Shawn Betts

Cognitive and metacognitive activity in mathematical problem solving: prefrontal and parietal patterns

Students were taught an algorithm for solving a new class of mathematical problems. Occasionally in the sequence of problems, they encountered exception problems that required that they extend the algorithm. Regular and exception problems were associated with different patterns of brain activation. Some regions showed a Cognitive pattern of being active only until the problem was solved and no difference between regular or exception problems. Other regions showed a Metacognitive pattern of greater activity for exception problems and activity that extended into the post-solution period, particularly when an error was made. The Cognitive regions included some of parietal and prefrontal regions associated with the triple-code theory of (Dehaene, S., Piazza, M., Pinel, P., & Cohen, L. (2003). Three parietal circuits for number processing. Cognitive Neuropsychology, 20, 487–506) and associated with algebra equation solving in the ACT-R theory (Anderson, J. R. (2005). Human symbol manipulation within an 911 integrated cognitive architecture. Cognitive science, 29, 313–342. Metacognitive regions included the superior prefrontal gyrus, the angular gyrus of the triple-code theory, and frontopolar regions.

Neural imaging to track mental states while using an intelligent tutoring system

Hemodynamic measures of brain activity can be used to interpret a students mental state when they are interacting with an intelligent tutoring system. Functional magnetic resonance imaging (fMRI) data were collected while students worked with a tutoring system that taught an algebra isomorph. A cognitive model predicted the distribution of solution times from measures of problem complexity. Separately, a linear discriminant analysis used fMRI data to predict whether or not students were engaged in problem solving. A hidden Markov algorithm merged these two sources of information to predict the mental states of students during problem-solving episodes. The algorithm was trained on data from 1 day of interaction and tested with data from a later day. In terms of predicting what state a student was in during a 2-s period, the algorithm achieved 87% accuracy on the training data and 83% accuracy on the test data. The results illustrate the importance of integrating the bottom-up information from imaging data with the top-down information from a cognitive model.

Brain networks supporting execution of mathematical skills versus acquisition of new mathematical competence.

This fMRI study examines how students extend their mathematical competence. Students solved a set of algebra-like problems. These problems included Regular Problems that have a known solution technique and Exception Problems that but did not have a known technique. Two distinct networks of activity were uncovered. There was a Cognitive Network that was mainly active during the solution of problems and showed little difference between Regular Problems and Exception Problems. There was also a Metacognitive Network that was more engaged during a reflection period after the solution and was much more engaged for Exception Problems than Regular Problems. The Cognitive Network overlaps with prefrontal and parietal regions identified in the ACT-R theory of algebra problem solving and regions identified in the triple-code theory as involved in basic mathematical cognition. The Metacognitive Network included angular gyrus, middle temporal gyrus, and anterior prefrontal regions. This network is mainly engaged by the need to modify the solution procedure and not by the difficulty of the problem. Only the Metacognitive Network decreased with practice on the Exception Problems. Activity in the Cognitive Network during the solution of an Exception Problem predicted both success on that problem and future mastery. Activity in the angular gyrus and middle temporal gyrus during feedback on errors predicted future mastery.

An fMRI investigation of instructional guidance in mathematical problem solving

Abstract In this fMRI study, students learned to solve algebra-like problems in one of the four instructional conditions during behavioral session and solved transfer problems during imaging session. During learning, subjects were given explanatory or non-explanatory verbal instruction, and examples that illustrated the problem structure or the solution procedure. During transfer, participants solved problems that required complex graphical parsing and problems that required algebraic transformations. Explanatory instruction helped in the initial phase of learning, but this benefit disappeared in transfer. The example type had little effect on learning, but interacted with problem type in the transfer. Only for algebraic problems, the structural example led to better transfer than the procedural example. The imaging data revealed no effect of verbal instruction, but found that participants who had studied structural examples showed higher engagement in the prefrontal cortex and angular gyrus. Activity of the right rostrolateral prefrontal cortex in the initial transfer block predicted future mastery.

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).

Individual differences and workload effects on strategy adoption in a dynamic task.

The current study investigated the effects of individual differences and workload on strategy adaptivity in a complex, dynamic task called the Space Fortress game (Donchin, 1989). Participants learned to use a strategy of flying a ship in circles around the fortress in a standard game environment. Once they mastered the strategy, they were assigned to different workload conditions and transferred to a nonstandard environment in which a strong wind was introduced that made it more difficult to achieve a circular orbit. About half of the participants continued with their prior circular strategy while the rest adopted a novel strategy that achieved comparable performance with less effort. With this novel strategy, rather than trying to complete orbits they flew into the wind and then allowed the wind to blow them back to achieve a pendulum-like path. Participants without a working-memory load were more likely to adopt the new strategy. Participants were also more likely to adopt the new strategy if their pattern of behavior exposed them more often to the potential of drifting with the wind. The results indicate that spontaneous changes in strategy occur when people are exposed to the potential of a new strategy and have the cognitive resources to understand its potential.

Not Taking the Easy Road: When Similarity Hurts Learning

Two experimental studies examined the effects of example format and example similarity on mathematical problem solving across different learning contexts. Participants were more successful inducing a correct problem-solving rule when they were provided with annotated examples rather than nonannotated examples. The effects of example similarity varied depending on learning context. In Experiment 1, by presenting an example and problem simultaneously, a direct comparison was possible between the cases. When the examples were similar, participants relied on superficial analogies that hurt learning. When an example was dissimilar from the given problem, participants appeared to study the example first to induce a solution procedure and then apply the rule to the problem, thus resulting in better learning and transfer. However, in Experiment 2 where the example and problem were presented in a sequential manner, the effect disappeared because the learning context did not support a direct comparison. We conclude that comparison is not inherently good for promoting learning and transfer, rather its effect depends on whether it supports relational mapping that is essential for schema acquisition.

End effects and cross-dimensional interference in identification of time and length: Evidence for a common memory mechanism

Memory plays a critical role in time estimation, yet detailed mechanisms underlying temporal memory have not been fully understood. The current functional magnetic resonance imaging (fMRI) study investigated memory phenomena in absolute identification of time durations and line lengths. In both time and length identification, participants responded faster to end-of-range stimuli (e.g., the shortest or longest items of the stimulus set) than to middle stimuli. Participants performed worse in the incongruent condition (mismatch between time and length in the stimulus position) than in the congruent condition, indicating cross-dimensional interference between time and length. Both phenomena reflect increased difficulty of retrieving information relevant to the current context in the presence of context-irrelevant information. A region in the lateral inferior prefrontal cortex showed a greater response to the middle stimuli and in the incongruent condition suggesting greater demands for controlled memory retrieval. A cognitive model based on the ACT-R (Adaptive Control of Thought – Rational) declarative memory mechanisms accounted for the major behavioral and imaging results. The results suggest that contextual effects in temporal memory can be understood in terms of domain-general memory principles established outside the time estimation domain.

Using Brain Imaging to Interpret Student Problem Solving

We have been exploring whether multi voxel pattern analysis (MVPA) of functional magnet resonance imaging (fMRI) data can be used to infer the mental states of students learning mathematics. This approach has shown considerable success in tracking static mental states such as whether a person is thinking about a location or an animal. Applying this to our case involves significant challenges not faced in many MVPA applications because it is necessary to track changing student states over time. The paths of states that students take in solving problems can be quite variable. Nevertheless, we have achieved relatively high accuracy in determining what step a student is on when solving a sequence of problems and whether that step is being performed correctly. Hidden Markov models can then be used to combine behavioral and brain-imaging data from an intelligent tutoring system to track mental states during students problem-solving episodes.