1. Leveraging the Rational Brain to Promote Fractions Competence
Edward M. Hubbard
Percival G. Matthews
Martina A. Rau

2. Outline
We will discuss how we combine three perspectives to create practicable, easily disseminable instruction for fractions:
A Feel for Fractions
A Head for Fractions
A Tutor for Fractions

3. Fractions as Gatekeeper
Fractions knowledge seems to play a gatekeeper role in supporting knowledge of algebra and more advanced forms of math.
5th grade fraction knowledge predicts algebra and overall math achievement in high school (Bailey et al., 2012; Siegler et al., 2012)
National Math Advisory Panel (2008) declared fractions knowledge to be “the most important foundational skill not presently developed in the school aged population” Ed

4. A Continuing Problem: Widespread Difficulties with Fractions
Both children and adults struggle to understand fractions
Typical 6th graders often claim that 1/8 is greater than 1/6
When a national sample of 17-yr-olds was asked whether
12/13 +7/8 ≈ a) b) c) d)21
More chose 19 and 21 than 2 (Carpenter, Corbitt, Kepner, Lindquist, & Reys, 1981)
Such estimates are off by more than a factor of 10!
UW Madison students err on these problems, too…
Matthews

5. Why Are Fractions So Difficult
Why Are Fractions So Difficult? The Dominant View: Our Brains Aren’t Built for Them
Some argue that innate constraints make fractions difficult:
The human system for processing number, the Approximate Number System (ANS), is designed to deal with discrete countable sets
Whole number concepts are supported by innate perception
Fractions are difficult because they lack such a basis…they must be built from whole number concepts (e.g., Feigenson, Dehaene & Spelke, 2004) Ed

6. A Perceptual Route to Fractions Framing Our Research
What if fractions are pretty natural too?
Emerging findings from developmental psychology and neuroscience suggest that innate perceptual abilities for fraction understanding do exist!
Duffy, Huttenlocher & Levine, 2005 ED
Vallentin & Nieder, 2008
OUR BIG GOAL: Let’s harness this nonsymbolic ability to teach about symbolic fractions

7. Why Use Perception? When Concepts Collide
Children do much better with the figure on the right
Why?
Discrete representations encourage counting, whereas continuous ones do not
PGM
(Jeong, Levine, & Huttenlocher, 2007)

8. The Value of Perceptual Expertise
A lot of expertise is fundamentally about repeated exposure and perceptual practice. Think about a few key examples:
Faces
X-Rays
Chess
Whole Numbers
We think we can similarly use perceptual exposure to teach fractions!
PGM

9. Evidence for the Building Blocks Perceiving Nonsymbolic Fractions
Adults can do it! -
4-yr-old kids can do it!
PGM

10. Our Model From Nonsymbolic Perception to Symbolic Math
This is all based on the ability to look at this figure and to tell that it’s about 2/3
Something that monkeys can do!
We want to forge nonsymbolic-to-symbolic links ED

11. A Head for Fractions
If fractions really do fit our brain, we should be able to identify where they are processed in the brain.
Adaptation Experiments
Habituated
Ratio
“Close”
Deviant
“Far”
Passively viewing
dot ratios [Jacob & Nieder, 2009b]
line ratios [Jacob & Nieder, 2009b]
Passively viewing symbolic fractions and fraction words
[Jacob & Nieder, 2009a]
1/6
One-fourth
One-half
Comparison Experiments
Symbolic fraction comparison
[Ischebeck et al., 2009]
2/5 3/8
3/7 6/8 ED

12. Neural Coding of Fractions
“A coding scheme for proportions has emerged that is remarkably reminiscent of the representation of absolute number. These novel findings suggest a sense for ratios that grants the brain automatic access to proportions independently of language and the format of presentation.” Jacob, Vallentin & Nieder, 2012

13. Our fMRI-Adaptation Paradigm

14. Neural Adaptation
Adapting Stimuli …
Symbolic Deviants 7 9 1 3
Near
Far
Distance
Effect
If the same neural circuitry represents symbolic fractions and nonsymbolic rational magnitude, we expect distance-dependent recovery across symbolic formats

15. Preliminary Results Brain Areas Showing a Distance Effect
Digits Only (26, -38, 44)
Lines and Digits (29, -37, 39)

16. Advantages of fMRI-A Does not require overt behavioral responses
Directly taps into neural representations
Not affected by cognitive strategy or skill level
Can be used both with children and adults
Developmental paradigm already tested with two children
Index of neural links between symbolic and nonsymbolic fractions
Can use to explore neural basis of individual differences and consequences of training

17. What Is the Fractions Tutor?
Intelligent tutoring system
Learning through problem solving
Individualized support
Highly effective [Koedinger & Corbett, 2006, Corbett et al., 2001]
Used in > 2,000 U.S. Schools
> 10h of supplemental materials
Conceptual learning through multiple graphical representations MR

18. Fractions Tutor Examples
Interactive problem solving

19. Fractions Tutor Examples
Connecting symbolic and unit-partitive representations

20. Fractions Tutor Examples
Perceptual fluency with unit-partitive representations

21. Fractions Tutor Effectiveness
4 classroom experiments with 3,000 4th-6th graders
> 50 teachers
16 schools
10 hours of supplemental instructional materials
Free & online:
Conceptual knowledge
pre
post
delayed
** d = .40
** d = .60

22. Planned Magnitude Learning Module
Becoming fluent with continuous representations

23. Conclusion: A Research Question
Neural architectures
Instructional activities
Cognitive Tutor
Non-symbolic abilities
Continuous
representations
Fractions
learning
How should we integrate activities with continuous representations into the Fractions Tutor to maximally enhance fractions learning?

24. Thanks! Behavioral and neuroimaging: Mark Rose Lewis
NSF REAL
NIH 1R03HD
Fractions Tutor:
NSF REESE
Wisconsin Alumni Research Foundation
Mark Rose Lewis
Elizabeth Toomarian
John Binzak
Ron Hopkins
Ryan Ziols
Joe Anistranski