# Tips for teaching A-level Maths/Further Maths students

I recently withdrew from a maths PhD to focus on other career options, and one of them that I'm very keen on is mathematics teaching.

I applied for a college A-level maths lecturer role (teaching students aged 16-18), in which I had to do a 20 minute micro-teach session, some exam questions and answer some interview questions about my general attitude and ethos to teaching. I wasn't offered the job but I was offered a very appealing teacher training opportunity which would allow me to train with the college and get a PGCE/CertEd at heavily reduced costs, a generous government bursary, and take part in guaranteed hours of teaching the A-levels partially due to needing to cover for a teacher who is going on maternity leave.

In the education system in the UK, students are very heavily tested and so there is a large tendency for teachers to get students through exams, more so than actually teaching the subject in a more general context. I know that as a college teacher I won't be able to reinvent the wheel with the education system in this country, and I may be pressured into solely focusing on getting good marks out of the students, but I would like to take steps in order to make my maths lessons engaging, rigorous and fulfilling.

Can someone suggest some ways in which I can teach a standard A-level course in a way that allows students to succeed in their exams, but also making it unique and exciting, and without diluting on the mathematical rigour?

• There are many things to consider here. The UK curriculum is undergoing major changes with the Mastery agenda. Do you have any topic in particular in mind as a start point? – Karl Jul 31 '15 at 22:03
• Which topic was the mini lesson? Briefly what did you do? How did you do it? I think asking for advice is a great start and very encouraging for the future. – Karl Jul 31 '15 at 22:19
• @Karl I taught a group of six A-level students about the compound angle formulae from C3/4 trigonometry. I created it in the form of a presentation using LaTeX, including colourful diagrams and engaging visuals. I started by asking students whether the sine function distributes over addition (the answer is no) and then stated the formulae, then providing a visual proof (for the case where the angles are acute and sum to less than $\pi/2$. I then went through some worked examples and then let students do an exam question at the end. Probably a bit too much for 20 minutes in hindsight! – omegaSQU4RED Jul 31 '15 at 22:25
• @Karl I know how to make aspects of the A-level course more rigorous, for example, providing the precise mathematical definition of a function - rather than just a rule $f(x)$ that does something to $x$ to give a value. Also, providing proofs of results that are often taken for granted (provided that this can be done without using techniques beyond the scope of A-level). I understand that there are major changes to the GCSE syllabus, and that there are changes in the A-level course structure, but do you know where I can find out more about this change (e.g. Department for Education/Ofqual)? – omegaSQU4RED Jul 31 '15 at 22:33

Here's how I would have attempted the lesson.

First I would have given the students various cards such as:

• $\sin(A+B)=\sin A+\sin B$
• $\sin(A+B)=\sin A \cos B + \sin B \cos A$
• $\sin(A+B)=\sin(B+A)$

As a rule of thumb 8 cards is usually good. Never do more than 12 It just gets messy. Ask the students to sort them in to piles of true and false. No calculator just discuss and provide a reason for your choice.

Next give them a calculator and ask them to see if they are correct. At the level you are talking about they should be able to reason to substitute values in. You could ask them whether or not they were initially right or what common misconceptions do they think others have? Do the rules always work or only sometimes?

To conclude I would have given them a fact or two such as $\sin30=0.5$ and$\cos45=0.707$ and challenged them to come up with as many other facts as they can.

I probably wouldn't have proved the results. Proof, in my opinion, needs delicate care in lessons. I'll elaborate on that if so wished later.

Best looking at exam boards themselves for latest info. AQA or Edexcel.

Hope this helps.

• The use of the cards at the start sounds like a great idea, that's something I'll have to consider for future reference. I was given feedback from my interviewer that I did try to fit too much into the lesson, but that the students really liked me and one of the observers (a non-mathematician) actually understood the topic after I finished the lesson! – omegaSQU4RED Aug 2 '15 at 1:00
• I'll edit my answer to give more info about the use of cards in lessons when I get chance. – Karl Aug 2 '15 at 4:10