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Eliane Stampfer Wiese, Kenneth R. Koedinger Human-Computer Interaction Institute

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.

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Eliane Stampfer Wiese, Kenneth R. Koedinger Human-Computer Interaction Institute

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  1. 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. Koedinger Human-Computer Interaction Institute Carnegie Mellon University stampfer@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? $25 Diagrams can … ? aid sense-making and become a bridge to abstract thinking NCTM 2013 Leinwand & Ginsburg 2007 harm performance if students can’t interpret them correctly Rittle-Johnson & Koedinger 2001 Booth & Koedinger 2011

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

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

  5. What Makes This So Hard? • Areas represent amounts • Bars represent fractions • Mapping the relationship between the pictures to the relationship between the fractions

  6. 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)

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

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

  9. Hint: True or false: + = FALSE

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

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

  12. Repeated Measures ANOVAs by Grade 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) Equivalence Addition Pictures Pictures & Numbers Half Pictures & Numbers Numbers 5th 6th 7th 5th 6th 7th

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

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

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

  16. Limitations: Population & Design 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

  17. 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 difficultywith each scaffold. With 6th and 7th graders, differences among the scaffolds with pictures decrease, but do not disappear

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

  19. 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

  20. 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

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

  22. 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

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