Investigating a Metacognitive Strategy for Solving Indefinite Integration Problems in Calculus: An fMRI study

Date of Completion

January 2011


Education, Mathematics|Biology, Neuroscience




Expertise and expert performance has long been a rich area of research. Experts differ from novices in several ways including the depth of their knowledge base, the ability to detect and recognize salient features of problems, more skilled and accurate performance, and strong self-monitoring skills. Advances in neuroscience methods such as functional magnetic resonance imaging (fMRI) now provide a means of investigating the changes in brain activation patterns as students progress along the continuum from novice to expert. ^ The purpose of this study is to use fMRI as a tool to extend the understanding of expertise in calculus by focusing on the first step of solving integration problems – selecting an appropriate and efficient technique using an expert-like metacognitive strategy. To accomplish this, research methods from education and neuroscience were employed. Eight right-handed, native English-speaking undergraduates who had completed Calculus II at local universities participated in this two-phase study. Phase 1 of the study consisted of a review of integration techniques followed by a block-design fMRI experiment. Phase 2 consisted of training to use and expert-like strategy (Schoenfeld, 1980) for selecting and integration technique followed by another block-design fMRI experiment. Data collection occurred at three points during this study – specifically during the pre-test prior to the Phase 1 training, and then during Phase 1 and Phase 2 post-testing and fMRI scanning sessions. Collected data included written pre- and post-tests, response time and accuracy data from the computer-based post-tests and fMRI experiments, fMRI signal data. Functional MRI data was preprocessed and the first-level data analyzed using SPM8 (Wellcome Trust Centre for Neuroimaging, 2009). For the second-level random effect model a region of interest approach was adopted and implemented in MarsBaR (Brett, Anton, Valabregue, & Poline, 2002).^ The results of this study identified six regions of interest (the bilateral hIPS, bilateral posterior superior parietal lobule PSPL, bilateral middle frontal gyrus excluding BA9) which were significantly active during the integration task as compared to the control task. Furthermore, three regions (bilateral insula, right precentral gyrus) were identified as deactivated during the integration task as compared to the control condition. The results also indicated that the strategy training improved participants' accuracy in selecting an appropriate technique as measured by the computer-based post-tests and in the fMRI experiment. This research provides a first step towards bridging the gap between education and neuroscience research in the area of calculus learning and adds to the limited body of literature exploring the neural basis of more complex mathematics. ^