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. (PsycINFO Database Record

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

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 (~10s) 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.

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.

Developmental and individual differences in understanding of fractions.

We examined developmental and individual differences in 6th and 8th graders' fraction arithmetic and overall mathematics achievement and related them to differences in understanding of fraction magnitudes, whole number division, executive functioning, and metacognitive judgments within a cross-sectional design. Results indicated that the difference between low achieving and higher achieving children's fraction arithmetic knowledge, already substantial in 6th grade, was much greater in 8th grade. The fraction arithmetic knowledge of low achieving children was similar in the 2 grades, whereas higher achieving children showed much greater knowledge in 8th than 6th grade, despite both groups having been in the same classrooms, using the same textbooks, and having the same teachers and classmates. Individual differences in both fraction arithmetic and mathematics achievement test scores were predicted by differences in fraction magnitude knowledge and whole number division, even after the contributions of reading achievement and executive functioning were statistically controlled. Instructional implications of the findings are discussed.