1. How useful are fraction bars for understanding fraction equivalence and addition? A difficulty factors assessment with 5 th, 6 th, and 7 th graders Eliane Stampfer Wiese, Kenneth R. Koedinger Human-Computer Interaction Institute Carnegie Mellon University stampfer@cs.cmu.edu

2. 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 & Koedinger 2001 Booth & Koedinger 2011

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7. 7 I’m Done!! 1-2 times per student per problem Stampfer & Koedinger 2013

8. How does fraction bar utility develop between 5 th and 7 th grade? 8

9. 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 5 th – 7 th graders at local public school (~150 students in each grade) 10

11. is bigger than is smaller than is equivalent to 11 a) b) c)

12. is bigger than is smaller than is equivalent to 12 a) b) c) 7 21 1 3 1 3 1 3 1 3 7 7 7 Pictures Only 1) Areas represent amounts Pictures & Numbers 2) Bars represent fractions Half Pictures & Numbers 3) Mapping relationships Numbers Only Control 82%88%90%81%86%93% 5 th 6 th 7 th 82%86%90%50%67% 1 0 57%

13. 13 Hint: True or false: += FALSE

14. Hint: True or false: 14 Pictures Only 1) Areas represent amounts 3 13 2 11 1 2 + 2 1 2 3 13 Pictures & Numbers 2) Bars represent fractions += Half Pictures & Numbers 3) Mapping relationships Numbers Only Control 5 th 6 th 7 th 82%87%79%75%84%64%46%70%77%52%64% 1 0 21%

15. Percent Correct by Grade & Scaffold 15 Students did 2 problems of each type ~150 students in each grade 5 th 6 th 7 th Pictures Pictures & Numbers Half Pictures & Numbers Numbers 5 th 6 th 7 th EquivalenceAddition

16. Repeated Measures ANOVAs by Grade 16 Pictures Pictures & Numbers Half Pictures & Numbers Numbers 5 th 6 th 7 th EquivalenceAddition Class Tracking Level x Scaffold x Task (repeated measures on scaffold and task) Main Effects: Scaffold and Task (p<.01) Interactions: 5 th grade Scaffold * Task (p<.01)

17. Repeated Measures for Equivalence 17 Grade x Scaffold, repeated measures on scaffold Pictures Pictures & Numbers Half Pictures & Numbers Numbers 5 th 6 th 7 th Main Effects: Scaffold and Grade (both p<.01) No Scaffold * Grade Interaction Difficulty depends presence of pictures Students improve from 5 th to 7 th grade

18. Repeated Measures for Addition 18 Grade x Scaffold, repeated measures on scaffold Pictures Pictures & Numbers Half Pictures & Numbers Numbers 5 th 6 th 7 th Main Effects: Scaffold and Grade (both p<.01) Scaffold * Grade Interaction (p<.01) Difficulty depends on scaffold Relative difficulty of scaffolds depends on grade Performance improves with grade

19. Separate ANOVAs on Scaffold by Grade 19 with Post-Hoc Tukey tests Pictures Pictures & Numbers Half Pictures & Numbers Numbers 5 th 6 th 7 th 5 th Grade: All differences (p<.01) 6 th and 7 th Grade: Numbers (p<.01) For 5 th graders, each scaffold type had a unique difficulty level For 6 th and 7 th graders, Numbers was more difficult than the others

20. 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 20

21. 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: 5 th graders have different levels of difficulty with each scaffold. With 6 th and 7 th graders, differences among the scaffolds with pictures decrease, but do not disappear 21

22. Diagram interpretation skills are sensitive to context, even when the domains are closely related and the diagrams being used are similar. 22 EquivalenceAddition

23. This Work Was Supported By: 23 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

24. Thank you! How useful are fraction bars for understanding fraction equivalence and addition? Eliane Stampfer Wiese, Ken Koedinger Carnegie Mellon University stampfer@cs.cmu.edu

25. + 1 4 Hint: 2 5 13 20 25 The top shaded parts are the same as the bottom True or false: + 1 4 2 5 13 20 = The amounts are the same, but is that what the question is asking? Confusion Because… Lack of Domain Knowledge?

26. Is it a Perceptual Problem? 26 Calculated disparities for each item ANOVAs showed no main effect of disparity, and no interaction between disparity and scaffold type