Teaching Problem-Solving Strategies - Moving Beyond 'Just Try Your Best'

“Just try your best” is well-intentioned but unhelpful advice for students facing challenging mathematics problems. Problem-solving is a skill that can be taught—not through practice alone, but through explicit instruction in strategies, systematic approaches, and metacognitive thinking. When we teach students how to approach problems, not just which procedures to apply, we develop mathematical power that transfers across contexts.

The Problem with “Just Solve It”

Why students struggle:

• Don’t know where to start
• Give up at first difficulty
• Lack systematic approaches
• Don’t monitor their thinking
• Can’t decide which strategy might work

What they need:

• Explicit problem-solving strategies
• Frameworks for tackling unfamiliar problems
• Metacognitive awareness
• Persistence skills
• Multiple approaches to try

Problem-solving isn’t about being “naturally good at maths”—it’s about having tools and knowing when to use them.

Polya’s Four-Step Framework

George Pólya’s classic framework provides structure for tackling any problem:

Step 1: Understand the Problem

What this means:

• Read carefully (multiple times)
• Identify what you know
• Identify what you need to find
• Restate in your own words
• Ask: “Do I understand what’s being asked?”

Guiding questions:

• What is this problem about?
• What information am I given?
• What do I need to find?
• Can I explain this problem to someone else?
• Are there any words I don’t understand?

Activities:

• Underline or highlight key information
• Circle what you need to find
• Cross out unnecessary information
• Draw a picture of the situation
• Rewrite the problem in simpler terms

Common mistake: Rushing to calculate before understanding. Slow down here.

Step 2: Make a Plan

What this means:

• Choose a strategy to try
• Think about similar problems you’ve solved
• Consider what operations or concepts might help
• Decide on a starting point

Guiding questions:

• Have I seen a problem like this before?
• What strategy might work?
• Can I break this into smaller parts?
• What’s a logical first step?

This is where explicit strategy instruction matters most.

Step 3: Carry Out the Plan

What this means:

• Execute your chosen strategy
• Show your work clearly
• Check each step as you go
• Stay organized

Guiding questions:

• Am I following my plan?
• Does this step make sense?
• Should I try something different?
• Am I keeping track of my thinking?

Important: If the plan isn’t working, return to Step 2 and try a different approach.

Step 4: Look Back

What this means:

• Check if your answer makes sense
• Verify calculations
• Ask if you answered the question asked
• Consider if there’s another way to solve it
• Reflect on what you learned

Guiding questions:

• Does my answer make sense?
• Did I answer what was asked?
• Can I check my answer a different way?
• What strategy worked? Why?
• What would I do differently next time?

Common mistake: Stopping once an answer is reached without verification.

Explicit Problem-Solving Strategies

These concrete strategies give students options when stuck:

Strategy 1: Draw a Picture or Diagram

When to use: Visual problems, spatial relationships, part-whole problems

How it works:

• Represent the problem visually
• Can be realistic or schematic
• Helps see relationships and patterns

Example: “A rectangular garden is 12m by 8m. What is its area?”

• Draw a rectangle
• Label sides: 12m and 8m
• Visualize: area = length × width = 12 × 8 = 96 m²

Why it works: Makes abstract concrete; reveals structure.

Strategy 2: Make a Table or Chart

When to use: Organizing information, finding patterns, comparing multiple items

How it works:

• Create columns and rows
• Systematically record information
• Look for patterns or relationships

Example: “Tickets cost $8 for adults and$5 for children. Find the cost for different group combinations.”

Why it works: Organizes complex information; reveals patterns.

Strategy 3: Look for a Pattern

When to use: Sequences, repeating situations, predictions

How it works:

• Identify what repeats or changes systematically
• Describe the pattern
• Use pattern to find answer

Example: “1, 4, 7, 10, … What’s the 10th number?”

• Pattern: add 3 each time
• Continue: 13, 16, 19, 22, 25, 28
• 10th number: 28

Why it works: Patterns reveal mathematical structure; enable prediction.

Strategy 4: Work Backwards

When to use: Know the end result, need to find starting point

How it works:

• Start with the answer
• Reverse each operation
• Work back to the beginning

Example: “I had some money. I spent $15, then earned$20. Now I have $32. How much did I start with?”

• End: $32
• Before earning $20:$32 - $20 =$12
• Before spending $15:$12 + $15 =$27
• Started with: $27

Why it works: Sometimes easier to reverse than work forward.

Strategy 5: Guess, Check, and Revise

When to use: Multiple constraints, trying possibilities systematically

How it works:

• Make an educated guess
• Test if it works
• Use result to make a better guess
• Repeat until finding the answer

Example: “Two numbers multiply to 48 and add to 13. What are they?”

• Guess: 4 and 12 → multiply: 4×12=48 ✓ add: 4+12=16 ✗
• Revise: 6 and 8 → multiply: 6×8=48 ✓ add: 6+8=14 ✗
• Revise: 3 and 16 → multiply: 3×16=48 ✓ add: 3+16=19 ✗
• Wait, try factors of 48: 1×48, 2×24, 3×16, 4×12, 6×8
• Try: 5 and… doesn’t work. Back to list.
• Actually: Need to be more systematic with all factor pairs

(Better approach would be guess-and-revise OR list all factor pairs then check sums)

Why it works: Systematic trial narrows possibilities; builds number sense.

Strategy 6: Break the Problem into Smaller Parts

When to use: Complex multi-step problems

How it works:

• Identify smaller questions within the big question
• Solve each part
• Combine to answer the whole

Example: “A school has 8 classes with 24 students each. If they sit in rows of 6, how many rows are needed?”

• Part 1: How many students total? 8 × 24 = 192
• Part 2: How many rows for 192 students in rows of 6? 192 ÷ 6 = 32
• Answer: 32 rows

Why it works: Makes overwhelming problems manageable.

Strategy 7: Simplify the Problem

When to use: Numbers too large or complex; pattern not clear

How it works:

• Use smaller, friendlier numbers
• Solve the simpler version
• Apply the same method to original

Example: “What is 15% of 240?”

• Too complex? Simplify: “What is 10% of 100?”
• 10% of 100 = 10 (move decimal point)
• Apply to original: 10% of 240 = 24
• Need 15%: 10% = 24, so 5% = 12, therefore 15% = 24 + 12 = 36

Why it works: Reveals structure without overwhelming numbers.

Strategy 8: Act It Out or Use Objects

When to use: Action-based or concrete situations

How it works:

• Use physical objects or role-play
• Actually perform the actions
• Count or observe the result

Example: “If 5 people each shake hands with every other person once, how many handshakes?”

• Use 5 students
• Actually shake hands
• Count: 10 handshakes

Why it works: Makes abstract situations concrete and observable.

Teaching Problem-Solving: The Instructional Sequence

Phase 1: Introduce Strategy Explicitly

Model the strategy:

• Choose a problem that fits the strategy
• Think aloud while solving
• Show every step of your thinking
• Name the strategy you’re using

Example: “This problem asks how many ways to make 50 cents. I’m going to use the ‘make a table’ strategy to organize all the possibilities…”

Phase 2: Guided Practice

Solve together:

• Present a new problem
• Students suggest which strategy might work
• Class implements strategy together
• Discuss what’s working and what’s not

Scaffolding:

• Provide partially completed tables/diagrams
• Offer strategy prompts: “What if you drew a picture?”
• Support decision-making about which strategy

Phase 3: Independent Practice with Support

Students solve with strategy choice:

• Provide problems that fit the newly learned strategy
• Students try independently
• Available support when stuck
• Share solutions and strategies

Phase 4: Strategy Integration

Mixed problems:

• Present problems requiring different strategies
• Students must choose appropriate strategy
• Emphasis on explaining why they chose that strategy

Phase 5: Metacognition and Reflection

Think about thinking:

• “What strategy did you use? Why?”
• “Did it work? Why or why not?”
• “What would you try differently next time?”
• “When might you use this strategy again?”

Building a Problem-Solving Culture

Create a Strategy Toolbox

Visual display in classroom:

• Poster with all strategies
• Icons or symbols for each
• Examples of when to use each

Reference cards:

• Students have personal reference cards
• Checklist of strategies
• Guiding questions for each

Use Problem-Solving Language

Instead of: “Did you get the answer?” Say: “What strategy did you use?”

Instead of: “That’s wrong” Say: “That strategy didn’t work. What else could you try?”

Instead of: “Let me show you how” Say: “What have you tried so far? What else is in your toolbox?”

Celebrate Multiple Strategies

After solving:

• “Who solved it a different way?”
• Display multiple solution paths
• Compare efficiency and elegance
• Value creativity and flexibility

Message: There are many ways to solve problems; flexible thinkers try multiple approaches.

Normalize Productive Struggle

Reframe difficulty:

• “This is a challenging problem” (not “this is hard for you”)
• “You haven’t figured it out yet” (growth mindset)
• “Struggle means your brain is growing”

Provide wait time:

• Don’t rescue too quickly
• Let students grapple productively
• Support ≠ removing the challenge

Develop Metacognitive Habits

Routine self-questions:

• “What am I trying to find?”
• “What do I know?”
• “What strategy should I try?”
• “Is this working?”
• “Does my answer make sense?”

Reflection protocols:

• After each problem, briefly reflect
• “What worked? What didn’t?”
• “What did I learn?”

Assessment of Problem-Solving

Look beyond correct answers:

• Can they explain their strategy?
• Do they try multiple approaches when stuck?
• Can they identify why a strategy didn’t work?
• Do they verify their answers?
• Can they tackle unfamiliar problems?

Rubric elements:

• Understanding (identified key information)
• Strategy selection (chose appropriate approach)
• Execution (carried out strategy correctly)
• Verification (checked answer)
• Communication (explained thinking)

The Bottom Line

Problem-solving isn’t a mystical talent—it’s a teachable set of strategies and habits of mind. When we teach students explicit problem-solving approaches, give them tools for their mental toolbox, model metacognitive thinking, and create a culture that values strategic thinking over quick answers, we develop mathematical problem-solvers.

The goal isn’t just to solve today’s problem—it’s to build problem-solving capacity that transfers to tomorrow’s unfamiliar challenge. That’s not just mathematics teaching—that’s empowering students with thinking skills for life.

And it starts with moving beyond “just try your best” to “here’s how to approach this systematically.” That’s the difference between mathematical helplessness and mathematical power.