Extending problem-solving procedures through reflection

•We develop a theory of how people extend their problem solving procedures through reflection.•The stages proposed by this theory are confirmed with use of HMM–MVPA analysis of fMRI data.•Differences between young adolescents (ages 12–14) and adults are localized in prior knowledge and computational fluency.•Young adolescents (ages 12–14) and adults do not differ in how they reflect on knowledge to extend their procedures. A large-sample (n=75) fMRI study guided the development of a theory of how people extend their problem-solving procedures by reflecting on them. Both children and adults were trained on a new mathematical procedure and then were challenged with novel problems that required them to change and extend their procedure to solve these problems. The fMRI data were analyzed using a combination of hidden Markov models (HMMs) and multi-voxel pattern analysis (MVPA). This HMM–MVPA analysis revealed the existence of 4 stages: Encoding, Planning, Solving, and Responding. Using this analysis as a guide, an ACT-R model was developed that improved the performance of the HMM–MVPA and explained the variation in the durations of the stages across 128 different problems. The model assumes that participants can reflect on declarative representations of the steps of their problem-solving procedures. A Metacognitive module can hold these steps, modify them, create new declarative steps, and rehearse them. The Metacognitive module is associated with activity in the rostrolateral prefrontal cortex (RLPFC). The ACT-R model predicts the activity in the RLPFC and other regions associated with its other cognitive modules (e.g., vision, retrieval). Differences between children and adults seemed related to differences in background knowledge and computational fluency, but not to the differences in their capability to modify procedures.