Structure in Elementary Mathematics

8/14/2019 Structure in Elementary Mathematics

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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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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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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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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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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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1.If students are proficient at solving,will they be equally proficient instructure identification?

Solving Trials vs. Structure Mapping Trials


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,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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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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1.If students are proficient at solving,will they be equally proficient instructure identification? Solving Structure Mapping


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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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,Task F ( , ) = . ,1 29 56 90 p


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,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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,Task F ( , ) = . ,1 29 319 94 p


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2.Does student performancedifferentiate according to structureor solution method (operation)?Structure matters!


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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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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?


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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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!

Documents

Structure in Elementary Mathematics