What mathematics can make of our intuition

The CogSci 2014 Proceedings have been posted and there are a number of links to interesting papers.

Here are some math-related investigations:

A neural network model of learning mathematical equivalence

The Psychophysics of Algebra Expertise:  Mathematics Perceptual Learning Interventions Produce Durable Encoding Changes

Two Plus Three is Five:  Discovering Efficient Addition Strategies without Metacognition

Modeling probability knowledge and choice in decisions from experience

Simplicity and Goodness-of-fit in Explanation:  The Case of Intuitive Curve-Fitting

Cutting In Line:  Discontinuities in the Use of Large Numbers by Adults

Applying Math onto Mechanism:  Investigating the Relationship Between Mechanistic and Mathematical Understanding

Pierced by the number line: Integers are associated with back-to-front sagittal space

Equations Are Effects: Using Causal Contrasts to Support Algebra Learning

One of the presentations I attended is represented by the paper:
Are Fractions Natural Numbers Too? This study challenges the argument that human cortical structures are ill-suited for processing fractions, a view which has been used to justify the well-documented difficulty that many children have with learning fractions.

Such accounts argue that the cognitive system for processing number, the approximate number system (ANS), is fundamentally designed to deal with discrete numerosities that map onto whole number values. Therefore, according to innate constraints theorists, fractions and rational number concepts are difficult because they lack an intuitive basis and must instead be built from systems originally developed to support whole number understanding.

…Emerging data from developmental psychology and neuroscience suggest that an intuitive (perhaps native) perceptually based cognitive system for grounding fraction knowledge may indeed exist. This cognitive system seems to represent and process amodal magnitudes of non- symbolic ratios (such as the relative length of two lines).

This particular study is fairly well-focused, however.  Researchers aim to demonstrate a link between our sensitivity to non-symbolic ratios and the acquired understanding of magnitudes represented by symbolic fractions.   Given this focus, the study looked at individual responses to “cross-format comparisons of various fractional values (i.e. ratios composed of dots or circles vs. traditional fraction symbols).  For example, a given symbolic ratio was given with a numerical numerator and a numerical denominator.  The dot stimulus would show an array of dots in the numerator, of a certain quantity, and another array, of a different quantity, in the demoninator.  The circle stimulus showed a blackened disc of a certain area in the numerator and another, of a different area, in the denominator.  With a reasonable amount of care taken in their analysis, the authors concluded that they had found evidence of “flexible and accurate processing of non-symbolic fractional magnitudes in ways similar to ANS processing of discrete numerosities.

Considered in concert with other recent findings, our evidence suggests that humans may have an intuitive “sense” of ratio magnitudes that may be as compatible with our cortical machinery as is the “sense” of natural number. Just as the ANS allows us to perceive the magnitudes of discrete numerosities, this ratio sense provides humans with an intuitive feel for non-integer magnitudes.

An important consequence of this kind of evidence is their suggestion that the widespread difficulty with fractions may be the result teaching fractions incorrectly –  with partitioning or sharing ideas that use counting skills and whole number magnitudes instead of encouraging the use of what may be our intuitive ratio processing system.  This point was driven home for me when I looked at the circle representations of ratios that were used in the study.  I found them very effective, very readable.

This view is certainly consistent with the proposal that a mental representation of continuous magnitudes predates discrete counting numbers  (as with Gallistel, et al).   But I also think that this initiative points to something likely to be important in cognitive science as well as math education.   My own hunch is that an intuitive sense of ratio is likely grounded in continuous magnitudes, like length and area or perhaps even in tactile sensations of measure, like in cooking, as was suggested by one of the paper’s authors.  And I think it plays some role in the long debate over the relationship between discrete and continuous numbers that can be seen in the history of mathematics.  One could argue that the ancient Greek’s rigorous distinction between number and magnitude contributed to their remarkable development of geometric ideas.  With the number concept isolated away from the geometric idea of magnitude, perhaps their geometric efforts were liberated, allowing a focused elaboration on that ‘intuitive sense of ratio,’ extending it, permitting manifold and deep results.  Understanding this cultural event in the light of cognitive processes might inform our ideas about how mathematics emerges as well as how to communicate that development in mathematics education.

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