Problem Solving Draw A Picture¶

Strategy · All strategies
Evidence · 2 claims (1 for, 1 against) · 5 studies (2 causal, 2 quant-synthesis, 1 review), q2–q4 · 2 of 5 report an effect size

Description¶

"Draw a picture" is a heuristic from Polya's problem-solving framework in which learners externalize a problem as a diagram — a picture, sketch, number line, or schematic — before or while solving it. The goal is not artistic rendering but a schematic representation that preserves the relational structure of the problem (quantities, relationships, actions), offloading working memory and making the solution path visible.

Design Implications¶

Drawing converts verbal/numeric problem statements into a second, visual representation, and dual representations support reasoning and recall Dual coding improves recall when verbal and visual codes are integrated. [+M]. The critical distinction is between pictorial drawings (literal images of surface features) and schematic drawings (relational diagrams); schematic representations predict solution success while pictorial ones do not, or actively interfere (Hegarty & Kozhevnikov, 1999) [+M]. Teaching the drawing step explicitly — including what to represent and how — matters more than simply instructing learners to "draw a picture."

Context¶

Requirements¶

• Problems whose structure can be meaningfully spatialized (comparison, part–whole, change, ratio, geometry, motion)
• Explicit instruction in diagram conventions (bars, arrows, labeled quantities), not just the exhortation to draw
• Time and norms that treat the drawing as part of the solution, not a preliminary chore
• Feedback that distinguishes schematic from merely pictorial representations

Constraints¶

• Learners with weak spatial or graphomotor skills may expend effort on the drawing itself rather than the mathematics [~M] — for these learners, pre-structured diagrams or digital tools reduce the production burden
• Instructing learners to draw without teaching how to represent relations produces pictorial drawings that add load without aiding solution cognitive overload degrades learning when extraneous load is added. [-M]
• For problems already presented visually, or for highly verbal/abstract problems with no natural spatial structure, forced drawing adds little and can distract [~W]
• Learners with high prior knowledge often solve faster without the intermediate step; the heuristic can become an unnecessary detour [~M]

Implementation Variability¶

• Bar/tape diagrams (Singapore Math) — standardized part–whole and comparison representations taught across grades
• Pre-drawn partial diagrams — learners complete or annotate a provided schematic, reducing production load
• Digital drawing tools — drag-and-drop or auto-labeling environments (e.g., Thinking Blocks) remove graphomotor demands
• Self-generated vs. provided diagrams — self-generation builds the transferable strategy; provided diagrams guarantee representational quality [~M]

Target Learners¶

• Novices and intermediate learners who have not yet internalized problem schemas [+M]
• Learners with learning disabilities in mathematics, when combined with explicit diagram-convention instruction [van Garderen & Montague, 2003] [+M]
• Less beneficial for experts, who typically represent problems internally [~M]

Target Learning Goals¶

• Problem representation: translating a problem statement into a solvable model
• Relational reasoning: making quantities and their relationships visible
• Metacognitive strategy use: building a general-purpose heuristic learners can deploy independently

Instructions¶

1. Read the problem and identify what is being asked and what quantities/relationships are present.
2. Choose a representation suited to the problem type (part–whole bar, number line, array, position sketch).
3. Draw a schematic diagram: label quantities, mark relationships with arrows or brackets — avoid decorative literal detail.
4. Map each element of the problem statement onto the diagram, checking nothing is missing.
5. Solve using the diagram as the working surface, updating it as the solution proceeds.
6. Compare the diagram to the answer for plausibility; discuss which representation worked and why.

Related Strategies¶

• Worked examples — worked examples can model the drawing step itself, showing what a good diagram looks like before learners generate their own
• Think-aloud modeling — pairing verbalized reasoning with diagram construction makes the representation decisions explicit
• Choose problems with multiple entry points to stimulate discussion of different solution approaches
• Teach a four-step verbalized problem-solving strategy (identify problem, propose solution, implement, evaluate)
• Provide explicit instruction in how to understand and use informative pictures if they are employed in word problem instruction

Examples¶

• Singapore Math bar models — a national curriculum sequence in which students progress from concrete objects to standardized bar diagrams for word problems (Thinking Blocks offers an interactive version)
• Cognitively Guided Instruction — teachers elicit and discuss students' own drawn representations of word problems, using the drawings to surface solution strategies
• Polya's How to Solve It (1945) — the original articulation of "draw a picture" as one of four phases: understand, plan (represent), carry out, look back

Key Sources¶

• Polya, G. (1945). How to solve it: A new aspect of mathematical method. Princeton University Press.
• Hegarty, M., & Kozhevnikov, M. (1999). Types of visual–spatial representations and mathematical problem solving. Journal of Educational Psychology, 91(4), 684–689. doi:10.1037/0022-0663.91.4.684
• van Garderen, D., & Montague, M. (2003). Visual–spatial representation, mathematical problem solving, and students of varying abilities. Learning Disability Quarterly, 26(1), 49–64. doi:10.2307/1593664
• Uesaka, Y., & Manalo, E. (2012). Task-related factors that influence Japanese schoolchildren's use of diagrams: The need for both elementary and deeper understanding of problem structure. Instructional Science, 40(4), 631–650.