### Abstract

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 language | English |
---|---|

Pages (from-to) | 163-177 |

Number of pages | 15 |

Journal | NeuroImage |

Volume | 97 |

DOIs | |

Publication status | Published - 2014 Aug 15 |

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### All Science Journal Classification (ASJC) codes

- Neurology
- Cognitive Neuroscience

### Cite this

*NeuroImage*,

*97*, 163-177. https://doi.org/10.1016/j.neuroimage.2014.04.031

}

*NeuroImage*, vol. 97, pp. 163-177. https://doi.org/10.1016/j.neuroimage.2014.04.031

**Discovering the structure of mathematical problem solving.** / Anderson, John R.; Lee, Hee Seung; Fincham, Jon M.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Discovering the structure of mathematical problem solving

AU - Anderson, John R.

AU - Lee, Hee Seung

AU - Fincham, Jon M.

PY - 2014/8/15

Y1 - 2014/8/15

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84901302103&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84901302103&partnerID=8YFLogxK

U2 - 10.1016/j.neuroimage.2014.04.031

DO - 10.1016/j.neuroimage.2014.04.031

M3 - Article

C2 - 24746954

AN - SCOPUS:84901302103

VL - 97

SP - 163

EP - 177

JO - NeuroImage

JF - NeuroImage

SN - 1053-8119

ER -