At a recent Year 11 Parents' Evening, I was guilty of hypocrisy. I reassured parents that their daughters would be very well prepared for their maths GCSE because we'd be doing half a term of revision lessons. Most parents nodded happily, but one dad challenged me. He asked why I didn't take the opportunity to teach his daughter some calculus. Good point! I've written many times about the importance of imparting mathematical knowledge without focussing on exams (a 'no teaching to the test' philosophy), and there I was promising an extended period of revision instead of mathematical enrichment.

I don't want to teach my students to pass exams. I want to teach them mathematics, and I want passing exams to be a welcome by-product of that. But what if some of my students fail to meet their potential in their GCSE exams this summer as a result of my decision to reduce revision time? It's a dilemma.

I don't want to teach my students to pass exams. I want to teach them mathematics, and I want passing exams to be a welcome by-product of that. But what if some of my students fail to meet their potential in their GCSE exams this summer as a result of my decision to reduce revision time? It's a dilemma.

**Bitesize enrichment**

The day after Parents' Evening I'd planned a standard lesson on algebraic proof with my Year 11s (by 'standard' I mean explanation, examples and practice). At the start of the lesson, by way of introducing the topic of proof, I mentioned that many mathematicians spend their lives working on proofs. As I said it I was reminded of the BBC Horizon episode Fermat's Last Theorem. So I told my students the story behind it. I drew a page with a margin on the board and wrote 'I have a proof of this theorem, but there is not enough space in this margin'. When I told them that Fermat dropped down dead before sharing his proof, they were enthralled! I googled the Homer Simpson picture below. One girl quickly tapped the numbers into her calculator - "oh my god, it works! Homer solved it!", she shouted. The class were genuinely interested. I explained that it was a 'near-miss' solution - I know about this because I've read Simon Singh's book The Simpsons and their Mathematical Secrets and I wrote a subsequent blog post about teaching a lesson on Fermat.

It was only ten minutes of spontaneity in a lesson, but so very worth it. My students went home from school having learnt something new about the world, and that's what education is all about.

The success of this unplanned tangent gave me the confidence to start diverting from my lesson plans all over the place. In a Year 10 lesson on percentages I got carried away explaining how banks make money from interest rate spread. I know a lot about finance from my previous career so I felt confident talking about this. But I admit that my knowledge of 'non-curricular' mathematics is limited. I have a pile of mathematical books that I'm desperate to read but never get time. Reading around your subject gives you the knowledge and confidence to go off on these interesting tangents. I

**must**read more! If you're on Twitter then do get involved in @MathsBookClub for encouragement and inspiration.

How well read are you? Book recommendations from matheminutes.blogspot.co.uk. |

**Barriers to spontaneity**

I don't deny that schemes of work are necessary. For one thing, they ensure essential coverage of all the topics required for public exams. In addition, synchronisation of teaching allows simultaneous assessment of whole year groups, and gives scope for departmental collaboration. Schemes of work are also a mechanism for implementing a maths department's teaching strategy or philosophy. So they are '

*a**good thing'*even though they may underpin a frustrating lack of freedom.
I was recently teaching 3D trigonometry to a top set Year
9 (very smart girls - many will take their GCSE in Year 10). I was going through the example below from CIMT. When we got to part (d) we
drew out the triangle in question and established that it was non-right angled. They haven't done the Sine and Cosine Rules yet so I said we'd skip it. I told them that they'd cover non-right angled triangles next year and they accepted that without question.

It's a bit sad isn't it? I was disappointed by
their lack of intellectual curiosity, but more disappointed with myself for
telling them they'd have to wait a year to finish solving the problem. Why didn't I just launch into teaching the Sine and Cosine Rules? Firstly, because of time pressure - the class would have fallen behind their peers if we'd spent an extra week or two on trigonometry, putting them at a disadvantage in their next Year 9 assessment (this would have upset them). Secondly, inconsistency of topic coverage would create problems for next year's teachers. Thirdly, I hadn't thought in advance about how I'd teach non-right angled trigonometry. It takes a very confident and experienced teacher to deliver a good lesson off the cuff. I need at least 10 minutes thinking time before I teach! So I left it at 'we'll do it next year'. Most unsatisfactory.

To some extent, schemes of work limit our ability to go off on interesting mathematical tangents. But it could be worse. We could be forced to use pre-written lesson plans, slides and resources. I enjoy designing tailor-made lessons for my students. I like having scope to be creative and I like being able to divert from my plan during a lesson if I want to. It's nice to have this autonomy. Ideally our schemes of work are flexible enough to accommodate these worthwhile diversions. If you're designing a scheme of work for next year, bear this in mind. Time for tangents is most welcome.

Thank you for this post. I think that being able to do this kind of thing is at the very heart of being able to communicate a love of learning, and is hence at the heart of why having a face to face enthusiastic teacher is so important.

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