Mathematical Language in New Spec Probability
Our role at MarkIt is to write fresh exam style questions that fit the purpose of the new A Level Maths syllabus so students get a true flavour of the exam and are not blindsided by the many subtle changes coming up.
So naturally, we need to think like examiners and write questions that do not just look like rehashed old spec questions. However, this week I was struck by just how much some new questions on Probability actually look unchanged. What has changed though, is the expected ‘meatiness’ of the answers students will need to compose. This comes from new requirements to test the use of Mathematical Argument and Language.
A subtle change in Probability
The questions below are very similar on first glance. In part (d), students are asked to find a conditional probability. In the new spec, the same part is worth more marks because it demands an explanation.
Ambiguous answers to part d :
Ambiguous answers are often correct, but incomplete. Students often struggle to understand what a 'good enough' answer is.
Neither of the above answers provide an explanation that can be given full marks. Students will need to understand that an explanation is not the stating of two facts, but the link between them. The link here is how being independent affects conditional probability, stating independence is an incomplete explanation that may have sufficed last year but may not cut the mustard this year.
A good answer to part d :
This student understands the meaning of independent events and the meaning of conditional events and is therefore able to correctly explain the link using very simple language. All students should be able to make this connection and we as teachers should encourage the learning and explaining of mathematical results as much as possible, now that we know it has a deliberate place in the new specification.
But how about a GREAT answer?
This answer is the 'grownup' version of the one above. We all want students to be able to write maths in maths and this answer exemplifies the portrayal of mathematical reasoning in a manner that cannot be debated. The use of notation is encouraged and admired in the new spec and the expectation is that we use it more regularly as part of our lessons so students become comfortably fluent in composing their own mathematical arguments.
How this mindset can be created
When asking a student a question about the conditional probability of independent events, do not tell them if they are right or wrong until they have fully explained why they think their answer is correct.
Another favourite of mine is asking why completing the square is a great way to find the vertex of a quadratic. Students typically learn the formula and learn which constants to pick - the application of curve transformations to explain the reason for the vertex adds a vital spotlight on how results can be explained mathematically.
A Level students have always known that they need to know the topic well before doing exam style questions. In the new specification there is an added need for students to be able to defend their reasoning. We can prepare them for this by demanding explanations from ourselves and our students as part of learning a topic.