How useful are fraction bars for understanding fraction equivalence

and addition?

A difficulty factors assessment with 5th, 6th, and 7th graders

Eliane Stampfer Wiese, Kenneth R. KoedingerHuman-Computer Interaction Institute

Carnegie Mellon Universitystampfer@cs.cmu.edu

2

Sam spent 5/7 of his money on a board game. The game cost $25. How much money did he have at first?

aid sense-making and become a bridge to abstract thinking

NCTM 2013 Leinwand & Ginsburg 2007

$25

?

harm performance if students can’t interpret them correctly

Diagrams can …

Rittle-Johnson & Koedinger2001Booth & Koedinger 2011

3

4

5

6

7

I’m Done!!

1-2 timesper student per problemStampfer & Koedinger 2013

8

How does fraction bar utility develop between 5th and 7th grade?

9

What Makes This So Hard?1) Areas represent

amounts

2) Bars represent fractions

3) Mapping the relationship between the pictures to the relationship between the fractions

10

Difficulty Factor Assessment• Paper & pencil quiz

• One question format per skill

• Tasks: addition and equivalence

• Within-subject design

• 5th – 7th graders at local public school (~150 students in each grade)

11

is bigger than

is smaller than

is equivalent to

a)

b)

c)

12

is bigger than

is smaller than

is equivalent to

a)

b)

c)

721

13

13

13

13

721

721

721

Pictures Only

1) Areas represent

amounts

Pictures &

Numbers2) Bars represent

fractions

Half Pictures &

Numbers3) Mapping

relationshipsNumbers OnlyControl

82% 88% 90%81% 86% 93%

5th 6th 7th

82% 86% 90%50%

67%

1

0

57%

13

Hint:

True or false:

+ =FALSE

14

Hint:

True or false:

Pictures Only

1) Areas represent

amounts

313

211

12+

211

12

313

Pictures &

Numbers2) Bars represent

fractions

+ =

Half Pictures

& Numbers3) Mapping

relationshipsNumbers Only

Control 5th 6th 7th

82% 87%79% 75% 84%64%46% 70% 77%52% 64%

1

0

21%

15

Percent Correct by Grade & ScaffoldStudents did 2 problems of each type~150 students in each grade

5th 6th 7th

Pictures

Pictures & Numbers

Half Pictures & Numbers

Numbers 5th 6th 7th 5th 6th 7th

Equivalence Addition

16

Repeated Measures ANOVAs by Grade

Pictures

Pictures & Numbers

Half Pictures & Numbers

Numbers 5th 6th 7th 5th 6th 7th

Equivalence Addition

Class Tracking Level x Scaffold x Task (repeated measures on scaffold and task)

Main Effects: Scaffold and Task (p<.01)Interactions: 5th grade Scaffold * Task (p<.01)

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Repeated Measures for EquivalenceGrade x Scaffold, repeated measures on scaffold

Pictures

Pictures & Numbers

Half Pictures & Numbers

Numbers 5th 6th 7th

Main Effects: Scaffold and Grade (both p<.01)

No Scaffold * Grade Interaction

Difficulty depends presence of picturesStudents improve from 5th to 7th grade

18

Repeated Measures for AdditionGrade x Scaffold, repeated measures on scaffold

Pictures

Pictures & Numbers

Half Pictures & Numbers

Numbers 5th 6th 7th

Main Effects: Scaffold and Grade (both p<.01)

Scaffold * Grade Interaction (p<.01)

Difficulty depends on scaffoldRelative difficulty of scaffolds depends on gradePerformance improves with grade

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Separate ANOVAs on Scaffold by Gradewith Post-Hoc Tukey tests

Pictures

Pictures & Numbers

Half Pictures & Numbers

Numbers 5th 6th 7th

5th Grade: All differences (p<.01)

6th and 7th Grade:Numbers (p<.01)

For 5th graders, each scaffold type had a unique difficulty levelFor 6th and 7th graders, Numbers was more difficult than the others

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• One school

• False addition questions all used same foil (adding numerators and denominators)

• Does not untangle diagram interpretation from skills with fractions – both improve with grade

Limitations: Population & Design

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Fraction Bar Utility Depends on Task and Develops Through Middle School

• Equivalence: within each grade, performance is equally high with diagrams and lower without them

• Addition: 5th graders have different levels of difficulty with each scaffold.

• With 6th and 7th graders, differences among the scaffolds with pictures decrease, but do not disappear

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Diagram interpretation skills are sensitive to context, even when the domains are closely related and the diagrams being used are similar.

Equivalence Addition

23

This Work Was Supported By:

A Graduate Training Grant awarded to Carnegie Mellon University by the Department of Education (R305B090023)

The Pittsburgh Science of Learning Center through NSF award SBE-0836012

Thank you! How useful are fraction bars for

understanding fraction equivalence and addition?

Eliane Stampfer Wiese, Ken KoedingerCarnegie Mellon University

stampfer@cs.cmu.edu

25

+14

Hint:25

1320

The top shaded parts are the same

as the bottom

True or false:

+14

25

1320=

The amounts are the same, but is

that what the question is asking?

Confusion Because…Lack of Domain Knowledge?

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Is it a Perceptual Problem?

• Calculated disparities for each item•ANOVAs showed no main effect of disparity, and no interaction between disparity and scaffold type