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Do Humans Have A Basic Capacity To Understand Fractions?

What basic cognitive tools lay the foundation for our number facility?

We know there are individual differences in math ability, even at a young age, but what basic cognitive tools lay the foundation for whether we are good with numbers? And specifically, what basic capacities influence our ability to do fractions? Fractions are thought by many in math education to be a central bottleneck that many students have trouble with, so much research has been directed at understanding the basic capacities essential to fraction understanding.

Now a series of research studies by Percival Matthews at the University of Wisconsin Madison and collaborators may be shedding some light on why some people may be better than others at doing fractions. Consider the following two panels:

Percival Matthews, used with permission
Source: Percival Matthews, used with permission

Most of us think of fractions entirely in terms of panel b, what is known as “symbolic processing.” But Matthews and Dana Chesney explored the importance of “nonsymbolic processing” of fractions, what you see in panel a, in a study conducted last year published in Cognitive Psychology. For example, people were asked to compare ratios of circle areas (left) with ratios of dot arrays (right) to decide which was greater. The researchers found that participants completed the nonsymbolic tasks of panel a faster than the numerical comparisons (5/6 to 2/3) of panel b. This was true even though the nonsymbolic ratios were compared across two different formats. They concluded that abstract “nonsymbolic-ratio comparisons are made without converting ratios to symbolic form.” Essentially, nonsymbolic processing may be a more basic fundamental capacity of humans.

In a more recent study published in Psychological Science, Matthews, Mark Lewis, and Edward Hubbard took this stream of research a critical step further, examining whether a variety of nonsymbolic ratio comparison tasks was actually linked to real world fraction knowledge and algebra performance in a sample of university undergraduates. The idea was to examine whether humans have a basic nonsymbolic-ratio processing system (RPS). In each of several ratio comparison tasks (panels a through d below) participants were asked to assess the ratio of white dots to black dots or white lines to black lines and to determine whether the left or right stimuli within each panel was greater. The control tasks (panels e and f) simply asked participants to compare the numbers of black dots and the lengths of individual lines.

Percival Matthews, used with permission
Source: Percival Matthews, used with permission

What the researchers found is that RPS predicts actual algebra performance and fraction knowledge, suggesting a link between the RPS and symbolic math abilities, and perhaps even more advanced thinking like algebra.

Matthews noted in an interview that this RPS appears to be “a primitive capacity that can help ground fraction concepts and that we currently don’t attempt to leverage it.” This new basic ratio processing system might be something we all have, just to different degrees, and might be integral in our capacity to learn fractions and hence higher level mathematics. “We miss out on the contributions it could make. Individual differences might moderate the effect of attempts to leverage it, but it might be that even folks on the low end could benefit quite a bit.” Of course, he also cautions that “we’ve got a long way to go before we know how practically useful these findings are.”

Matthews also argues that we might even be missing a number of talented students who have incredible ratio-processing abilities but lack identification and hence development of such talent. Given that being able to perform fractions appears critical not only to math education but also performance in many STEM areas, perhaps the discovery of this new system might eventually lead to much better understanding of how we can develop interventions to help all students to improve their math abilities.

© 2016 by Jonathan Wai. You can follow me on Twitter or Facebook.

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