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

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

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

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) 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… Matthews

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

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

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)

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

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

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

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

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

Our fMRI-Adaptation Paradigm

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

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

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

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

Fractions Tutor Examples Interactive problem solving

Fractions Tutor Examples Connecting symbolic and unit-partitive representations

Fractions Tutor Examples Perceptual fluency with unit-partitive representations

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/ Conceptual knowledge pre post delayed ** d = .40 ** d = .60

Planned Magnitude Learning Module Becoming fluent with continuous representations

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?

Thanks! Behavioral and neuroimaging: Mark Rose Lewis 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