Research News

Abstract math concepts may be grounded in basic non-symbolic processing abilities

January 06, 2016

UW-Madison’s Percival Matthews and Edward Hubbard have co-authored a research article that sheds new light on the question of whether there are basic abilities that lay the foundation for people to better understand math concepts such as fractions and algebra.

The article, “Individual Differences in Nonsymbolic Ratio Processing Predict Symbolic Math Performance,” was recently made available online by the journal Psychological Science.

Matthews, an assistant professor with the School of Education’s No. 1-ranked Department of Educational Psychology, notes that there already is plenty of research indicating fractions knowledge is both difficult for many people to grasp and also vitally important for those hoping to gain an advanced understanding of mathematics.

Ratio non-symbolic and symbolic
These are​ examples of (a) non-symbolic and (b) symbolic
ratio comparisons. In testing, participants were asked to
indicate whether the ratio expressed on the left or the right
is greater.
The study conducted by Matthews and Hubbard, which included 181 undergraduates at a large American university, points to human beings possessing a certain innate, non-symbolic ratio processing ability that varies from person to person. And it’s differences in this ratio-processing system (RPS), their research indicates, that has meaningful effects on one’s development of advanced mathematical thinking.

The authors report that “individual differences in RPS acuity predict performance on four measures of mathematical competence, including a university entrance exam in algebra. We suggest that the nonsymbolic RPS may support symbolic fraction understanding much like the (approximate number system) ANS supports whole-number concepts. Thus, even abstract mathematical concepts, such as fractions, may be grounded not only in higher-order logic and language, but also in basic nonsymbolic processing abilities.”

Ratio comparisons
 Here are sample stimuli from the non-symbolic comparison
tasks. Participants were asked to determine whether the
ratio of white dots or white line lengths, to black dots or
black line lengths was greater on the left or right.
“This basic perceptual ability seems to have consequences even for complex mathematical thinking like algebraic reasoning, which is quite counterintuitive,” says Matthews, who leads the Math Education Learning and Development Lab at UW-Madison. “At the end of the day, we want to encourage psychologists, math education researchers and educational practitioners to consider how actively trying to leverage these abilities might help improve fractions learning.”

Hubbard also is an assistant professor and is the director of the UW Educational Neuroscience Lab in the Department of Educational Psychology and the Waisman Center. Hubbard’s lab explores questions at the intersection of education and neuroscience.

The other co-author of the research paper is Mark Lewis, an assistant researcher with the Wisconsin Center for Education Research (WCER).

In their conclusion, the authors write: “Our findings indicate that individual differences in the RPS — a basic nonsymbolic processing capacity that until recently has received little study — predict outcomes on three different measures of symbolic fraction ability and performance on algebra entrance exams at a major selective university. This stands in stark contrast to the assertion that fraction concepts are unsupported by primitive architectures. Indeed, the RPS may prove to be an underappreciated neurocognitive start-up tool that serves as a primitive ground for advanced mathematical concepts.”

Matthews says the team plans to conduct similar RPS studies with children and to dig deeper into the nature of the ratio processing system.

Says Matthews: “How does performance vary over a wide range of tasks? How might we formally model these abilities? What brain regions are implicated in processing nonsymbolic ratio magnitudes? How does, and can, this ratio system contribute to learning fractions? Can individual differences in these abilities at early ages predict outcomes across large time spans?”

“At this point, we have far more questions than answers, and we plan to get to work coming up with more answers,” he says.

To learn more about this topic, make sure and check out the research article on the Psychological Science website.