Keith Devlin H-STAR Institute, Stanford University THE SYMBOL BARRIER Using Video Games to Overcome the Greatest Obstacle to Good Mathematics Learning
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
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
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