Teaching Problem-Solving Explicitly: Why Method Choice Cannot Be Learned by Osmosis

There is a student every tutor knows: hands in flawless homework on blocked exercises, executes any method once told which to use, and then sits paralysed in front of an exam question that does not announce its topic. The usual diagnosis is not enough practice. The better diagnosis is that the student has been taught every method and never taught how to choose one. Method choice — recognising what kind of problem you are facing and selecting an approach — is a separate skill from method execution, and in most classrooms and tuition sessions it is never taught at all. It is merely demonstrated, invisibly, and students are left to absorb it by osmosis. The strong ones do. Everyone else concludes they are bad at problem-solving.

The expert blind spot

The reason this layer goes untaught is that experts cannot see themselves doing it. When an experienced teacher reads a question, the recognition happens in milliseconds — this is a ratio problem wearing a recipe costume — and the visible teaching starts at step one of the solution. The student watches a solution that begins after the hardest part was already done. Over hundreds of demonstrations, students learn that solutions apparently start themselves, and that the ability to start is something you either have or lack. The first job in teaching problem-solving explicitly is therefore self-observation: catching your own selection process and dragging it into the open.

Think-alouds: making the invisible audible

The core technique is the think-aloud: solving a problem in front of the student while narrating not the steps but the decisions. The difference is easy to hear. A step narration says 'first we write the equation'. A decision narration says 'I see the word remainder, which makes me suspect this is a division-with-leftovers problem — let me test that suspicion before committing'. Good think-alouds include the unglamorous parts experts usually edit out: the false start, the check against a rough estimate, the moment of rereading the question. Students need to see that hesitation is part of the method, not evidence of its absence.

• Narrate recognition: what features of the question you noticed first, and what they suggested.
• Narrate rejection: the method you considered and discarded, and why — rejected options teach as much as chosen ones.
• Narrate monitoring: the running checks — does this number look plausible, am I answering what was actually asked.
• Then reverse the roles: the student solves and thinks aloud while you stay silent. Their narration is the best diagnostic instrument you own.

Name the strategies, then post the names

Heuristics only become usable when they have names. 'Draw a diagram', 'work backwards from the answer', 'try a simpler number first', 'restate the question in your own words', 'find what connects the given and the wanted' — these are teachable moves, and Polya laid them out decades ago. Singapore's own syllabus makes heuristics explicit at primary level for exactly this reason. Naming does two things: it makes the strategy discussable ('which heuristic did you use there?') and it makes it retrievable under pressure, because a student can search a short list of named tools in a way they cannot search a fog of past experiences. Keep the list short — six or eight moves — and refer to it by name every single lesson until the names appear in the student's own talk.

Practise the choosing, not just the doing

Explicit teaching must be matched by practice that actually exercises selection, and blocked exercises cannot do it — twenty questions under the heading Pythagoras never require anyone to recognise a Pythagoras problem. Two formats target the choosing directly.

1. Mixed sets with a classification step: eight questions from four topics, and before solving anything, the student sorts them — which method, and which clue told you. On a tight day, the sorting alone is a complete and valuable exercise.
2. Method-choice discussions without execution: show a question, ask only 'how would you start, and why?', then move to the next. Ten starts practised in the time one full solution would take, aimed at exactly the skill exams punish.

A student who owns five methods but cannot choose between them effectively owns none of them on exam day. Selection is the skill; execution is its servant.

Patience with the payoff curve

One warning for tutors and the parents watching over their shoulders: explicit problem-solving teaching looks slower at first. Lessons contain more talking, more sorting, more half-solved problems, and fewer pages of completed sums. The payoff arrives later but compounds — because method choice, once taught, transfers across every topic that follows, while another sheet of blocked practice improves exactly one. Hold your nerve for a few weeks, track performance on mixed questions rather than blocked ones, and let that number make the argument for you.