Quality of Written Communication in Maths: What Examiners Expect

Three Types of Written Response in Maths

Not all written communication marks are the same. There are three distinct types, and each requires a different kind of response.

• Give a reason: This requires you to state the specific mathematical fact that justifies your calculation. For angle questions, this means naming the theorem: angles on a straight line add to 180° or vertically opposite angles are equal. For number questions, it might mean stating a property such as a prime number has exactly two factors. The reason must be the mathematical theorem or rule, not a description of what you calculated.
• Explain: This requires you to connect the mathematics to the context of the question. In a statistics question comparing two data sets, a valid explanation names the measure, gives both values, and interprets the difference. The median of Group A is 52, which is higher than the median of Group B at 45, showing that Group A scored higher on average is an explanation. Group A did better is not.
• Prove or Show that: This requires a complete, gap-free logical chain from the given information to the required conclusion. Every step must be shown and justified. Using specific numbers as examples does not constitute a proof — algebra must be used. Writing Let n represent any integer before your working signals to the examiner that you understand the generality required.

Worked Example 1 — Geometric Reasoning

Question (3 marks): Find angle x, giving a reason for each step. Lines AB and CD are parallel. Angle AEF = 55°, angle EFG = 70°. Student A (earns all 3 marks): x = 180° − 55° − 70° = 55°. Reason for step 1: angles on a straight line add to 180°. Reason for step 2: alternate angles are equal because AB and CD are parallel. Student B (earns 2 out of 3): x = 55°. No reasons given. The student earns M1 for a correct method and A1 for the correct answer, but loses B1 because no reason is stated for either step. Student C (earns 1 out of 3): x = 55° because the lines are parallel. This student has identified that parallel lines are relevant but has not named the specific theorem. Because the lines are parallel is not a reason — it restates the given information. The mark scheme requires the specific theorem name: alternate angles are equal or co-interior angles add to 180°. The Cambridge IGCSE 0580 examiner report (Jun 2023) notes this directly: when finding angles in geometry, candidates must state the reason for each step. The reason carries marks that are completely independent of whether the numerical answer is correct.

Worked Example 2 — Statistical Interpretation

Question (4 marks): Class A median: 58, IQR: 12. Class B median: 52, IQR: 20. Write two comparisons between the two distributions. Student A (earns 4 marks): The median of Class A (58) is higher than the median of Class B (52), showing that Class A scored higher on average. The IQR of Class A (12) is lower than the IQR of Class B (20), showing that Class A's scores were more consistent. Student B (earns 0 marks): Class A did better than Class B and was more consistent. Student B has the right idea but earns nothing. Did better and more consistent are not mathematical comparisons. The mark scheme requires three elements in each comparison: the correct measure named (median, not average; IQR, not spread), the specific values from both data sets, and a word or phrase interpreting what the difference means in context. Missing any one of these three elements typically costs the mark.

Worked Example 3 — Algebraic Proof

Question (4 marks): Prove that the sum of any three consecutive integers is always divisible by 3. Student A (earns 4 marks): Let the three consecutive integers be n, n + 1, and n + 2, where n is any integer. Their sum is n + (n + 1) + (n + 2) = 3n + 3 = 3(n + 1). Since 3(n + 1) is 3 multiplied by an integer, the sum is always divisible by 3. Student B (earns 1 out of 4): 1 + 2 + 3 = 6. 6 is divisible by 3. 4 + 5 + 6 = 15. 15 is divisible by 3. Therefore it is always true. Student B has demonstrated the result with examples but not proved it. Examples cannot prove a general statement. The mark scheme awards B1 for a specific correct example but gives no further marks because the algebraic generalisation is absent. The AQA 8300 specification is explicit: proof questions require all steps shown in algebraic form. The key phrase in Student A's proof is where n is any integer. This establishes that the result applies to all integers, not just the examples chosen. Without this phrase — or something equivalent such as Let n be an integer — the proof is technically incomplete.

Board-Specific Notes on Written Communication

Each exam board has specific patterns and conventions for written communication marks.

• AQA 8300: Written communication marks are flagged with an asterisk on the question paper. On these questions, the mark scheme gives explicit guidance on what constitutes a clear, logical, and linked response. Questions about ratio, best value, and comparing data sets frequently carry these marks at Higher.
• Edexcel 1MA1: Hence or otherwise instructions appear on proof and derivation questions. Using the hence method following on from the previous part typically earns all available marks. Some mark schemes specify hence only, meaning an alternative method earns no credit.
• CIE 0580: State the geometric reason is explicit in every angles and circle theorem question. The exact theorem name must be given. Abbreviated reasons such as alt angles instead of alternate angles are equal are usually accepted — check past papers for the accepted phrasing.
• OCR J560: Explain your answer questions require contextual interpretation. For statistics questions, always reference the specific measure and both values. For number questions, connect the calculation back to the original question context.

Building the Habit

Written communication marks are not difficult once you know what is required. The challenge is building the habit of writing reasons automatically, even under exam pressure. The most effective practice is to complete past paper geometry questions and angle problems with one rule: never write a calculation without also writing the theorem name that justifies it. After a few weeks of practice papers with this rule, the habit becomes automatic. For proof questions, practise writing Let n be an integer at the top of every algebra proof before you begin the calculation. For statistical comparisons, use the template: the [measure] of [Group A] is [value], which is [higher/lower] than the [measure] of [Group B] at [value], showing that [Group A] [interpretation]. The marks are there. The language is learnable. Practise writing it until it is as automatic as the calculation itself.