Procedural fluency is knowing how to do the maths (algorithm, calculation). Mathematical reasoning is understanding why it works. Both matter. A student might fluently multiply two-digit numbers but struggle to explain why standard algorithm works or to choose an efficient strategy.
Australian curriculum emphasises reasoning. That's right. Students need to think, not just compute.
Choose Rich Problems: Look for problems with multiple entry points and solution strategies. "How many different rectangles can be made with 12 square tiles?" admits multiple approaches: systematic listing, factors, trial-and-error with drawing.
Avoid Tricks: Don't give students a problem to which you've taught a specific solution path. Real problem-solving involves figuring out which strategy applies.
Give Thinking Time: Real problem-solving takes time. Allocate 20+ minutes for exploration, not 5 minutes of rushed calculation.
1. Understand the Problem: Restate the problem in your own words. Draw a picture. Identify what you know and what you need to find.
2. Make a Plan: What strategy might work? (Draw, make a table, guess-and-check, work backwards, break it into simpler problems.)
3. Solve: Carry out your plan. Try multiple approaches if the first doesn't work.
4. Check and Reflect: Does your answer make sense? Can you solve it a different way? What did you learn?
Create a visual reference students can access independently:
Teach these explicitly. Model one strategy with a problem, then let students choose strategies for the next problem.
Embrace Mistakes: Wrong answers are learning opportunities. "You got a different answer than I did. Let's see where our thinking diverged. What can we learn?"
Require Justification: "How do you know? Explain your thinking. Prove it's correct using a different method."
Value Efficiency: Celebrate students who find elegant or efficient solutions. "I like how you grouped those numbers to make the calculation easier."
Look beyond correct answers. A student might get a problem wrong but show solid reasoning in their working. Another might get it right but through trial-and-error rather than logical progression.
Ask Why: "Can you explain how you solved this? Why did you choose that strategy? Can you solve it another way?"
Look at Working: The process matters more than the answer. A messy working that shows clear thinking is better than a correct answer with no visible thinking.
Embed mathematical reasoning in authentic contexts. "Design a garden bed. The perimeter is 20 metres. What dimensions minimise cost of edging? What if you want maximum growing space?"
Science inquiry also develops mathematical reasoning. Students measure, analyse data, find patterns, predict. That's maths thinking in action.
