College of Arts & Sciences
Department of Psychology
Department of Psychology
Publications

Cognitive Explanations of the Problem Size Effect in Mental Arithmetic in Neuropsychologia (Ian Lyons)

July 12, 2019 – Professor Ian Lyons, in collaboration with researchers at Ghent University and KU Leuven (Belgium), has published a paper in Neuropsychologia.

Description of study findings

Not all researchers interested in human behavior remain convinced that modern neuroimaging techniques have much to contribute to distinguishing between competing cognitive models for explaining human behavior. Here, we took up this challenge in an attempt to distinguish between two competing cognitive accounts of the problem size effect (PSE). The PSE occurs when people solve arithmetic problems, and it is one of the most robust finding in investigations of mathematical cognition. Specifically, the PSE refers to the fact people tend to solve numerically large problems (e.g., 7×8) more slowly and erroneously than small problems (e.g., 3×4). Cognitive explanations for the PSE can be categorized into two separate categories: representation-based and process-based views. According to the representation-based account, larger problems are represented less precisely, which is why they are harder to compute, leading to poorer performance relative to small problems. From the process-based point of view, larger problems are solved using a wider range of relatively inefficient strategies, which is why performance tends to suffer on large relative to small problems.

Traditional cognitive approaches (i.e., behavioral measures such as reaction-times and error-rates) and standard fMRI approaches (i.e., univariate contrasts) have struggled to distinguish between these two accounts of the PSE. By contrast, a representational similarity analysis (RSA) approach with fMRI data provides competing hypotheses that can distinguish between accounts without recourse to reverse inference. Specifically, the representation-based view predicts that reduced precision of larger problems is due to overlapping neural representations, and smaller problems are more precise due to more distinct neural representations. The neural patterns observed for larger problems should therefore be more similar to one another (due to overlapping representations), and the neural patterns observed for smaller problems should be less similar to one another (due to more distinctive representations). Processing-based approaches predict the opposite. According to this view, solving larger problems uses a wide range of strategies, so the neural patterns seen for large problems should be relatively distinct from one another. By contrast, solving smaller problems relies on a narrow range of highly efficient strategies, implying that neural patterns for smaller problems should be quite similar to one another.

In the case of solving multiplication problems, RSA results provided clear evidence in favor of the representation-based over the process-based account of the PSE. A clear majority (>95%) of brain regions that showed a traditional PSE also showed greater similarity for large relative to small problems (none showed the opposite, which would have favored the process-based view). For solving addition problems, results also favored the representation-based view, albeit less overwhelmingly: 28% of regions favored the representation-based view, and none favored the process-based view. Finally, post-hoc analysis of the multiplication data distinguished still further between competing representation-based theoretical accounts: Individual multiplication problems appear to be stored as individual memory traces sensitive to input frequency, independent of the actual magnitudes being represented.

Together, these data expand our understanding of the neural basis of mental arithmetic. More broadly, these results provide an example of how human neuroimaging evidence can directly inform cognitive-level explanations of a common behavioral phenomenon, the problem size effect.