I’ve written a couple of times already this year about MEI’s Teaching Advance Mathematics course. Someone on Twitter recommended it to me last year and it’s definitely the best course I’ve been on for professional development. I had my second (and final) observation today and I was thrilled with how well it went.
I was worried that the observation would have to be rearranged at the start of the lesson, as five sevenths of my class were out on a trip that I didn’t know about. But actually it meant I could just work really closely with the two that were there.
The lesson was the first on probability so would be the first time the year 12s had encountered it since GCSE last year. By the end of the lesson, I wanted my students to have a good understanding of two-way tables, probability trees and Venn diagrams and how they can be used simultaneously to represent the same set of data. Once deciding on my desired outcome, I went straight to Integral to look at the resources there. If this is a website you haven’t seen yet, check it out! The majority of the resources are part of a paid-for subscription but there are some resources that you can access for free.
It took a while to plan the lesson, but eventually I was happy with what I had. Prior to the TAM course, I would have done very little planning for an A Level lesson. I used to just introduce a concept then ask my students to practise a process, which is ridiculous as it’s not how I teach any other key stage! Many of my lessons now are resource-heavy and use resources that prompt students to ask questions, rather than give them questions to answer. I also try to include scaffolding in my A Level lessons in the same way I do in my other lessons.
The feedback from the lesson was generally positive and it was great to have an opportunity to reflect on a lesson with someone else. Most observations I’ve had this year have come with little or no feedback, so it was brilliant to have so much detail for this one. I had asked my observer to focus specifically on my questioning as that’s an area of pedagogy I’ve been working on all year. Closed questions are easy because they feel so ‘safe’, although I’m getting more confident with open questions as my subject knowledge develops. Generally I’m finding that “convince me” questions are developing the understanding of my students and this was reflected in today’s feedback. Such a simple phrase is having a big impact on discussion in my classroom – previously, students would often answer my questions with a question, for example “it is…?” and “could you…?”, whereas now they have to have more confidence with themselves. I also say, “are you asking me or telling me?” to encourage them to think more carefully about what they’re saying. My observer suggested I add, “what do you know for sure?” to my repertoire as this is a great tool to use when students are completely stuck and scaffolding has not helped.
Following the observation, my observer and I discussed the fine line between scaffolding and funnelling, or giving too much away. I’ve been advised to read Hewitt’s 1999 paper on arbitrary and necessary maths where he suggests that arbitrary maths is anything that cannot be learned (for example calling a square ‘a square’) and necessary maths is anything that can be found (for example the area of a square). It’s on my desk and I’m looking forward to reading it over the next couple of days. I’m hoping this will help me further develop my questioning.
We discussed the language of probability too. ‘And’ and ‘or’ can be particularly confusing because ‘or’ in maths means ‘and’ in English!
Student: So this means right-handed with blue eyes and right-handed with brown eyes?
Me: Yes… but we use ‘or’ instead of ‘and’ in maths…
The student was correct in his observation – in order to obtain a probability of 18/29, we did need both of these categories. But mathematically we needed to say “right-handed with blue eyes OR right-handed with brown eyes”. It seemed the more we discussed it, the more complicated it became! Especially when you include the union and intersection symbols, their names and that they mean add or multiply respectively. Throw the complement of a set into the mix and it’s quite a literacy-heavy lesson.
One part of the lesson that made me feel a little uncomfortable was when one of the two students compared two Venn diagrams, one showing A’ U B and the other showing A ∩ B’. He asked, “are these two inverses? You know, one would be a reflection of the other if you drew them on a graph?” Part of me was thrilled that he was making connections with topics that we had previously covered, and obviously I was pleased to see that he had recognised they were complements of each other. I clarified with him that actually they were mutually exclusive events. When we looked at probability trees again later in the lesson, I reiterated that each pair of branches summed to one which showed mutual exclusivity. Later on, during the post-observation discussion, we discussed whether it could be graphed or not (we didn’t actually come to a conclusion) or whether a Venn diagram was already a graph.
If I were to teach the lesson again, I would start with something more concrete to start with, such as a sorting activity. I was a little bit miffed with myself as I’d included a task in the lesson earlier on then taken it out, which required students to sort the numbers into a Venn diagram where A is the set of prime numbers and B is the set of odd numbers. I took it out as I thought it would be too easy for a year 12 lesson, whereas it would have actually given a concrete example to keep referring back to. I’m going to start my next lesson with this, then ask the class to sort the data into a two-way table and a probability tree too. I’ll then start using set notation from there.
My observer asked me to identify two areas that I’m going to continue to work on. One was the use of scaffolding in relation to Hewitt’s work on arbitrary and necessary maths. I’m going to start unpicking lesson objectives into what is arbitrary and what is necessary so I can move towards only giving my students arbitrary information and getting them to work out the necessary information using what they already know. The second thing to work on is adapting lessons, and I’m hoping this will develop more with my subject knowledge.
I’m really pleased with how my lesson went and how it’s identified new areas to work on. I really enjoyed the feedback and discussion afterwards. We really don’t have enough time in our busy working days to just talk maths and pedagogy any more. If you haven’t already participated in the TAM course, try and get on it for next year. Not only will it benefit you and your students, but your school will get paid to send you. Seriously, it’s fantastic CPD. Thank you MEI!