Seven transitions, and the four that get skipped

1  Understanding: real situation to situation model

The solver builds a mental representation of what is going on. This step is invisible, and it is skipped in analysis far more often than it is skipped by students. Failures here look like misreading but are usually a plausible-but-different reading of the situation.

2  Simplifying: situation model to real model

Deciding what matters and what to ignore. This is where assumptions are made, and it is the step school tasks most often perform for the student. A problem that says "a tap fills a tank at a constant rate" has done the whole of step 2 already. The phrase closes off the one interesting question, which is whether the flow stays constant as the head of water drops.

3  Mathematising: real model to mathematical model

Choosing variables, relations and notation. Failures here are the ones teachers see most, because they surface as not knowing which formula to use, but the cause is frequently an incomplete step 2. The informal model is not definite enough to be written down.

4  Working mathematically

The only step school mathematics reliably teaches, and the only one most exercises contain.

5  Interpreting: mathematical results to real results

Turning a number back into a statement about the situation, with units and meaning. Cheap to require, and requiring it changes what students attend to.

6  Validating

Is this plausible? Would a practitioner accept it? If not, the cycle turns again, and that repetition is the point of drawing it as a cycle. It is also the hardest blockage to see, because a solver who has stopped validating does not look stuck. They look finished.

7  Presenting

Communicating the result in the situation. Often dropped as non-mathematical. It is the step that makes the difference between a solved problem and an answered question.

This is statutory content, not enrichment

At A level the cycle is written into the subject content. Overarching theme 3 defines five modelling competencies to be applied across the whole qualification:

• OT3.1: "Translate a situation in context into a mathematical model, making simplifying assumptions"
• OT3.2: "Use a mathematical model with suitable inputs to engage with and explore situations"
• OT3.3: "Interpret the outputs of a mathematical model in the context of the original situation"
• OT3.4: a model can be refined by considering its outputs and simplifying assumptions
• OT3.5: "Understand and use modelling assumptions"

Department for Education, GCE AS and A level subject content for mathematics. © Crown copyright, Open Government Licence v3.0. OT3.4 is paraphrased rather than quoted.

OT3.1 to OT3.4 are the cycle: translate, use, interpret, refine. A task that stops at "compute" delivers two of the five and abandons the rest. At GCSE the same idea appears in one sentence, that models "may be more or less effective depending on how the situation has been simplified and the assumptions that have been made", which names the two transitions school tasks most often skip.

What actually transforms a problem

Across the paired examples in the library, the effective moves reduce to a small set, and none of them make the mathematics harder:

1. Make the given quantity the unknown. Give a deadline and ask when to leave, rather than giving the journey and asking for the speed.
2. Delete an idealising condition. Remove "constant rate", "level ground", "rectangular". Each deletion hands back one assumption.
3. Make the site real. A wall that is already there, fence panels of a fixed length, a drainpipe. Real constraints break the symmetry of tidy answers.
4. Introduce an asymmetric consequence. Late is worse than early. This generates genuine openness more reliably than anything else, because it requires a judgement no calculation supplies.
5. Ask for a recommendation, not a value. Changes the output into something a named person can act on, which pulls in interpretation and validation together.

The move that reliably does not work is changing the surface of the situation while leaving every decision supplied. That is a costume.

See the four paired examples →