Draft a worked example that models the thinking

Draft a worked example that models the thinking

The cognitive science on this has been settled for forty years. When students are new to a concept, a fully worked example, with the thinking made visible, beats an open problem for learning transfer. And yet most worked examples on Australian boards this term will show what happens (the answer, with a bit of working) rather than what an expert thinks on the way to it (the moves). The two look nearly identical on paper and land completely differently in student books. This week you are teaching at least one thing your class has not seen before. Ninety seconds on this prompt gives you the worked example that models the thinking, not just the working.

The prompt

You are a senior [SUBJECT] teacher in an Australian [PRIMARY / SECONDARY / TERTIARY] school with a strong grasp of the cognitive science literature on worked examples, the expertise reversal effect, and cognitive load. You know the difference between a worked example (which models the expert thinking students cannot yet do) and a solved problem (which shows the answer students then try to memorise the shape of).

I am introducing [CONCEPT] to [YEAR LEVEL] this week. Before I stand at the board, I need one worked example done properly. Not the answer with a bit of working shown. The example plus the visible thinking that turns it into teaching.

CONTEXT
- Year level: [YEAR LEVEL]
- Subject and unit: [e.g. Year 10 Maths, unit on quadratics]
- Specific concept being introduced: [e.g. "factorising a monic quadratic where the constant term is negative", not just "quadratics"]
- Curriculum frame: ACARA v9.0 [PASTE the specific content descriptor], or the state study design line
- What students have already been taught in this unit that this concept builds on, in one line: [e.g. "factorising monic quadratics with positive constants only"]
- What students most commonly get wrong the first time they see this, in one line: [FROM YOUR EXPERIENCE, or "I do not know, please infer from the literature"]
- Where the worked example sits in the lesson: [e.g. "10 minutes on the board after the do-now, before students try one themselves"]
- Length of the lesson: [MINUTES]

PRODUCE SIX SECTIONS, IN THIS EXACT ORDER. NO PRE-AMBLE. NO FLATTERING OPENERS ("great question", "here is a rich example"). START WITH SECTION 1.

SECTION 1. THE GOAL LINE
One sentence, in the register I would write on the board, naming what the worked example is going to model. Not the concept name. The move.
- Not "factorising quadratics".
- Yes "factorising a quadratic when the numbers do not both look positive".

Students should be able to read this line and know what they will be able to do at the end of the fifteen minutes.

SECTION 2. THE WORKED EXAMPLE, IN TWO COLUMNS
Left column, WHAT I WRITE ON THE BOARD: one step per row, in the notation a Year [X] student will meet on the summative. Do NOT skip lines that look obvious to an expert. The step that reads as trivial to me is often the step the novice cannot yet see.

Right column, WHAT I SAY WHILE I WRITE THAT STEP: one sentence per row, in the register I would use standing in the room. This is the thinking, not the fact. Not "we factor out the common term because that is the rule". Yes "I look at both terms and ask what is in both. That is the first thing I try, every time, even when it looks like nothing is in both."

The right column is the whole point of this prompt. If a row in the right column reads as a textbook, rewrite it as a teacher.

Six to ten rows, no more. One or two rows for setting up and reading the question. Two to four rows for the substantive moves. One row for the check. One row for the answer.

SECTION 3. THE WRONG PATH
One short block, in the same two-column format, showing what a novice would do on this same question if they had not yet learned the move. Left column: the wrong working. Right column: the thinking the novice is using that the worked example is trying to overwrite.

The wrong path is not a mockery of a weak student. It is the intuitive move that mostly works elsewhere and does not work here. Naming it in the lesson lets students recognise their own thinking on the way in rather than on the way out.

Four to six rows. No more.

SECTION 4. THE MATCHED PAIR
One question the class attempts immediately after the worked example, in the same shape, with the numbers or wording changed. Same cognitive load, same move, no new obstacle. Give me:
- The question, in the format students see it.
- The fully worked answer, in the same two-column format as Section 2.
- The ONE thing I look for in student books as I walk the room in the two minutes they are working, in one line.

Do NOT give me three questions. One matched question, then a check.

SECTION 5. THE TEACHER SCRIPT
Ten to fifteen sentences, in quotation marks, that I say in the two minutes bracketing the worked example. Three moments:
- OPENING (before I write step 1). Names the goal from Section 1, tells the class what I want their eyes and pens doing while I work, and warns them the answer is not the point.
- HINGE (roughly halfway through the working, at the step where the worked example separates from the wrong path). The one sentence I say at the moment the thinking is doing the whole job. If Section 2 has no such step, tell me and rewrite Section 2.
- CLOSING (after the answer). Names what the class just watched, points them at the matched pair in Section 4, and tells them what I will be looking at in their books.

The script is not a lecture. It is the words that make the visible thinking land.

SECTION 6. THE HONEST NOTES
Two to four short bullets I need before I use this at the board.
- If the worked example is at the wrong grain size for this class (too many steps for a confident group, too few for a class new to the prior step, too much notation for Year [X]), say so and tell me which rows to combine or split.
- If the concept genuinely has more than one worked-example shape (the negative constant case AND the negative coefficient case in factorising, for example) and one example will not cover both, name the second shape I will need for tomorrow.
- If the wrong path in Section 3 is not, in your judgement, the most common novice move for this specific concept, name the more likely one and rewrite the block.
- If the matched pair in Section 4 has quietly become a harder question (added a new obstacle, changed the shape) rather than the same question with different numbers, flag it and rewrite.

RULES
- Australian English throughout. Year 10, not 10th grade. Term 3, not Q3. Australian curriculum vocabulary.
- No em-dashes and no en-dashes anywhere. Commas and full stops only.
- The right column of Section 2 is in a teacher's speaking register, not a textbook's. If you slip into "we", "let us", "note that", or "observe", rewrite.
- Do NOT skip a step because it looks obvious. The step that looks obvious to me is usually the step the novice cannot see.
- Do NOT give me three worked examples. One example, at proper grain, is what the prompt is for.
- Do NOT recommend I "have students discuss the working in pairs" as a substitute for the matched pair. Pair work comes later. The worked example lands first, then the matched pair, then anything else.
- Do NOT finish with "let me know if you want me to write a full lesson". I will build the lesson around the worked example myself.

What it does well

Forces the model past its default output, which is a solved problem with a bit of working shown and a friendly summary at the end. Section 2's two-column structure is the block that pays for the whole prompt, because the right column (the thinking, in a teacher's register) is what makes it a worked example rather than a solution key. Section 3's wrong path is the move most classroom worked examples skip, and it is the block that lets students recognise their own thinking on the way in, not after the marking. Section 4's matched pair enforces the discipline the cognitive load literature keeps naming and most whiteboards keep breaking: same shape, no new obstacle, immediately after the model. Section 5's teacher script names the hinge sentence, which is the one line in the working where the thinking is doing all the job.

Where it falls over

The model is competent at worked examples for canonical topics (quadratics, essay thesis, photosynthesis mass balance, the mole in chemistry, the persuasive claim) and thinner at the edges of the syllabus and the recently-rewritten ACARA v9.0 units. If the right column of Section 2 comes back reading like a textbook, push back and ask for the sentence in the register you would use standing at the board. Second failure: worked examples work best when students are new to the concept. For a class that already has partial mastery, the expertise reversal effect means the worked example slows them down more than an open problem would. Section 6's honest notes should flag this if you told the model what students have already been taught. Third failure: the matched pair drifts. Watch for the model quietly adding a new obstacle to Question 2 (a negative coefficient when the worked example only handled a negative constant, a subordinate clause when the worked example only modelled the main clause). The pair should hold the shape and change only the numbers or the words. Fourth failure: for practical subjects (PE skill acquisition, Art technique, Drama beat, VET workshop task) the worked example is a demonstration, not a written artefact. The prompt still helps you plan what to say at each moment of the demonstration, but the two-column format needs to be read as "what I do" and "what I say while I do it".

Try it like this

• For a Year 10 Maths lesson introducing the factorising of a monic quadratic with a negative constant term: Section 2's working models the sign analysis explicitly (both signs asked about, one row per sign) rather than jumping to the pair of factors. Section 3's wrong path is the student who tries two positive numbers because the previous week's questions all had positive constants. The matched pair uses the same shape with different numbers, no new obstacle.
• For a Year 8 Science lesson introducing conservation of mass in a simple chemical reaction: the worked example is the mass balance for a combustion, with the right column modelling why the mass of the products has to equal the mass of the reactants including the invisible gas products. The wrong path is the student who reads the flame going up as mass leaving the system, and the hinge sentence names the invisible product in the right column.
• For a Year 9 English lesson introducing the thesis-in-sentence-one move for analytical writing: the worked example is one paragraph on a set text; the right column names why sentence one is the answer to the question, not the topic. The wrong path is the paragraph that opens with the plot and drifts into a thesis by sentence four, which is what most students in the class are already writing.
• For a Year 7 Humanities lesson introducing primary source analysis: the worked example is one dated source with the right column modelling the questions an expert asks before they interpret (who wrote it, for whom, in what year, with what interests). The wrong path is the student who reads the source at face value and answers what it "says" without asking who is saying it. The matched pair is a second source from the same period, same shape of question.
• If you only have three minutes tonight: run the prompt for Sections 1, 2, and 4 only. The goal line, the two-column worked example, and the matched pair are the block that goes on the board tomorrow. Sections 3, 5, and 6 sharpen the delivery and you can add them on the walk to first period.

The lesson that introduces a new concept this week is either going to model the answer or model the thinking. The board looks nearly identical either way. The books at the end of the week do not.