Having outlined in part 1 (here) and part 2 (here) how I used to approach teaching surds (what a surd is, multiplying, dividing, addition/subtraction, expanding brackets, rationalising the denominator), alongside why these approaches were unsuccessful, I’m now going to share what I do now, and why I think it is contributing to high levels of long-term learning.
When I now begin teaching a class about surds, I will always have assessed that they have certain prerequisites in place: the most important of these is that they are fluent in recognising all squares and square roots up to 225. If a pupil is unable to do this, simplifying surds will become highly inaccessible. My strategy for doing this remains the same as discussed in part 2.
Where my approach starts to change, however, is in that very first lesson. Previously, I would have introduced the topic via Pythagoras’ Theorem, before giving them the official definition of what a surd is. However, although students could repeat this definition, they often lacked a deeper understanding of the concept of a surd. As such, in that first lesson, I present them with this sequence of questions. On a practical level, I display each question on the whiteboard one at a time. I then silently circle the correct answer. I complete the first slide in silence. Pupils, during this time, are entirely silent, and are not writing anything down. They are simply watching, and thinking about what might be going on.
Once this is complete, pupils will complete these 5 practice questions, one at a time, on mini-whiteboards. This will allow me to see who is correctly spotting the rule for what a surd is/is not, alongside who is not. Depending on how this goes, I will make a decision on what to do next. Sometimes, it will make sense to show some more examples; sometimes, where the majority of pupils are confident, we will discuss a definition.
The real advantage of structuring the lesson in this way, is that pupils are often fairly adept at pattern-spotting and identifying rules, and so I’m allowing the opportunity for that to be utilised. By showing both examples and non-examples, pupils will begin to define for themselves: ‘okay, so it’s not a surd when it’s a square number?’. Of course this is not the only definition I want my pupils to have! I will then carefully explain the definition, and we will discuss rational and irrational numbers, but because they already have a concept forming of what a surd is, I have found they are far more able to recall and use this definition.
Following on from this, I wanted my pupils to be able to simplify surds. The procedure for this is not something which is likely to be discerned by pupils from viewing examples and non-examples; I have found that is more beneficial to begin with an example-problem pair (please buy and read How I Wish I’d Taught Maths by Craig Barton on this!). I show them an example, on the board, in silence, which pupils watch. After that, I narrate the steps, and pupils then copy it down, before beginning a very similar question independently. If a significant proportion of pupils are struggling to complete these independently, I will complete a second example. Once most pupils are showing that they can replicate the procedure (and initially, this is all I am assessing), they will work through these questions independently.
I really like these questions. The first one is straightforward. The second one is no more difficult, but here I want pupils’ attention to be drawn to the fact that they are both in the form ‘a root 2′. Why is this? How could we have inferred this from the question before carrying out the procedure? This is followed by another question where the answer follows this same pattern. Again, this is useful to discuss with pupils: hang on, root 8 simplifies to 2 root 2, and 32 is 4 times bigger than 8, so shouldn’t it be 8 root 2? Why is that not the case?
(It is worth pointing out at this stage that my pupils (who by now are fairly well practised in reflect, expect, check) found the expectation stage really hard this time round! They kept finding that what they expected to have happened wasn’t happening, hence it was so important to retrospectively expect. Why did this happen? What is happening here?)
Following that, question 4 obviously simplifies to 8. This is only obvious to an expert! With my class, who were all more than able to inform me that 8*8=64 and that the square root of 64 is 8, continued with the procedure that they had just been shown. Upon reaching answers of 8 root 1 or 4 root 4, I then encouraged them to use what they already knew – oh yes, of course that’s 8. This allowed them to see that 8 root 1 and 4 root 4 and 8 are all equivalent, which I would never have previously exposed them to explicitly.
After this, there are three questions in the form 8 root a. Again, this needs to come with follow up questions and discussions. What is it about these surds that means they must simplify in this way? Write me another surd which will simplify in this way. I then want to expose them again to the fact that root 80 is half of root 320. This is curious and surprising to a novice learner of surds! Why is that? What had you expected? Why was this expectation incorrect?
Root 40 still feels like it must be half of root 80, except it isn’t. How can we tell that it isn’t from the simplified forms? This is a really common misconception – I want to draw it out in the earliest stages of instruction, before it becomes embedded and thus harder to correct.
There are then a few questions which allow pupils to refer back to their knowledge of square numbers, and hopefully receive some confirmation that their expectations are correct. This obviously does not always happen. So, root 16 is 4, so root 160 must be 10 root 4! Are you sure? Let’s check. Ah, what mistake have you made? Root 1600 has then been intentionally included – wait, our answer is no longer in surd form – therefore what type of number must 1600 be?
The final few questions give pupils the opportunity to practice and consolidate their understanding.
Now, this set of questions are often completed far more quickly by pupils than a ‘random’ set of simplifying surds questions, because they can use the patterns that they begin to spot to help speed up the procedure. Pupils, throughout, will be encouraged to write down what they are spotting and when they come across a surprising answer – we will discuss these as a class. This has promoted high quality mathematical thinking, reasoning and communication as being really key to success in my classroom: pupils are not just thinking about how to copy a procedure, but they are also thinking about the underlying mathematical structure.
This, however, is most definitely not the end of the story. Pupils will continue to be exposed to ‘simplifying surds’ questions in Do Now activities over the coming lessons, months and year. This opportunity to regularly retrieve knowledge is fundamental if pupils’ long-term memory is going to be affected. They will also meet this topic in other contexts – Pythagoras’ Theorem is the one that springs most readily to mind. However, I don’t just want my pupils to practise simplifying surds – while procedural fluency is key, I want them to think mathematically and so I need to provide them with opportunities to do so.
In recent months, I have started making use of Venn Diagrams – primarily from www.mathsvenns.com.
This is a fantastic task. It is open ended and really allows for creativity, and as a tool of assessment, means I can more accurately assess who is happy with the procedure, and who is happy with the underlying structure. I may also use this task when revisiting surds – e.g. with a year 10 or year 11 class who have previously been exposed to the key concepts and procedures. I do not want to use this with pupils before they have the relevant content knowledge in place, or it is likely to cause confusion and leave pupils perceiving surds in a negative way, if they are not given the chance to experience success quickly.
I will also use a variety of problems taken from the fantastic www.openmiddle.com, such as this one below.
Again, this is incredibly powerful. Pupils do not view maths lessons as simply ‘the teacher does examples, we do questions, we get them right or wrong’, but as the opportunity to create and question and conjecture. It is certainly an example of a ‘problem worth solving!’.
Overall, I think this approach to teaching surds has been my best yet. Obviously, it is not perfect and without need of refinement. Some of my pupils still hold misconceptions which I will need to continue to discover and challenge. Some of my pupils will still make ‘silly’ mistakes. Some of my pupils will still forget with alarming regularity. However – on the whole, it feels like we’re getting there.