# 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.