In part 1 (available here), I outline how I taught surds for the very first time. This is part 2.
The second time I taught surds, I had put significantly more thought into my planning. I had thought about common misconceptions and mistakes that I had seen students make previously, and so I had considered how I would address these. I had discussed the topic with more experienced colleagues, and had drawn upon their expertise. I had decided on a logical progression through the content which needed to be covered, and so I was feeling pretty pleased about how this would go.
In the weeks leading up to introducing surds to my new class, I had consistently included questions in their Do Now to ensure that they were fluent with their square numbers. For those that weren’t, I had explained to them that they would find our next topic significantly easier to access if they were able to recall these without even having to think, and provided them with a homework task on our online platform to help with this.
The first lesson arrived. By the end of this lesson, I wanted all pupils to be able to explain what a surd was, as well as being able to simplify surds. This felt achievable. I began by showing them an isosceles right-angled triangle, where both legs had length 1. Asking how we could determine the length of the hypotenuse, a significant proportion recalled that we could use Pythagoras’ Theorem. A couple of pupils correctly talked me through the procedure, which I modelled on the whiteboard, and we concluded that the hypotenuse was of length root 2. One pupil commented ‘okay, so the answer is 1.414213562 then, so 1.41 to 2 decimal places’.
This obviously led nicely into me discussing what surds were and why they were useful – we spoke about the story of Hippasus, which they were pleasingly interested in (or possibly, just humouring me), and then we looked at a couple of surds and non-surds. (root 3 is a surd, root 4 is not a surd). I then presented them with root 12 , we discussed if it was/wasn’t a surd, and when I felt convinced that all pupils could articulate why it was a surd, I explained and modelled the process of simplifying.
After this, we went through a few more examples as a class of how to simplify surds, and pupils completed some questions on mini-whiteboards, which meant I knew exactly who would initially quire some additional input from me. Following that, pupils did some practice of the skill. I had given them a set of questions (and I genuinely can’t remember which questions – either a physical textbook or an online worksheet) and they got on fairly well with this. In fact, the majority of our lessons on surds seemed to go quite well.
Surds are a higher GCSE topic, and they are something that a lot of pupils find conceptually difficult, but through assessing my groups’ performance in class (by using mini-whiteboards and multiple choice questions and just looking at their written work), I was so impressed with how they were doing. I felt quite happy with my surds planning and delivery this time round – pupils had made great progress, I reasoned – bring on the next topic!
As you may well have realised, things were not quite this straightforward. I try to regularly incorporate topics from previous lessons/weeks/terms into current lessons as much as possible, to ensure that pupils’ learning is retained and not forgotten over time (still got some work to do on improving this). Now, a curious thing was happening: whenever they came across a surds version, they would start manipulating them in all sorts of weird (for which you can read incorrect) ways.
For example, I once presented them with this question:
Simplify root 20.
All pupils had excelled with this in class time, but when exposed to it later, and out of context, less than half of all pupils obtained the right answer. Several had written that the answer was root 5, which I was baffled by. When I queried this, I encountered this response: ‘well, that’s how you simplify, isn’t it? You just keep on dividing by 2’. Ah.
Evidently, there was still some work to be done. Even more commonly, pupils would simplify root 20 to 5 root 4 – this is not an uncommon misconception, but still an alarming one. These errors that were being made by pupils were serious, as they suggest that the fundamental concept of what a surd is and the basic rules for manipulating surds had possibly never been understood by my pupils. Looking back now, I think that my careful modelling and examples and practice time had resulted in a class who could replicate a procedure, but that was it.
I am a huge advocate for procedural fluency in mathematics, on the basis that when key procedures are automatized, working memory can handle complicated parts of a problem with ease, and I would have argued that my initial series of lessons on surds with that class was about developing procedural fluency. The main objection to that, though, is that it had failed. My pupils were not procedurally fluent.
Worse than this, was that I had always intended for them to be procedurally fluent alongside having a deeper understanding of the concepts at hand. Conceptual understanding, after all, had to be our ultimate goal. I had really tried hard to communicate the concept of a surd clearly, and whenever I was modelling a process, I was full of questions to try and promote this conceptual understanding. John, why do I need to find a factor that is also a square number? Ally, which of these is a surd? Which isn’t a surd? How do you know? Convince me. Convince me. Convince me.
Despite this, it hadn’t worked. This is more difficult to admit than with the first time I taught surds. This time round I had thought much more carefully about how best to communicate relevant mathematical ideas, and yet it had still resulted in pupils thinking that to simplify a surd, you can just divide by 2. Admittedly, some of the class had continue to demonstrate a good level of performance over time (and hence, I feel reasonably confident in asserting that learning had taken place), but this is not a good enough outcome – I wanted all of my pupils to be at this stage.
As such, I went away, did some reading and some hard thinking and some more discussion with colleagues about what had and what not worked.
Part 3 will be the story of what I do now, and why I think it is the most successful way (spoiler: minimally different questions may be mentioned).