Problem Solving and Communication
The Problem Solving and Communication domain tests the process of tackling unfamiliar quantitative situations and explaining the solution clearly. It rewards candidates who define quantities, choose representations, carry out a plan, interpret the result in context and communicate each step precisely.
Core rules and concepts
- A reliable cycle is: understand the problem, devise a plan, carry out the plan, and review the result. The review stage checks units, reasonableness and whether the question asked has actually been answered.
- Translate words into symbols by naming unknowns and stating units. A diagram, table or equation is a communication tool, not decoration.
- Rate problems add as rates only when the quantities are rates of the same job: combined rate = sum of individual rates.
- Keep exact values until the final step where possible; round only after deciding the required precision.
- Communication requires showing the reasoning, not just the answer: state the rule used, label intermediate results and include units in the final sentence.
- Check by estimating, substituting back, or solving the problem by a second method.
Worked example
One painter can paint a room in 6 hours; another can do it in 4 hours. Working together, how long will they take? Let the room be 1 job. The first painter’s rate is 1/6 job per hour; the second’s is 1/4. Combined rate = 1/6 + 1/4 = 2/12 + 3/12 = 5/12 job per hour. Time = 1 ÷ (5/12) = 12/5 = 2.4 hours, or 2 hours 24 minutes. The answer is reasonable: together they are faster than the faster painter alone but not instantaneous.
Traps that cost marks
- Averaging the two times instead of adding the rates.
- Giving 2.4 hours as ‘2 hours 4 minutes’; 0.4 hour is 24 minutes.
- Rounding a rate too early and carrying the error into the final answer.
- Solving for an intermediate quantity and failing to answer the question asked.
- Omitting units or giving a bare number when the context requires interpretation.
Pedagogical Content Knowledge and Instructional Practice in Mathematics
The Pedagogical Content Knowledge and Instructional Practice in Mathematics domain tests how well a mathematics teacher can turn subject knowledge into learning. It examines lesson planning, task design, representations, questioning, mathematical discourse, misconceptions, differentiation, formative assessment and feedback in the mathematics classroom.
Core rules and concepts
- A learning objective should name observable mathematical learning, not an activity. ‘Pupils will explain why equivalent fractions have the same value’ is assessable; ‘pupils will do a worksheet’ is not.
- Tasks should be chosen to expose mathematical structure. Multiple representations - concrete, pictorial, symbolic and verbal - help pupils connect ideas rather than memorise procedures.
- Misconceptions are predictable and logical from the pupil’s point of view. Diagnose the reasoning before reteaching; a wrong answer may reveal a sound but misapplied rule.
- Questioning should move from recall to reasoning: ‘What did you notice?’, ‘Why does that method work?’, ‘Can you find a counterexample?’
- Formative assessment is used during learning to adjust teaching. Exit tickets, mini-whiteboards and targeted probing questions give evidence of understanding.
- Differentiation can change support, representation or pacing while keeping the same mathematical goal; it should not lower the goal by giving only easier procedures.
Worked example
A pupil writes 2/3 + 1/4 = 3/7, adding numerators and denominators. The teacher’s reasoning should be: the pupil is applying a whole-number addition pattern to fractions and has not yet internalised that the units must be equal. A suitable response is to compare the claim with benchmarks using an area model: 2/3 is greater than 1/2, so the sum must be greater than 1/2, but 3/7 is less than 1/2. Then partition rectangles into twelfths to show 2/3 = 8/12 and 1/4 = 3/12, so the sum is 11/12. The pupil explains why the denominators must name the same-sized parts before adding.
Traps that cost marks
- Equating a busy activity with mathematical learning.
- Correcting the answer without diagnosing the reasoning.
- Using an assessment that does not match the stated objective.
- Asking only closed questions and then assuming silence means understanding.
- Planning differentiation that removes the core concept from some pupils.