EJMT Abstract

When two concepts contain a common concept, overgeneralization (the phenomenon of overgeneralizing specific rules or semantic features) may occur in the process of learners gaining an understanding of the two concepts in relation to each other. In order to analyze the overlap singularity and elimination singularity phenomena in singular regions, we constructed a learning system that performs simulations on loss surfaces using neural networks. We first defined four learning stages in the process of symbolic comprehension of the concepts of “permutations” and “combinations” in high school and analyzed the change in training loss and the dynamics on the loss surface. From this analysis, we propose learning guidance for mathematics teachers. As an application of the learning system, we trained a neural network by inputting test data for a technical college and analyzed the understanding of three of that college’s classes.