Mathematics Teachers’ Subject Matter Knowledge and Pedagogical Content Knowledge in Problem Posing

TL;DR

This qualitative study of four Korean middle/high school mathematics teachers found that although they could define problem posing and distinguish problem generation from reformulation, their actual posed problems often did not reflect that subject matter knowledge and were less varied or lower-level than expected. Teachers valued problem posing but reported barriers to classroom implementation, and they differed in how they assessed students’ incorrect posed problems, pointing to professional development needs.

Key findings

• All four teachers defined problem posing and identified problem generation and problem reformulation as types, but their actual problem-posing results did not match their stated SMK; three of the four posed problems requiring a low level of cognitive demand, and Table 4 shows fewer problem-posing types in reality than in their expectations.
• For KCT, teachers viewed problem posing as important but cited barriers including insufficient class time and limited textbook examples; all except Yoon expressed positive perspectives, while Yoon said problem posing was important but “not easy to apply” and lacked usable assessment standards.
• For KCS, teachers linked problem-posing proficiency to mathematical proficiency and stressed students’ prior concept knowledge, but they disagreed about the role of problem solving in problem posing and about evaluating incorrect posed problems: Kim and Yoon evaluated the student’s posed problems negatively, whereas Seo and Hyun inferred conceptual understanding even when the answer was incorrect.
• The authors conclude that professional development is needed for teachers’ pedagogical knowledge in problem posing.

Methods

• Qualitative interview design with four Korean intermediate-level middle/high school mathematics teachers; each participated in one semi-structured, open-ended interview of approximately one hour, with two mathematics questionnaires—one for problem generation and one for problem reformulation—and 10 minutes to pose problems.
• Interviews were audio recorded, transcribed, supplemented with researchers’ observation notes, and analyzed with inductive analysis and open coding using problem-posing codes and teacher-knowledge codes.
• Data were triangulated for internal validity; researchers coded independently, resolved disagreements, and reported an inter-rater agreement of 0.95.

Limitations

• The paper states that the participants each had no more than five years of teaching experience, that the limited number of posed problems may have constrained the variety of problems observed, and that the study is a preliminary analysis requiring follow-up research.
• The paper also notes that teachers’ practices did not fully reflect their KCT and KCS because of educational-setting constraints such as limited time and textbook content.
• Evident from the design: the sample is only four volunteer teachers in Korea, so findings are not presented as broadly generalizable, and the data rely on interviews, questionnaires, and observation notes rather than extended classroom observation.