What Research Says About Teaching Mathematics Through Problem Posing

TL;DR

This synthesis reviews what research says about teaching mathematics through problem posing (P-PBL), illustrating practice with a Grade 1 Chinese lesson on two-digit addition and subtraction and proposing a four-step P-PBL instructional model. It concludes that P-PBL has strong theoretical and empirical support for fostering student learning while flagging numerous unanswered questions about processes, tasks, and teacher learning.

Key findings

• Problem posing is defined as four specific activities: students posing problems from given situations, students reformulating existing problems, teachers generating problem-posing situations, and teachers predicting problems students might pose (Cai & Hwang, 2020).
• Undesirable responses are common: Silver and Cai (1996) found nearly 30% of problems posed by middle school students were nonmathematical problems or nonproblem statements, despite directions asking for problems.
• Problem-posing tasks are scarce in curricula: Cai and Jiang (2017) reported that both Chinese and U.S. elementary curricular materials include less than 3% problem-posing tasks.
• Empirical studies report positive effects: Kopparla et al. (2019) found both problem-posing and problem-solving groups improved, with problem-posing improvement stronger in the problem-solving group; Akben (2020) found improved problem-solving skills and metacognitive awareness among 61 chemistry and 40 physics students; Yang and Xin (2021) found an intervention effective for three students with learning disabilities; Li et al. (2020) found gains after three workshops.
• Zhang and Cai's (2021) analysis of 22 problem-posing teaching cases found teachers skipped posed problems irrelevant to lesson goals, quickly handled easy problems in whole-class discussion, assigned very challenging problems as homework, and focused discussion on moderately challenging problems.

Methods

• Design: a synthesis/review of research on teaching mathematics through problem posing, explicitly "not intended to be comprehensive with a systematic literature search" but tracing work the author has engaged with.
• Illustration: a detailed Grade 1 lesson from Chinese elementary teacher Ms. Yang (Yang & Cai, 2016) with four episodes on multiple representations, number relationships, posing addition problems, and posing subtraction problems.
• Output: a proposed four-step P-PBL instructional model (present situation; provide prompt; students pose problems individually or in groups; teachers analyze, select, sequence, and solve posed problems).

Limitations

• The paper states the review is not comprehensive and lacks a systematic literature search; it is a "sister paper" to Cai (2003) tracing the author's own engaged work.
• The paper repeatedly flags unanswered questions: why students pose nonmathematical or suboptimal problems, the absence of a definitive problem-posing process framework, whether problem-solving task variables adapt to problem posing, how teachers can use posing to understand student thinking, and how teachers learn to teach through problem posing.
• The proposed P-PBL model requires further verification and elaboration, especially Step 4 (analyzing, selecting, and sequencing posed problems); the paper notes it needs more teaching cases to revise the model.
• Evident from the text: the empirical support cited is largely drawn from the author's own projects and a single illustrative lesson, and the paper reports no new primary data of its own.