Leveraging the Rational Brain to Promote Fractions Competence

Edward M. HubbardPercival G. Matthews

Martina A. Rau

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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

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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”

Fractions as Gatekeeper

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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) 1 b) 2 c) 19 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…

A Continuing Problem:Widespread Difficulties with Fractions

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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)

Why Are Fractions So Difficult?The Dominant View: Our Brains Aren’t Built for Them

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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

OUR BIG GOAL: Let’s harness this nonsymbolic ability to teach about symbolic fractions

A Perceptual Route to FractionsFraming Our Research

Vallentin & Nieder, 2008

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Children do much better with the figure on the right• Why?• Discrete representations encourage counting, whereas

continuous ones do not

Why Use Perception?When Concepts Collide

(Jeong, Levine, & Huttenlocher, 2007)

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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!

The Value of Perceptual Expertise

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Adults can do it! -

4-yr-old kids can do it!

Evidence for the Building BlocksPerceiving Nonsymbolic Fractions

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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

Our ModelFrom Nonsymbolic Perception to Symbolic Math

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A Head for Fractions

Adaptation ExperimentsHabituated

Ratio“Close”Deviant

“Far”Deviant

Passively viewing dot ratios [Jacob & Nieder, 2009b]

Passively viewing line ratios [Jacob & Nieder, 2009b]

Passively viewing symbolic fractions and fraction words [Jacob & Nieder, 2009a]

1/6 One-fourth One-half

Comparison Experiments “Close” “Far”

Symbolic fraction comparison [Ischebeck et al., 2009]

2/5 3/8 3/7 6/8

If fractions really do fit our brain, we should be able to identify where they are processed in the brain.

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“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

Neural Coding of Fractions

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Our fMRI-Adaptation Paradigm

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Neural Adaptation

If the same neural circuitry represents symbolic fractions and nonsymbolic rational magnitude, we expect distance-dependent recovery across symbolic formats

Adapting Stimuli

…

…

Symbolic Deviants

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Near Far

Distance Effect

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Preliminary ResultsBrain Areas Showing a Distance Effect

Digits Only (26, -38, 44)Lines and Digits (29, -37, 39)

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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

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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

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Fractions Tutor ExamplesInteractive problem solving

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Fractions Tutor ExamplesConnecting symbolic and unit-partitive representations

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Fractions Tutor ExamplesPerceptual fluency with unit-partitive representations

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Fractions Tutor Effectiveness

4 classroom experiments with 3,000 4th-6th graders• > 50 teachers• 16 schools

10 hours of supplemental instructional materials

Free & online: https://fractions.cs.cmu.edu/

Con

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ual k

now

ledg

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pre post delayed

** d = .40

** d = .60

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Planned Magnitude Learning Module

Becoming fluent with continuous representations

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Conclusion: A Research Question

Neural architectures

Instructional activities

Non-symbolicabilities

Continuousrepresentations

Cognitive Tutor

Fractionslearning

How should we integrate activities with continuous representations into the Fractions Tutor to maximally enhance fractions learning?

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Thanks! Behavioral and

neuroimaging:• NSF REAL 1420211• NIH 1R03HD081087-

01 Fractions Tutor:

• NSF REESE-21851-1-1121307

Wisconsin Alumni Research Foundation

Mark Rose Lewis Elizabeth Toomarian John Binzak Ron Hopkins Ryan Ziols Joe Anistranski