This study aimed to examine the content common and specific processes in the relationship between abilities in high school math problem solving and motivation. A questionnaire survey was administered to 171 first-year students at a private high school. The following four findings were obtained: (1) students with higher mastery goal in learning both vector and sequence had a higher emotional engagement, resulting in higher problem-solving abilities common to both vector and sequence; (2) students a with higher performance goal in learning both vector and sequence had a higher behavioural engagement, resulting in higher-problem solving abilities common to both vector and sequence, but the contribution of performance goal to the abilities was smaller than that of mastery goal; (3) students with a higher mastery goal in learning vector had a higher emotional and behavioural engagement, resulting in higher vector-specific problem-solving abilities; (4) sequence-specific problem-solving abilities were nor associated with achievement goals and engagement.

In this study, we aim to characterize students’ uses of counter-examples and their explanations by counter-examples, through a qualitative analysis based on a questionnaire survey targeting sixth graders in elementary school. For attaining this, we first reviewed the literature related to the notion of generic examples, and the role of counter-examples for explanation in mathematics education research. Then, an analytical framework, consisting of two basic aspects, “generic use of counter-examples” and “scope of explanatory power by counter-examples”, was constructed. To illustrate this framework, we conducted a questionnaire survey using a proving task (called “number pyramid”), analyzed the students’ writings, and classified them into each aspect of the framework. As a result, students’ different responses to the tasks are characterized by the aspects of the explanatory power of the counter-examples as well as the aspects of generic use of counter-examples. This study implies that the students have more difficulties with increasing explanatory power rather than increasing the genericity of use of counter-examples. Some implications for teaching and learning in lower secondary schools are also discussed.

In sixth grade arithmetic class, in order to clarify the characteristics of group learning of the solution search-type group, in which group members find the solution while discussing, and the characteristics of group learning of the presentation-type group, in which each other’s solution is presented and examined after self-resolution, an interpretive analysis of an utterance case of one session of group learning was carried out. Compared to the presentation-type group, there were more “opinions” in the solution search-type group, in which group members added their opinions while finding solutions. As to differences in interaction, in solution search-type group learning, interaction by a chain of extended utterances was found in the activity of describing and verbalizing the solution, whereas in the presentation-type group learning, interaction by changing questions and explanations was observed among group members with different solution methods.