Discovering the structure of mathematical problem solving

John R. Anderson, Hee Seung Lee, Jon M. Fincham

Research output: Contribution to journalArticlepeer-review

26 Citations (Scopus)


The goal of this research is to discover the stages of mathematical problem solving, the factors that influence the duration of these stages, and how these stages are related to the learning of a new mathematical competence. Using a combination of multivariate pattern analysis (MVPA) and hidden Markov models (HMM), we found that participants went through 5 major phases in solving a class of problems: A Define Phase where they identified the problem to be solved, an Encode Phase where they encoded the needed information, a Compute Phase where they performed the necessary arithmetic calculations, a Transform Phase where they performed any mathematical transformations, and a Respond Phase where they entered an answer. The Define Phase is characterized by activity in visual attention and default network regions, the Encode Phase by activity in visual regions, the Compute Phase by activity in regions active in mathematical tasks, the Transform Phase by activity in mathematical and response regions, and the Respond phase by activity in motor regions. The duration of the Compute and Transform Phases were the only ones that varied with condition. Two features distinguished the mastery trials on which participants came to understand a new problem type. First, the duration of late phases of the problem solution increased. Second, there was increased activation in the rostrolateral prefrontal cortex (RLPFC) and angular gyrus (AG), regions associated with metacognition. This indicates the importance of reflection to successful learning.

Original languageEnglish
Pages (from-to)163-177
Number of pages15
Publication statusPublished - 2014 Aug 15

Bibliographical note

Funding Information:
The research reported here was supported by the Institute of Education Sciences, U.S. Department of Education , through Grant R305A100109 to Carnegie Mellon University. The opinions expressed are those of the authors and do not represent views of the Institute or the U.S. Department of Education. Correspondence may be directed to John Anderson, Department of Psychology, Carnegie Mellon University, 5000 Forbes Ave., Pittsburgh, PA 15213 ( ). We would like to thank Jelmer Borst, Julie Fiez, Aryn Pyke, and Katerina Velanova for their comments on the paper. We would also like to thank Jelmer Borst for his help in constructing the illustrations in Fig. 10 .

All Science Journal Classification (ASJC) codes

  • Neurology
  • Cognitive Neuroscience


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