How I first taught surds
During my NQT year, I taught a top set Year 9 class. During our first few weeks, I became quite frustrated with the fact that whatever I tried to teach them, they had seen before. Multiplying decimals? Check. Product of prime factors? Check. Fractional and negative indices? Check. Although I was able to challenge and extend their existing understanding, I was desperate for the opportunity to teach them something new. As such, I was delighted to see ‘Surds’ (simplifying, multiplying, dividing, expanding brackets, rationalising the denominator) appear on our SOW – finally, here was my chance!
Disappointingly, it was a disaster.
For some context, I had spent my training year in that same school and things had generally gone reasonably well. For more context, a big focus on that school was on students’ independence and learning for themselves and using technology to support this as much as possible. A lesson in which a topic was explained carefully and precisely by the teacher, before giving students the opportunity to practise independently would not have been well regarded (or, at least, this was a widely held perception at the time).
It just so happened that my first lesson on Surds with this Year 9 class was going to be observed by my NQT professional mentor, and as you would expect, I wanted to impress. I went to Google, typed in any combination of ‘surds’ ‘introduction’ ‘fun’ ‘investigation’ ‘discovery’ and ‘lesson’, and came across a carousel activity. This seemed perfect. It would minimise teacher talk and allow the students to be really active in their learning. I downloaded it, made a few small tweaks and printed it, confident in the fact that I was doing the exact right thing. (I should add: this is not intended as a criticism of this resource – more that I used it incredibly poorly).
The day of the lesson arrived. I was astonishingly well-resourced: I had a booklet made for each student, and hint cards ready to hand out if anyone was struggling. I had put my students into carefully considered groups of 3. I had a timer on the board, to provide students with a visual reminder of how long they had to complete each stage of the carousel. My learning objectives with differentiated three-ways (‘All, Most, Some’). I had an exit ticket all prepared, and lots of mini-whiteboards on hand. This could only go well.
However. In addition to being astonishingly well-resourced, I was astonishingly badly planned. I had not considered what prior knowledge my students would need to access ‘surds’, and I hadn’t thought about how to assess if that prior knowledge was in place. I hadn’t thought of the sequencing of the content. I hadn’t thought about common misconceptions, and if it would be better to tackle them explicitly or respond to them as and when they occurred. I hadn’t thought about how best to draw links between surds and other areas of the curriculum. Given this, it is unsurprising things did not develop as I had hoped.
In the lesson itself, I began by telling my class ‘today, we’re going to do things a bit differently. I’m not going to tell you anything about surds, but you’re going to use the information I’ve given out to learn about them for yourself.’ There would have been five different stages to the carousel activity (‘what is a surd?’ ‘simplifying surds’ ‘multiplying and dividing surds’ ‘expanding brackets’ ‘rationalising the denominator’). Each group was a given a stage of the carousel, and they had approximately 10 minutes to read the notes and examples, before trying some practice questions. I was constantly flitting between groups and handing out hint cards where appropriate.
Very quickly, students went off task. Although I wanted them to be able to discuss the maths, and so I knew the class wouldn’t be silent, I knew a significant proportion were not focused on what they were being asked to do. I continue to move around the classroom. Students were telling me ‘I don’t get it, I don’t understand’ – but, I reasoned, this was because they were not engaging in the way that they should be. Students weren’t even bothering to read the hint cards. If they actually read the hint cards, they would understand! How could they not?
A few students (who were female, very conscientious and extremely highly-attaining) were working really well and persevering. ‘Fantastic!’, I thought. This shows that this is an appropriate task, but it is the behaviours of the others which means they are not learning as well as they could be. These students would want to ask questions: ‘miss, with expanding brackets, I don’t understand which terms we multiply?’, but I felt unable to stop and help deal with misconceptions because I needed desperately to re-engage the pupil who had just thrown a pen on the other side of the classroom.
The lesson ended. I felt fairly certain that only a minority of my pupils had learnt anything whatsoever in that hour. Despite this, the feedback from my observer (a non-Maths specialist) was overwhelmingly positive. He remarked that giving them the opportunity to learn independently was ‘exactly what our students need’ and that it was a case of persevering with similar tasks to build up their resilience. Yes, the students were noisy, but that was a sign that they were engaging well in the task. However, when I collected in books, it was very apparent that despite this ‘engagement’, learning (or, to be more precise, performance) had been minimal.
In particular: the students who had completed the carousel activity in the order outlined earlier had completed far more of the practice questions than those who had been given the ‘expanding brackets’ or ‘rationalise the denominator’ questions first. This is entirely unsurprising. In order to begin attempting rationalising the denominator, pupils will already need to be fluent with simplification of surds and multiplying surds; they had not been given the opportunity to develop this fluency. I regret so much about how I delivered this lesson, but particularly the fact that I had not thought about this sequencing, and consequently, disadvantaged multiple students.
In our next lesson, having reflected upon just why the first had been so unsuccessful, I made an attempt to go ‘back to basics’ with surds. I would introduce what surds were, and demonstrate the process of simplifying them. This still did not go well. The majority of pupils in this class (who were typically highly motivated, and would normally display a positive attitude towards learning maths) had decided that surds were difficult, and their mind-set was very much ‘I can’t do surds.’ I tried my best to work around this: I would make everything as simple as possible, by breaking down each step in any procedure. I checked that they could quickly recall all the square numbers up to 225. I told them the story of the Pythagoreans and root 2. Ultimately, it was still a real challenge. I tried countless strategies over the next few weeks, but that initial exposure had left them believing that surds were ‘basically impossible’.
We moved on with the SOW, and I was lucky enough to have some spare at the end of that academic year, in which I retaught surds. Luckily, this was for more successful! So successful, in fact, that one student commented to another (who had joined midway through the school year), ‘honestly, Miss made us do these before and we all hated them but I’m not sure why’.
More to follow in Part 2!