Pupil-Generated Word Problems¶

Strategy · All strategies
Evidence · 1 claim (1 for) · 4 studies (2 quant-synthesis, 1 causal, 1 associational), q2–q4 · 1 of 4 report an effect size

Description¶

Learners author their own word problems — selecting quantities, operations, contexts, and constraints — and then exchange, solve, and critique one another's problems. Problem posing reverses the usual direction of instruction: instead of decoding someone else's problem, learners must construct the semantic structure themselves, which forces them to attend to how quantities, relationships, and question sentences fit together.

Design Implications¶

Generating problems requires learners to analyze the deep structure of word problems rather than pattern-match surface features, and this structural awareness transfers to problem solving Problem posing improves both posing skill and problem-solving performance. [+M]. The activity functions as a form of generative self-explanation: deciding what makes a problem solvable, what makes it hard, and what makes it realistic requires articulating relationships that passive solvers often skip Self-explanation improves conceptual understanding. [+M]. Posing is most effective when paired with solving and critique, not used as a standalone activity.

Context¶

Requirements¶

• A recently taught concept or operation to anchor the problems (posing works best when it extends, not precedes, instruction on the target skill)
• Structured prompts or constraints (e.g., "must involve two steps," "use a money context") so problems align with learning objectives
• Time and a protocol for sharing, solving, and discussing peer problems (Peer Discussion)

Constraints¶

• Unconstrained posing produces trivial or unsolvable problems; learners may write problems that rehearse the wrong skill or embed incoherent quantities [-M]
• Learners with weak basic skills often cannot pose problems at all, since posing presupposes some fluency with the operations involved [-M]
• Peer-authored problems vary widely in quality and difficulty, which can waste time or mislead if the teacher does not curate them [-W]
• Some learners are reluctant to share their problems or unmotivated to solve peers' problems without an authentic audience or purpose

Implementation Variability¶

• Semi-structured posing: learners complete a problem stem ("Tom has ___ marbles. He gives ___ to ___. How many…?") — lowers the load for novices
• Transformation tasks: learners rewrite a given problem (change the operation, add a step, make it multi-step) — a middle ground between solving and full generation
• Pose-then-solve: learners must solve their own problem, which catches incoherent problems and closes the loop
• Problem-swapping gallery: problems posted and solved by peers, with the author giving feedback on solution errors

Target Learners¶

• Elementary and middle school learners who have basic fluency with the target operations [+M]
• Learners who rely on surface-feature keyword strategies ("altogether means add"); posing exposes why keywords fail [~M]
• Less suitable as an introduction for novices who have not yet solved problems of the target type — pose after, not before, initial instruction [~M]

Target Learning Goals¶

• Structural understanding of word problems: how givens, relationships, and questions combine
• Discrimination between problem types and operations (which structure demands which operation)
• Real-world connection: mapping everyday contexts onto mathematical relationships
• Communication: justifying why a problem is solvable, realistic, and appropriately difficult

Instructions¶

1. Anchor the task in a recently taught concept and review one worked example, naming its structural parts (Problem Presentation)
2. Give posing constraints tied to the objective (operation, number of steps, context) and have learners draft one or two problems (Application)
3. Have learners solve their own problems first to verify solvability
4. Exchange problems for peers to solve, then discuss discrepancies between intended and actual solutions (Peer Discussion)
5. Curate the best problems for later practice, quizzes, or an authentic audience (another class, a class problem book)

Related Strategies¶

• Activating Prior Knowledge — reviewing familiar problem structures before posing gives learners raw material to work from
• Peer Discussion — the critique phase where structural awareness is made explicit

Examples¶

• Cognitively Guided Instruction classrooms use problem-posing extensions after children solve story problems, asking them to invent new problems for the same number relationship (Cognitively Guided Instruction (CGI) for Math)
• A grade-4 class studying multiplication poses "shopping" problems (3 items at $1.25 each), swaps them, and the teacher selects several for the unit test — learners see their own problems become assessment items
• Illustrative Mathematics-style curricula include "write your own problem" tasks as lesson warm-ups, often with a stem to complete

Key Sources¶

• Silver, E. A. (1994). On mathematical problem posing. For the Learning of Mathematics, 14(1), 19–28.
• Cai, J., & Hwang, S. (2002). Generalized and generative thinking in US and Chinese students' mathematical problem posing and problem solving. Journal of Mathematical Behavior, 21(4), 401–421. doi:10.1016/S0732-3123(02)00142-6
• Singer, M., Ellerton, N., & Cai, J. (2013). Problem-posing research in mathematics education: New questions and directions. Educational Studies in Mathematics, 83(1), 1–7. doi:10.1007/s10649-013-9478-2
• Kilpatrick, J. (1987). Problem formulating: Where do good problems come from? In A. H. Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 123–147). Lawrence Erlbaum.