Write Your Own Word Problem¶

Strategy · All strategies
Evidence · 1 claim (1 for) · 2 studies (1 causal, 1 quant-synthesis), q3 · 0 of 2 report an effect size

Description¶

Learners are assigned to author their own word problems that incorporate key vocabulary, operations, or structures discussed in class. Writing a problem requires inverting the usual task: instead of translating a situation into an equation, learners must understand a mathematical relationship well enough to embed it in a coherent story. The resulting problems can then be exchanged, solved by peers, and discussed.

Design Implications¶

Problem posing sits at the deep end of mathematical understanding: to write a solvable, sensible problem, learners must grasp the underlying relations, not just surface keywords Problem-posing instruction improves problem-solving achievement. [+M]. The activity doubles as an elicitation of Self-Explanation — deciding which numbers and phrasing make a problem work forces learners to articulate why a structure is what it is Eliciting self-explanations improves understanding. [+S]. Silver argues that instruction rich in problem posing fosters both creativity and flexibility in mathematical thinking [+M].

Context¶

Requirements¶

• Prior exposure to worked examples of the target problem type, so learners have a model to imitate and vary
• Explicit criteria for what makes a problem valid (solvable, realistic numbers, unambiguous wording)
• A follow-up structure — peer solving, gallery walk, or class discussion — so authored problems are actually used

Constraints¶

• Learners with weak understanding tend to write superficial keyword-matching problems ("add" appears, so add), rehearsing the misconception rather than repairing it [-M]
• Without validity criteria and feedback, learners may practice producing incoherent or trivially easy problems, yielding little learning [-W]
• Open-ended posing can overload novices; constraining the task (e.g., "write a problem that uses these three numbers and requires two steps") preserves the benefit while managing load [~M]

Implementation Variability¶

• Structured posing: give the answer or the equation and ask for a matching story — lower floor, strong focus on structure
• Semi-structured posing: give a scenario or image and ask learners to add the mathematical question
• Free posing: learners invent everything; highest ceiling, best for learners with solid prior knowledge
• Error-based posing: ask learners to write a problem that looks solvable but contains a trap, then explain the trap — connects to Erroneous Examples

Target Learners¶

• Elementary and middle-grades mathematics learners, where problem-posing programs have shown achievement gains [+M]
• Learners who can solve routine problems but treat keywords mechanically; authoring exposes the gap between surface features and structure
• Less suitable as an initial activity for complete novices, who lack the schema needed to construct a coherent problem [~M]

Target Learning Goals¶

• Conceptual understanding of problem structure and the relationship between situations and operations
• Transfer: recognizing that different surface stories can share the same mathematical deep structure
• Metacognition and mathematical communication: judging what makes a problem clear, solvable, and realistic

Instructions¶

1. In class, work through exemplar problems and name the key words and structures (Activation).
2. Model the inversion: take an equation and show how a story problem can be built around it (Cognitive Apprenticeship).
3. Assign learners to write 2–3 of their own problems using the key words, with stated validity criteria.
4. Have learners exchange problems and solve a peer's, then give the author feedback on clarity and solvability.
5. Debrief by comparing several authored problems that share a deep structure, prompting learners to explain the commonality (Self-Explanation).

Related Strategies¶

• Solve Before You Teach — the complementary direction of the same inversion
• Peer Instruction — a structure for using learner-authored problems in class

Examples¶

• Cognitively Guided Instruction classrooms (CGI for Math) use children's own story problems as a window into their thinking and as material for class discussion.
• NCTM problem-posing tasks — the National Council of Teachers of Mathematics explicitly includes "pose problems" in its Principles and Standards; teachers ask students to generate questions from a given data set or image (NCTM).
• A homework cycle in which each student submits one problem, the teacher curates a set, and the next day's quiz is drawn entirely from student-authored problems — raising both effort and audience awareness (Authentic Audiences & Purposes).

Key Sources¶

• Silver, E. A. (1996). Fostering creativity through instruction rich in mathematical problem solving and problem posing. ZDM, 3, 75–80. doi:10.1007/s11858-997-0003-x
• Cankoy, O., & Darbaz, S. (2010). Effect of a problem posing elementary mathematics program on problem solving achievement. Procedia — Social and Behavioral Sciences, 2(2), 2827–2831.
• Chi, M. T. H., de Leeuw, N., Chiu, M.-H., & LaVancher, C. (1994). Eliciting self-explanations improves understanding. Cognitive Science, 18(3), 439–477. doi:10.1207/s15516709cog1803_3