The Architecture of Problem Solving
Think of problem solving like designing a building. The structure is the kind of problem you're working with. The tools are the heuristics that help you make progress. The blueprint is metacognition.
- Problem Structures that shape the thinking required
- Heuristics (3Rs) - Representing, Reasoning, Regulating
- Metacognition that guides strategy choice
The Structure: Problem Types
Problem structures are the architecture of a task—what kind of thinking it demands. Naming the structure helps students choose strategies intentionally.
Additive & Multiplicative
Join/separate/compare • equal groups • scaling
Functional / Pattern
Growing patterns • covariation • recurring cycles
Combinatorial
Combinations • arrangements • "How many ways?"
Constraint & Logical
Clues • rules • elimination • must-be-true
Optimisation
Best/most/least • efficiency • trade-offs
Spatial & Geometric
Visualising • transforming • decomposing space
Real-World / Modelling (Integrated Inquiry)
Assumptions • measurement • validation • decision-making
"How many tennis balls would fit into a classroom?"
The Tools: Heuristics (3Rs)
Heuristics are the tools students use to make progress. The 3Rs organise those tools into making thinking visible, reasoning with structure, and regulating decisions.
Draw a diagram
Sketch the relationships so the structure is visible.
Make a table or organised list
Organise information to track change or reveal structure.
Use materials or act it out
Model the situation with objects/actions to explore what's happening.
Use a simpler or smaller case
Try a smaller version to see how the problem works before scaling up.
Look for a pattern
Notice repetition/growth and use it to predict or generalise.
Work systematically
Check cases in an ordered way so nothing is missed or repeated.
Use logical reasoning
Use constraints to rule options in/out based on what must be true.
Use a mathematical model
Use a structured model (array/number line/grid/ratio table) to show relationships.
Write an equation
Use a number sentence to express the relationship and test ideas.
Estimate and refine
Start with a sensible benchmark, then improve it as you learn more.
Trial and improve
Try, check, adjust—repeat with purpose (productive guess-and-check).
Check for reasonableness
Decide whether your result makes sense in the context.
The Blueprint: Metacognition
Metacognition is the blueprint: how students plan, monitor, and evaluate their strategy choices. It helps them use the tools (heuristics) intentionally within the structure (problem type).
Plan
What's my goal? What structure might this be? What tool could help first?
Name the structure. Ask students to justify their first tool choice.
Monitor & Adapt
Is this working? What's confusing? Should I change tools or representation?
Prompt a tool-switch: 'What model/diagram could make this clearer?'
Evaluate
Does my answer make sense? How do I know? Would this work in another case?
Press for sense-making and generalisation: 'How could we check?'
Quick prompts (use during the struggle)
- What stays the same? What changes?
- What information matters? What can be ignored?
- How can you make your thinking visible?
- If you're stuck, try a different tool: diagram, model, table, smaller case.
- How do you know you haven't missed a possibility?
When students choose a tool, link it back to the structure. When they get stuck, invite a tool-switch from the tools.
Ready to apply these ideas?
Explore our weekly problems and see these structures and tools in action.