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Abstract Using data from a problem-posing project, this study analyzed the characteristics of middle school students’ responses to problem-posing prompts that did not match our assumptions and expectations to better understand student thinking. The study found that the characteristics of middle school students’ unexpected responses were distributed across three different problem-posing processes: 1) orientation responses related to different interpretations of the problem-posing prompt or situation accounted for the majority; 2) connection responses related to making connections among pieces of information accounted for the second most common type; and 3) generation responses related to generation of problems only accounted for a very small proportion. Additionally, it was found that the problem-posing prompts influenced the distribution of types of unexpected responses. These findings contribute to our understanding of problem-posing processes and have implications for the design of problem-posing tasks. Most importantly, this analysis reveals that even though these responses are unexpected, students’ responses make sense to them and our objective should be to make sense of their responses. 

This study used three pairs of problem-posing tasks to examine the impact of different prompts on students’ problem posing. Two kinds of prompts were involved. The first asked students to pose 2–3 different mathematical problems without specifying other requirements for the problems, whereas the second kind of prompt did specify additional requirements. A total of 2124 students’ responses were analyzed to examine the impact of the prompts along multiple dimensions. In response to problem-posing prompts with more specific requirements, students tended to engage in more in-depth mathematical thinking and posed much more linguistically and semantically complex problems with more relationships or steps required to solve them. The findings from this study not only contribute to our understanding of problem-posing processes but also have direct implications for teaching mathematics through problem posing.