Aryn Pyke

Hidden Stages of Cognition Revealed in Patterns of Brain Activation.

To advance cognitive theory, researchers must be able to parse the performance of a task into its significant mental stages. In this article, we describe a new method that uses functional MRI brain activation to identify when participants are engaged in different cognitive stages on individual trials. The method combines multivoxel pattern analysis to identify cognitive stages and hidden semi-Markov models to identify their durations. This method, applied to a problem-solving task, identified four distinct stages: encoding, planning, solving, and responding. We examined whether these stages corresponded to their ascribed functions by testing whether they are affected by appropriate factors. Planning-stage duration increased as the method for solving the problem became less obvious, whereas solving-stage duration increased as the number of calculations to produce the answer increased. Responding-stage duration increased with the difficulty of the motor actions required to produce the answer.

Visuospatial referents facilitate the learning and transfer of mathematical operations: Extending the role of the angular gyrus

Different external representations for learning and solving mathematical operations may affect learning and transfer. To explore the effects of learning representations, learners were each introduced to two new operations (b↑n and b↓n) via either formulas or graphical representations. Both groups became adept at solving regular (trained) problems. During transfer, no external formulas or graphs were present; however, graph learners’ knowledge could allow them to mentally associate problem expressions with visuospatial referents. The angular gyrus (AG) has recently been hypothesized to map problems to mental referents (e.g., symbolic answers; Grabner, Ansari, Koschutnig, Reishofer, & Ebner Human Brain Mapping, 34, 1013–1024, 2013), and we sought to test this hypothesis for visuospatial referents. To determine whether the AG and other math (horizontal intraparietal sulcus) and visuospatial (fusiform and posterior superior parietal lobule [PSPL]) regions were implicated in processing visuospatial mental referents, we included two types of transfer problems, computational and relational, which differed in referential load (one graph vs. two). During solving, the activations in AG, PSPL, and fusiform reflected the referential load manipulation among graph but not formula learners. Furthermore, the AG was more active among graph learners overall, which is consistent with its hypothesized referential role. Behavioral performance was comparable across the groups on computational transfer problems, which could be solved in a way that incorporated learners’ respective procedures for regular problems. However, graph learners were more successful on relational transfer problems, which assessed their understanding of the relations between pairs of similar problems within and across operations. On such problems, their behavioral performance correlated with activation in the AG, fusiform, and a relational processing region (BA 10).

There Is Nothing So Practical as a Good Theory

Abstract Children from low-income families begin school with less mathematical knowledge than peers from middle-income backgrounds. This discrepancy has long-term consequences: Children who start behind usually stay behind. Effective interventions have therefore been sought to improve the mathematical knowledge of preschoolers from impoverished backgrounds. Some curriculum-based interventions have met with impressive success; two disadvantages of such interventions, however, are that they are quite costly in terms of the time and resources they require and their multifaceted lessons make it impossible to determine why they work. In this chapter, we describe a theoretical analysis that motivated the development of a simple, brief, and inexpensive intervention that involved playing a linear number board game. Roughly an hour of playing this game produced improvements in numerical magnitude comparison, number line estimation, counting, numeral identification, and ability to learn novel arithmetic problems by preschoolers from low-income backgrounds. The gains were greater than those produced by playing a parallel nonnumerical game or engaging in other numerical activities. Detailed analyses of learning patterns indicated that this theoretically motivated, game-based intervention exercises most of its effects by enhancing and refining childrens representations of numerical magnitudes.

A Computational Model of Fraction Arithmetic.

Many children fail to master fraction arithmetic even after years of instruction, a failure that hinders their learning of more advanced mathematics as well as their occupational success. To test hypotheses about why children have so many difficulties in this area, we created a computational model of fraction arithmetic learning and presented it with the problems from a widely used textbook series. The simulation generated many phenomena of children’s fraction arithmetic performance through a small number of common learning mechanisms operating on a biased input set. The biases were not unique to this textbook series—they were present in 2 other textbook series as well—nor were the phenomena unique to a particular sample of children—they were present in another sample as well. Among other phenomena, the model predicted the high difficulty of fraction division, variable strategy use by individual children and on individual problems, relative frequencies of different types of strategy errors on different types of problems, and variable effects of denominator equality on the four arithmetic operations. The model also generated nonintuitive predictions regarding the relative difficulties of several types of problems and the potential effectiveness of a novel instructional approach. Perhaps the most general lesson of the findings is that the statistical distribution of problems that learners encounter can influence mathematics learning in powerful and nonintuitive ways.

When math operations have visuospatial meanings versus purely symbolic definitions: Which solving stages and brain regions are affected? ☆

Abstract How does processing differ during purely symbolic problem solving versus when mathematical operations can be mentally associated with meaningful (here, visuospatial) referents? Learners were trained on novel math operations (↓, ↑), that were defined strictly symbolically or in terms of a visuospatial interpretation (operands mapped to dimensions of shaded areas, answer = total area). During testing (scanner session), no visuospatial representations were displayed. However, we expected visuospatially‐trained learners to form mental visuospatial representations for problems, and exhibit distinct activations. Since some solution intervals were long (˜10 s) and visuospatial representations might only be instantiated in some stages during solving, group differences were difficult to detect when treating the solving interval as a whole. However, an HSMM‐MVPA process (Anderson and Fincham, 2014a) to parse fMRI data identified four distinct problem‐solving stages in each group, dubbed: 1) encode; 2) plan; 3) compute; and 4) respond. We assessed stage‐specific differences across groups. During encoding, several regions implicated in general semantic processing and/or mental imagery were more active in visuospatially‐trained learners, including: bilateral supramarginal, precuneus, cuneus, parahippocampus, and left middle temporal regions. Four of these regions again emerged in the computation stage: precuneus, right supramarginal/angular, left supramarginal/inferior parietal, and left parahippocampal gyrus. Thus, mental visuospatial representations may not just inform initial problem interpretation (followed by symbolic computation), but may scaffold on‐going computation. In the second stage, higher activations were found among symbolically‐trained solvers in frontal regions (R. medial and inferior and L. superior) and the right angular and middle temporal gyrus. Activations in contrasting regions may shed light on solvers’ degree of use of symbolic versus mental visuospatial strategies, even in absence of behavioral differences. HighlightsLearners solved identical problems via symbolic or visuospatial mental strategies.HSMM‐MVPA analysis applied to fMRI data revealed 4 distinct solving stages.Group differences were not evident overall but rather were stage‐specific.Stages were encode, plan, compute, respond – with group differences in the first 3.The visuospatial group had more active imagery/semantic areas to encode & compute.

The influence of object size on judgments of lateral separation

The size of a previously experienced object has a significant effect on depth judgments in visually sparse environments. The present research explored whether familiar object size also significantly influences judgments of lateral separation. Experiment 1 required participants to detect changes in lateral separation using a one-shot change detection paradigm with a closer/same/farther forced-choice response. Participants accurately detected changes in lateral separation, although concurrent changes in object size significantly reduced the accuracy of their judgments such that increases (or decreases) in object size caused a relative underestimation (or overestimation) of separation. Experiment 2 replicated these results using a distance reproduction methodology. These results indicate that familiar object size exhibits a significant effect on judgments of lateral separation.