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.

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.

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.

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.

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.