrier

The Symbol BarrierClassic symbolic instruction

Classic symbolic instruction, different mediumThe Symbol Barrier

Use the compelling engagement of video games to motivate practice and drive time-on-(traditional)-task

Most math ed video games overlay traditional text-based curriculum onto video games

The Symbol BarrierClassic symbolic instruction, different medium

STILL TRADITIONAL, TEXT-BASED INSTRUCTION

The Symbol BarrierClassic symbolic instruction, different medium

The Symbol Barrier

Math is not intrinsically about symbols

The Symbol Barrier

It is a way of thinking

Math is not intrinsically about symbols

The Symbol Barrier

It is a way of thinking

It is something you do

Math is not intrinsically about symbols

The Symbol Barrier

What is the best way to learn how to do something?

It is a way of thinking

It is something you do

Math is not intrinsically about symbols

The Symbol Barrier

Just do it!

What is the best way to learn how to do something?

It is a way of thinking

It is something you do

Math is not intrinsically about symbols

The Symbol Barrier

Learning by doingLook, no books

Learning by doing (simulator learning)Look, no books

So we should design learning environments like this, right?

So we should design learning environments like this, right?

THIS IS EXPENSIVE

So we should design learning environments like this, right?

THIS IS EXPENSIVE

COST ~ $10M+

Do not necessarilyneed all of this

The real world can provide much of the motivating context

Blend the real and the virtualto mutually leverage each other

Good interaction designtrumps many features

Do not necessarilyneed all of this

Meet Jiji

The real world can provide much of the motivating context

Blend the real and the virtualto mutually leverage each other

Good interaction designtrumps many features

✦ Good interaction design trumps many features

✦ The real world can provide ✦ much of the motivating context✦ Blend the real and the virtual to

mutually leverage each other

Meet Jiji

✦ Good interaction design trumps many features

✦ The real world can provide ✦ much of the motivating context✦ Blend the real and the virtual to

mutually leverage each other

The hard part✦ Find novel representations of mathematics that are native

to a digital (or physical) interactive environment✦ — not necessarily native to a static flat surface (stone

tablet, clay tablet, sandbox, slate, parchment, blackboard, whiteboard, paper, and bathroom mirrors)

Meet Jiji

Casual games: find mathematical equivalents to

Start with this question

eUCLIDʼs approach

What is the best way to learn the piano?

Start with this question

eUCLIDʼs approach

What is the best way to learn the piano?

eUCLIDʼs approach

What is the best way to learn the piano?

The piano is a great interface to music

eUCLIDʼs approach

What is the best way to learn the piano?

The piano is a great interface to music

eUCLIDʼs approach

✦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

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

eUCLIDʼs approach

?

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

eUCLIDʼs approach

?

Instruments on which to play matheUCLIDʼs approach

Grade 2, Operations & Algebraic Thinking #2, Grade 2, Number & Operations in Base Ten #2, #8; Grade 3, Operations & Algebraic Thinking #1, #4,; Grade 4, Operations & Algebraic Thinking #5;Grade 6, Number System #5, #6

Grade 4, Operations & Algebraic Thinking #5; Grade 6, Expressions & Equations #2, #4, #5, #6; Grade 8, Expressions & Equations #7, #8

Grade 3, Operations and Algebraic Thinking #4, #5, #6, #7;Grade 4, Operations and Algebraic Thinking #3, #4;Grade 6, The Number System #4, Expressions and Equations #7

Grade 6, Ratios & Proportional Relationships #1, #3; Grade 7, Ratios & Proportional Relationships #2, #3

Back-end analytics track studentsʼ progress, adjust the difficulty level and pace, and provide student- and teacher-feedback.

Placeholder art; images are not the actual eUCLID games

Mathematicalthinking

Mathematicalthinking

Mathematicalthinking

Mathematicalthinking

Placeholder art; images are not the actual eUCLID games

Back-end analytics track studentsʼ progress, adjust the difficulty level and pace, and provide student- and teacher-feedback.

eUCLIDʼs approach

Instruments on which to play math

Video games offer educators a tool – to supplement other forms of instruction and learning – that we have never had before, and which can overcome the single biggest obstacle to mastering everyday math: the 98% v 37% Symbol Barrier.

Used wisely, video games can totally transform the learning of everyday math, so all children hit that 98% level, and we impact all five of the NRC’s strands.

MY DREAM

Devlin makes the case with care, repeatedly drawing on documented studies and educational principles.—Bill Wood, MAA Reviews, September 2011

Keith Devlin’s highly readable book sets the foundation for a new approach to learning mathematics ... The book is based on empirically well supported and lucidly explicated theories of learning, teaching, and gaming. It will become a classic.—James Paul Gee, Arizona State University and author of What Video Games Have to Teach Us About Learning and Literacy

Mathematics Education for a New Era connects Devlin’s deep understanding of mathematics education to the new research in digital-games-based learning to pave a path for re-energizing mathematics education.—Kurt Squire, University of Wisconsin, author of Video Games & Learning: Teaching and Participatory Culture in the Digital Age

General Background

devlin@stanford.edu

Website: profkeithdevlin.comBlog: profkeithdevlin.org

Blog: MOOCtalk.orgMAA blog: devlinsangle.blogspot.com

Devlin makes the case with care, repeatedly drawing on documented studies and educational principles.—Bill Wood, MAA Reviews, September 2011

Keith Devlin’s highly readable book sets the foundation for a new approach to learning mathematics ... The book is based on empirically well supported and lucidly explicated theories of learning, teaching, and gaming. It will become a classic.—James Paul Gee, Arizona State University and author of What Video Games Have to Teach Us About Learning and Literacy

Mathematics Education for a New Era connects Devlin’s deep understanding of mathematics education to the new research in digital-games-based learning to pave a path for re-energizing mathematics education.—Kurt Squire, University of Wisconsin, author of Video Games & Learning: Teaching and Participatory Culture in the Digital Age

General Background

devlin@stanford.edu

Website: profkeithdevlin.comBlog: profkeithdevlin.org

Blog: MOOCtalk.orgMAA blog: devlinsangle.blogspot.com

AVOID: Confusing mathematics (a way of thinking) with its (symbolic) representation on a static, flat surface. (cf. music and musical notation.)

AVOID: Presenting the mathematical activities as separate from the game action and game mechanics.

AVOID: Relegating the mathematics to a secondary activity when it should be the main focus.

AVOID: Reinforcing the perception that math is an obstacle that gets in the way of doing more enjoyable things.

AVOID: Reinforcing the perception that math is an arbitrary hurdle to be overcome, or circumvented, in order to progress .

AVOID: Encouraging the student to try to answer quickly, without reflection. AVOID: Reinforcing the belief that math is just a large bag of isolated facts and

tricks. AVOID: Reinforcing the perception that math is so intrinsically uninteresting

it has to be sugar coated.

THINGS TO AVOIDeUCLiD design principles