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Structure in Elementary
Mathematics
Ji Son Learning Lab KP Thai Human Perception
Lab
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Collaborators
Philip Kellman
Genius Programmers
Joel Zucker Tim Burke
Department of Education
Ellen Clarke
Julia PhelanIES Cognition and Student Learning
GrantIntegrating Conceptual Foundations in Mathematics through theApplication of Principles of Perceptual Learning
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What is mathematics?
Mathematics
m tM a th e m a tician S tu d en t
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What is mathematics?
Mathematics
m tM a th e m a tician S tu d en tStructure
,PatternsRelationships
, Change
roblemSolving
Getting the,right answer
Computing
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What is mathematics?
Mathematics
m tM a th e m a tician S tu d en tStructure
,PatternsRelationships
, Change
roblemSolving
Getting the,right answer
Computing+ =3 4 7 .+ = +3 4 5 2
=7 7
+ =3 4 ____
& , ; - & ,M cN eil A liba li 20 05 R ittle Joh nso n A lib ali 19 9 9
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Why?
m tMathematician Student. .U S textbooks
- ( %)explanations of mathematics 36- ( %)unsolved exercises 45- -interest grabbing illustrations
( %)irrelevant to lesson 19
- Japanese textbooks- ( %)explanations of mathematics 81- ( %)unsolved exercises 19- -interest grabbing illustrations( %)0
, , & ,Mayer Sims Tajika 199
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The silent assumption
In order to solve, one must (at leastimplicitly) understand structure True?
1.If students are proficient at solving,will they be equally proficient instructure identification?
2.Does student performance
differentiate according to structureor solution method (operation)?3.How do we train students to
appreciate structure?
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Method
31 homeschooled participants fromvarious states (roughly in 3 rd grade)
Completed sessions from homecomputers
11.87 sessions (range 3-34) 383.42 trials total (range 93-757) (37.42 trials per session)
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Types of trials
Two tasks: Solving and StructureMapping
Four operations: Addition,subtraction, multiplication, division
Four contexts: objects, distance,money, measurement
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Problem Structure learning
What kinds of problems? Two tasks: Solving and Structure
Mapping Four operations: Addition,
subtraction, multiplication, division Four contexts: Objects, Distance,
Money, Measurement Short sentences and algebraic
equations
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Francis puts 15 tomato slices in 3.salads evenly How many tomato slices are in each salad?
Solve - Object
.E n ter an sw e r
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Francis puts 15 tomato slices in 3.salads evenly How many tomato slices are in each salad?
Structure Map - Object
__ slices per salad plus 3 salads equals 15 slices
3 salads times 15 slices equals __ slices per
salad
15 slices minus 3 salads equals __ slices per salad
15 slices evenly in 3 salads equals __ slices per salad
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Francis puts 15 tomato slices in 3.salads evenly How many tomato slices are in each salad?
Structure Map - Object
15tomatoslices
=/ 3salads
omato sl ices er salad
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.Ellen walks 3 miles per hour How many hours will it take her to walk 12 miles?
3 x __ =12
12 - 3 = __
Structure Map - Distance
+ =__ 31 2
/ =1 2 __3
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.Garrett earns 6 dollars per hour How many dollars does Garrett earn in 5 hours?
6 + 5 = __
6 x 5 = __
Structure Map - Money
- =6 5 __ / =6 5 __
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. Henry s cactus is 5 feet tall Sarah s.cactus is 10 feet tall How many times
taller is Sarah s cactus than Henry s cactus?
Structure Map - Measurement
5 fee t tim es ___ eq u a ls 1 0feet
5 feet plus 10 feet equals__ 5 times 10 feet equals __feet
10 feet minus __ equals 5feet
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The silent assumption
In order to solve, one must (at leastimplicitly) understand structure True?
1.If students are proficient at solving,will they be equally proficient instructure identification?
Solving Trials vs. Structure Mapping Trials
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Solving Trials vs. Structure Mapping Trials
,O p e ra tio n F ( , ) = . ,1 29 86 14 p
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Breaking down mapping problems
Some mapping problems had the unknown as the solution
+ =A __ C+ =__ B C
- =C __ B- =C __ A
+ =A B __+ =B A __
- =C A __- =C B __
=A x __ C=__ x B C
=C __ B =C __ A
=A x B __=B x A __
=C A __ =C B __
U nkn ow n onle ft
U nkno w n onR ig h t
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Solving Trials vs. Structure Mapping Trials
ll mapping problems with the unknown as the solution be easi
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Breaking down Mapping problems
,Operation F ( , ) = . ,1 29 85 28 p
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The silent assumption
In order to solve, one must (at leastimplicitly) understand structure True?
1.If students are proficient at solving,will they be equally proficient instructure identification? Solving Structure Mapping
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The silent assumption
In order to solve, one must (at leastimplicitly) understand structure True?
1. If students are proficient at solving,will they be equally proficient instructure identification? Solving Structure Mapping
2.Does student performancedifferentiate according to structureor solution method (operation)?
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Addition and Subtraction structure
sta rt ch an g e
en d
& , ; ( ) Willis Fuson 1988 Cognitively Guided Instruction CGI
+ =Start Change End+ =Change Start End
=End Start Change =End Change Start
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Addition and Subtraction structure
,Task F ( , ) = . ,1 29 56 90 p
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Addition and Subtraction structure
,Task F ( , ) = . ,1 29 56 90 p
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Multiplication and Division Structure
( )Cognitively Guided Instruction CGI
UnitsGroups
Units per Group
:Quantity 1:Quantity 2
:Q1 per Q2
=Groups x Units per Group Units=Units per Group x Groups Units
=Units Groups Units per Group =Units Units per Group Groups
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Multiplication and Division Structure
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Multiplication and Division Structure
,Task F ( , ) = . ,1 29 319 94 p
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The silent assumption
In order to solve, one must (at leastimplicitly) understand structure True?
1. If students are proficient at solving,will they be equally proficient instructure identification? Solving Structure Mapping
2.Does student performancedifferentiate according to structureor solution method (operation)?Structure matters!
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The silent assumption
In order to solve, one must (at leastimplicitly) understand structure True?
1. If students are proficient at solving,will they be equally proficient instructure identification? Solving Structure Mapping
2. Does student performancedifferentiate according to structureor solution method (operation)?Structure matters!
3.How do we train students toappreciate structure?
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Training to see structure
Explicitly teach procedure for a wordproblem?
Explicitly teach abstraction?e.g., a + b = c word problems
has and gets new .How many does have now?
was and increased . How is now?
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Instead of explicit teachingimplicit Perceptual Learning
VaryingViewpoints
InvariantStructure
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Instead of explicit teachingimplicit Perceptual Learning
VaryingInstances Evan has 10 stickers and gets 8 new
.stickers How many stickers does Evan have now?
Martha has 11 bananas and gets
.3 new bananas How many bananas does Martha have
now?
Max has 2 sweaters and gets 7 new
.sweaters How many sweaters does Max have now? InvariantStructure
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VaryingInstances Evan has 10 stickers and gets 8 new
.stickers How many stickers does Evan have now?
Martha has 11 bananas and gets
.3 new bananas How many bananas does Martha have
now?
Max has 2 sweaters and gets 7 new
.sweaters How many sweaters does Max have now?
Instead of explicit teachingimplicit Perceptual Learning
InvariantStructure
has and gets new . How many does have now?
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Assessment
Two problems (one mapping, onesolving) from each structural typewas presented on the computer
40 problems total on pre- and
posttest
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Accuracy By Problem Type
Test , F ( , ) = . ,1 30 7 01 p
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Reaction Time
t ( ) = . ,30 3 94 p
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Next step: Rigorous testing
Will solving improve given practiceon mapping?
Will mapping improve given practiceon solving?
Transfer to new tasks: Writing down an equation for a given
word problem.
Choosing the correct word problemthat goes with a particular equation
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The silent assumption
In order to solve, one must (at leastimplicitly) understand structure True?
1. If students are proficient at solving,will they be equally proficient instructure identification? Solving Structure Mapping
2. Does student performancedifferentiate according to structureor solution method (operation)?Structure matters!
3. How do we train students toappreciate structure? Maybe
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What is it to learn mathematics?
Mathematics
m tM a th e m a tician S tu d en tExperts
,See Patterns Represent
,Relationships Att end to Change
ecoming an Expert Learning to See
,Patterns Represent
,Relationships Attend to Change
Keepour eye on the prize Do we went to create solution experts or structural
experts?
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Thank you!
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