Reflections on Dividing Fractions

Since I first started teaching, I’ve always been somewhat unsure of how to approach the topic of dividing fractions by fractions. While the procedure is relatively easy to perform, giving students the opportunity to see why and how it works always proved much more challenging. This week, I taught this in my favourite way yet, which I will share in this post.

 

Attempt 1

The very first time I taught this, I was keen to avoid sharing any form of ‘keep, flip, change’ mnemonic or ‘just flip the second fraction upside down and multiply’, because I wanted my students to understand what they were doing beyond following a procedure. As such, I did a bit of research (back before I’d joined Twitter!) and decided to present students to the topic by using models such as these. After all, I reasoned, it is always best to start with a pictorial representation.

division

One of the biggest issues is that I was asking far too much of the students I was teaching: I had incorrectly believed that the simple presentation of a pictorial method would enable the students to see the underlying mathematical structures, but without excellent teacher questioning and exposition, this is unlikely to happen. I (at this point in my career) certainly wasn’t able to make effective use of this model. As such, I think the main outcome of this particular lesson was that students had simply acquired more procedural knowledge – but this procedure had even more opportunities for students to make errors than the standard ‘keep, flip, change’ algorithm.

Moreover, while it worked quite nicely for small fractions, and is excellent for exploring what is happening when fractions with the same denominator are divided, it became quickly convoluted when dealing with mixed numbers and less straightforward problems. Again, this isn’t necessarily an issue with the model itself, but I had failed to plan for an ‘exit strategy’ (that is – a way for students, over time, to be able to answer similar questions without the need for the model). I want to be clear that the reason this sequence of lessons was not as successful as I had hoped was more due to a lack of pedagogical knowledge on my part, rather than an inherent problems with the model, but the consequence of this experience was that I completely changed my approach when I next taught division of fractions.

Attempt 2

The second time I taught division of fractions by fractions, I had made up my mind that I wasn’t going to spend much time on why the procedure that I taught worked. My reasoning was that my first attempt had been unsuccessful, and I was currently exploring the idea that it might be better to teach the how before the why. The rationale was that if students become procedurally fluent with how to divide fractions, and through practice, are able to automatise the procedure, then this success would enable me to return to why it worked afterwards. I’d also been considering the idea that asking students to understand why is often more cognitively demanding than understanding how, so I made the decision to begin with how.

The lesson in question ran relatively successfully: I modelled through some examples, and then students did some practice in a variety of ways. Obviously, multiple students did want to know why this procedure was working, but I promised them that we would return to this later. In reality, though, this never really happened. After that initial lesson, students stopped asking why.

Moreover, when we returned to dividing fractions in subsequent lessons/homeworks/assessments, it became clear that the majority of my class were failing to retain this procedure. I think there are multiple reasons why this was happening, but I think that one important reason is that students spent most of the initial lesson on dividing fractions not thinking. They were answering plenty of questions successfully, but then again, they only needed to think ‘well, I flip this upside down’ – the multiplication of the numerators and the denominators was something that students in that class were already confident with. As such, once they’d completed the first few questions, the rest of the lesson was simply more mindless practice, as although the numbers became more difficult to manipulate, they didn’t have to think about the important key features of dividing fractions by fractions. As before, this attempt had not been successful as it needed to be.

 

Attempt 3

Next time round, I decided to approach the idea from ‘pattern spotting’ and allowing students to hopefully deduce the need to ‘flip the second fraction upside down’. We started with this question:

fractions 1

All students were quite happy with the idea that this would be 2, once it had been shown diagrammatically. We moved onto related questions:

fractions 2

Again, this all seemed to be fairly straightforward. We then discussed this question: ‘if 4 divided by ½ is 8, what other ways can you state the relationship between 4 and 8?’ From this, I tried to draw out the necessary truth that dividing by ½ must be the same as multiplying by 2, and that dividing by 2/3 must be the same as multiplying by 3, and so on. Some students (those of higher prior attainment) grasped this idea really quickly, which was excellent. For other students (particularly those of lower prior attainment), this explanation was evidently not complete. While they were confident with how they could divide whole numbers by unit fractions using simple diagrams, they really struggled with making the link to how this process related to questions such as:

fractions 3

They could replicate the changing of ¼ to 4, and the division sign to a multiplication sign, but I still think something was missing.

 

More Recently

In this last week, I taught division of fractions to my Year 7 class. For some context, my Year 7 class are completely mixed in terms of their prior attainment. In class, we had already looked at addition and subtraction of fractions, multiplication, simplifying and converting between mixed numbers and improper fraction. I am continually astonished at their mathematical capabilities and in awe of their primary school teachers, as the vast majority of this class are remarkably fluent with the manipulation of fractions.

I had a quick look through the Key Stage 2 National Curriculum (always my first point for teaching anything to Year 7 these days!), and noted that between Years 3 and 6 they would have covered:

  • Recognising unit fractions as a division of a quantity
  • Solving problems involving fractions to divide quantities, including non-unit fractions where the answer is a whole number
  • Dividing proper fractions by whole numbers

This gave me a really useful starting point.

I began the lesson by giving them these three related statements:

fractions 4

I then gave them a sequence of multiplication statements, and asked them to write any related division statements. While the first couple of questions involved the use of only positive integers, I also included questions such as:

fractions 5

(This was particularly interesting, as one of my highest attaining students then asked: ‘miss, I’ve written 4 divided by 1/2 = 8, but wouldn’t 4 divided by 1/2 be 2?). I also included questions such as:

fractions 6

 

This was a nice introduction – the last couple, in particular, caused all sorts of questions and to be asked and conjectures to be made. After this, I brought the class back together, and we looked at questions such as:

fractions 7

After multiplying and simplifying, they noticed that when you have:

fractions 8

Or questions in that form, the answer must be 1. This was a lot of fun: there was a noticeable ‘buzz’ in the room, and at this point, I introduced them to the word reciprocal. I would never have bothered sharing this with a Year 7 class previously, as I mistakenly believed it would have overcomplicated things, and after all, they would go on to meet this terminology later. This was a mistake – it is so important that our students have the opportunity to express things in precise mathematical language as early as possible, and they were so excited about this new word to add to their collection. After this, we did some practice of finding reciprocals.

I brought the class back together again, and explained we were going to look at how this all fitted together. Returning to the earlier question of:

fractions 9

I challenged them to write a related division statement. After a few moments of thinking, pupils shared the following ideas:

fractions 10

At this stage, there were quite a few murmurs around the room: some students were beginning to spot the relationship between division by 4/3 and multiplication by ¾, and so on. Obviously, some students were not yet there, but through some discussion and questioning, we drew out the implication that division by a fraction must necessarily be the same as multiplying by its reciprocal. We went through some examples together, students did some more practice, and every single student left being able to confidently use the word reciprocal, with most being able to articulate the relationship between division and multiplication of a reciprocal.

 

Obviously, I don’t think this lesson was perfect! I’ve still got a lot of learning to do about how best to bring in different representations in order to support all students with their understanding, and I still don’t know how this approach will prove to be in the long-term. Nonetheless, it was one of my favourite lessons I’ve taught this year, and I think it was definitely the best I’ve ever taught this topic. I’m going to continue to reflect on how best to improve this, and I’m definitely going to continue to develop my use of precise mathematical language with all classes and all topics.

Tackling Re-Teaching and Overcoming Familiarity

On October 13th 2018, I presented a workshop at #MathsConf17 entitled ‘Tackling Re-Teaching and Overcoming Familiarity’. In this post, I will discuss the ideas that I shared on the day itself!

So, I began by looking at the mistakes I had made when re-teaching topics in the past.

These, I think, are the following:

  1. I hadn’t assessed the prior knowledge of the students.
  2. I was teaching topics as though they were seeing them for the first time.
  3. I failed to consider the thoughts/feelings/perceptions that my students experienced, when they encountered topics with which they were already familiar.

 

Problem 1: I hadn’t assessed the prior knowledge of the students.

I always used to think that when I did make the decision to re-teach a topic, I did so based on prior assessment data. Specifically, if the majority of a class that I taught underperformed on a particular question in an exam, then I would reteach that particular topic.

I shared the example of how my Foundation Year 11 class had all answered part (b) of the following question incorrectly in their first GCSE Mock Exam, and explained how I had re-taught adding and subtracting fractions in light of this.

mc1.JPG

However, there is a huge problem associated with assessing based solely on high-stakes, summative assessments. David Putwain’s study in 2007 focused on KS4 students in the UK and the effect that test-anxiety can have on performance; he (unsurprisingly) found that students who are most affected by exam-anxiety are the students who are most likely to under-perform, which is due to the amount of space in working memory that the ‘worry’ is occupying. Even more interestingly, mock exams are often likely to cause even more significant anxiety than ‘the real thing’, as students are often acutely aware of how close (and yet how far!) they are to sitting the high stakes GCSE.

However, even if we discount exam-anxiety as causing my students to under-perform on this particular question, the fact still remains that one question, one time, doesn’t tell us very much (and this is why I think it is worth being wary of exactly what we can infer from Question Level Analyses).

For a student to be really successful in adding fractions, I would suggest they need to be secure in the following pre-requisites:

  • Adding fractions with the same denominator
  • Identify fractions as the proportion shaded
  • Being able to place fractions on a number line (they need a sense of how big 2/7 is compared to 8/9).
  • Identify the LCM (or even just a common multiple) of two numbers
  • Simplifying fractions

There’s probably far more than these, but ultimately, if a pupil isn’t secure in these fundamentals, then trying to take them through the whole procedure of adding and subtracting fractions is quite likely to lead to them coming unstuck. I couldn’t have told you which of these skills my class could or could not do – I had just grouped all these errors into ‘can’t add fractions’ and tried to re-teach accordingly.

My approach now is to always begin with some multiple choice diagnostic questions. Sometimes, depending on the class (or topic), I will do a printed paper copy pre-test; sometimes I’ll get pupils to answer on mini-whiteboards, and sometimes I’ll just ask them to show me 1, 2, 3 or 4 fingers depending on which option they’d like to vote for. I always source these from www.diagnosticquestions.com, because the quality of questions is really high, and importantly, it should be impossible for students to guess incorrectly while still holding onto a misconception.

However, this is only the very first step on the journey! After all, what you are most likely to find within a typical class is that some pupils know certain prerequisites, some don’t. This process of planning, therefore, can become extremely difficult and can quickly result in students working on different activities as you attempt to plan for all of their starting points. I’ll come back to this!

 

Problem 2: I was teaching topics as though they were seeing them for the first time.

When I began my lesson with my Year 11 class on Adding and Subtracting Fractions, I started as though they were seeing this for the very first time. During the teacher exposition section, I may well have uttered these words: ‘oh, I know you’ll have done adding and subtracting fractions before, but…’ and then continued with the explanations I had planned. The problem was that that sentence had gone unfinished, and I have no doubt that some of my students finished it for themselves:

‘oh, I know you’ll have done this before, but you couldn’t do it’

‘I know you’ve done this before, but you’ve forgotten’.

‘I know you’ve done this before, but you failed.’

This, in itself, is incredibly demoralising. While I would never say any of the above to a class, and while I would always attempt to be positive and supportive and encouraging (‘come on, we’ll get there! You can do this!’), my students had probably heard those statements of encouragement already.

Now, as you’ll be aware, students have met fractions quite a few times before. If we take a look now at the National Curriculum expectations for ‘Fractions’ for Key Stage 1 and Key Stage 2:

mcyear1mcyear2mcyear3mcyear4mcyear5mcyear6

By Year 3, pupils are expected to be able to order unit fractions! In year 5, pupils are first taught how to add fractions with different denominators. By year 6, pupils are expected to add and subtract mixed numbers. We can be quite clear that by Year 11, pupils have studied fractions quite a lot.

This, I suppose, leads to the question: well, why are re-teaching so much? And the answer, I think, is obvious: because students forget! This, however, is so far from an ideal situation. It is such a shame and such a waste of so much of their time at school that they are being re-taught relatively basic mathematics again and again and again. My Year 11 class were absolutely not atypical, and so I think it is so important that we focus on strategies to avoid us being stuck in a cycle of re-teaching.

To avoid having to re-teach, I think we need to do the following:

  • Cover things in a good level of depth from the beginning.

This is obviously key – if pupils are exposed to procedure after procedure after procedure in a superficial way, then we can expect them to forget completely, or to remember misconceptions. I’m certainly not an expert at this (although I attended excellent sessions by @MrMattock and @berniewestacott on the day, which has given me plenty of food for thought!) but the more that we can do to look in depth at concepts and procedures, and the more we can use multiple representations to strengthen pupils understanding, the better. Moreover, don’t move on if your class are not ready to!

This is easier said than done – I am fully aware of time pressures in certain schools being to cover a lot of content, but the more you can do to slow down to ensure success first time round, the better.

  • Ensure topics are revisited regularly – in starter activities/homework/low-stakes quizzes

This again, seems obvious – don’t teach them adding fractions, leave it for a year, and then be surprised that they’ve fallen back to ‘add the tops, add the bottoms’ again! You need to give them the opportunity to practice it – put some questions in with starter activities and in their homework. Interleave it when looking at other topics – the most straightforward example is always related to perimeter, but what about when teaching the order of operations? What about collecting like terms?

Finally, it’s really important that we take time to look at our schemes of work and the sequencing of content. The nature of our subject means that revisiting old content is inevitable – it would be ludicrous to claim that we can teach EVERYTHING to do with fractions in one go, as fractions is effectively an unlimited concept – there is always something more you can do with it. I’m certainly not an expert at the best ways to structure a scheme of work, but I think a greater awareness of what students have done before/will be going on to do can only help if we are to avoid the constantly re-teaching problem.

However, even when we are not having to reteach something that should be Year 5 or Year 6 knowledge, we are always going to re-cover things – specifically, you will need to extend students’ knowledge on a topic they have encountered before, which leads onto problem 3.

 

Problem 3: I failed to consider the thoughts/feelings/perceptions that my students experienced, when they encountered topics with which they were already familiar.

It is worth, at this point, drawing a distinction between the Year 11 Foundation class previously mentioned and the Year 10 top set that I was teaching at the same time. The reason for this is that I want to demonstrate that re-teaching and familiarity and how we approach this is something that is relevant to students of all different levels of prior attainment – it may be easy to see how what I have spoken about so far as something that affects only middle and bottom sets, but this is not always the case.

I was having to teach this Year 10 class the GCSE Higher topic of ‘Ratio and Proportion’; while I was not directly re-teaching simplifying into ratios or sharing into a ratio, they were certainly familiar with the fundamental ideas. Now, when I was teaching my Year 10s ratio, or my Year 11s fractions, I would regularly hear comments along the following lines (importantly, these comments would be consistent, despite the differing prior attainment of both classes and despite the re-teaching/revisiting distinction):

  • ‘We’ve done this before – this is easy’.
  • ‘I can’t do this, I always get this wrong’.
  • ‘We’ve done this before, I’m really good at this’
  • ‘I’ve never seen this before in my life’.

 

Now, within a typical class, you will most likely have a mixture of these types of students at any one point. The question is, then: why are they thinking like this and how can we best respond?

Perhaps even more than that, you might be thinking ‘well, why does this even matter?’

To be honest, for the first few years of my teaching I’d have been thinking very much this. For some context, I was one of those people who liked learning Maths at school – and I would imagine that this is the same for most people reading this blog! Maths was always something that I was quite good at; I didn’t have to work that hard, and so I didn’t really mind revisiting topics. Now, I’ve always been empathetic to my students who don’t feel the same way as I did, and so I am (hopefully) supportive and encouraging but equally, when it came down to it, it didn’t matter that how they felt about it; what mattered was whether they could do it or not. However, I think if we ignore the way in which our students feel when they encounter a familiar topic, we are unlikely to get them to make as much progress as is possible.

At this, point, then, it is worth making a distinction between prior knowledge and prior experience. Prior knowledge is that which we can assess relatively easily; prior experience cannot be. Of course, the longer you teach a class for, the more of an insight you have into this. With the classes that I’ve taught for more than one consecutive year, I’ve got much more of an awareness of how easy or difficult they found a topic the first time round, and I can make an attempt to realistically gauge how they might feel when they see solving equations or standard form for the second or third time. For a class that you’re teaching for the first year, this is going to be tricky! I’m not sure I’ve particularly found a way around this yet, but I think it is so important to be aware.

For the student who is thinking: ‘I always get this wrong, I can’t do this’, or has any similarly negative mind-set, you need to ensure they achieve success quickly. We need to be aware that motivation doesn’t necessarily lead to success, but success frequently leads to motivation. No matter how positive you are feeling and no matter how encouraging you try to be, it will not matter if they still cannot do the necessary skill.

For the student who is thinking, ‘we’ve done this before, I’m good at this’ or ‘this is easy’ – you need to be careful. I think this is where things get really interesting. The reason that this is so interesting is that often the students who proclaimed to find something easy, or became offended or patronised when faced with familiar content were not students who could demonstrate they had the necessary knowledge! They would tell me ‘oh, I know how to do this’, and mean it, but I would know that they couldn’t – i.e. they’d answered the prerequisites incorrectly, and when talking to those students one-on-one, it became really clear they had either misconceptions or insufficient knowledge. For example, I taught an amazingly bright Year 10 student, and the moment I’d set the class off on some ratio practice, she’d proclaim that ‘she got it’ and she didn’t need to do any more. When speaking to her, she simply couldn’t do the more challenging questions, but felt that she did. With my Year 11 class, I had one student who was adamant that he did not need to practise any more adding fractions, as he’d understood and he wanted to move on. Now we all know that practise is necessary for students to retain anything long term – we know that ‘practice makes permanent’ – but if I couldn’t get them to practise, then we were back to the beginning.

So, I was really fascinated by what was going on: why did I teach so many students who were adamant that they’d ‘got it’ when I could tell this wasn’t the case? Again, went away, did some thinking, did some reading, and came across the following that really helped me to make some sense of what was going on.

Daniel Willingham identifies that there are two main ways in which we determine if we know something – familiarity and partial access.

He states that “familiarity is the knowledge of having seen or otherwise experienced some stimulus before, but having little information associated with it in your memory.” Now, familiarity is generally quite a good guide for us to check if we know something! However, this would mean we would never make a mistake of when we decide if we know something, and of course, we sometimes do. One study that was done into this involved giving participants a variety of word pairs to study (e.g. golf and par), before participants answered some trivia questions. Where the words in the questions were familiar to the participants, they were significantly more likely to say that they ‘knew’ the answer, even if this was not the case. Where the words in the questions were not familiar, participants could assess their own knowledge more accurately.

The implications for this in the classroom seem fairly obvious: we are constantly providing pupils with cues – whether in the form of key words, lesson titles, images, the things we say. These are helping to familiarise our pupils with the content we need to cover – but this familiarity can also be leading them to a false sense of knowing, and I think this needs to be taken seriously in our planning.

The other way in which we determine if we know something is by ‘partial access’, which Willingham defines as: “the knowledge that an individual has of either a component of the target material or information closely related to the target material.” Again, partial access is a fairly good guide to knowing things in most circumstances! If you asked me a question about late 00s reality TV, for example, I would immediately feel like I knew it, due to a high level of knowledge of this particular topic – regardless of whether I’d ever learnt that given fact before. The example Willingham cites is of ‘Who composed the ballet for Swan Lake?’ Regardless of whether you know this or not, the fact that you can probably name several composers means you may feel quite certain that you do know it. If instead, I asked, ‘who choreographed the ballet Swan Lake?’, you’d probably feel much less certain and be more likely to say you don’t know, because you are less likely to have access to a list of choreographers. Again, this has important implications for us in the classroom.

Our students, all the time, have lots of partial access to lots of different areas of mathematics; some much more than others! Nuthall’s claim is that our students, at any one time, know between 40 and 50% of what we are trying to teach them. This is what makes our job incredibly difficult – not only will our students actually have different knowledge to each other, but they will almost certainly have different (and often incorrect) perceptions about the knowledge that they have.

 

In terms of what this means for us in the classroom, then:

We need to ensure we are providing students with the opportunity to practise, but this practice needs to feel and look different. If the practice that we give our students is very much the same as the practice they have experienced before, we are likely to be inducing that feeling of familiarity, which can be so damaging. For the confident student, they are likely to switch off (‘I can do that – I know how to do this – I’ve done it before’). For your less confident student, their experience is very likely, at least on some level, to be ‘well, I’ve failed at this before – I’ll probably fail again’.

Before discussing possible resources, it is definitely worth acknowledging that this will be context dependent. If, for example, you use similar tasks with classes in Year 7, Year 8, Year 9 and Year 10, then the possible impact is likely to be lost when using with Year 11, and they will experience that same false sense of knowing which comes from familiarity.

This is one task (taken from http://www.openmiddle.com/adding-fractions-7/), which I have since used when re-teaching.

mcopen middle.JPG

Now, when I first came across this problem, I thought it had answered all of my problems! All students, having been re-taught adding fractions, would be able to practise, but there would be a wider sense of purpose.

When I used it, however, it wasn’t quite that simple. Students, and this is, perhaps, more of a reflection on the nature of my classroom environment than of the task itself, wanted to find the right answer. Some felt that it would be impossible, and gave up – when I suggested they just plugged some numbers in to see where this would get them, it became clear they didn’t see much point. Some other students went about reasoning about what the denominators must be, but didn’t actually get that much practise done, and so failed to achieve the outcome that I was hoping for. I think this is because this task was missing something that is key for revisiting topics: pupils need to feel successful early on. If not, it is all too easy for students to give up.

Now, this is something I’ve really struggled with! With more ‘open problems’ compared to more routine practice, the success often comes later. At least; the students have a perception that it will come later, or more frequently, won’t come at all, no matter how much we promote the very true narrative that making mistakes and learning from them is excellent and valuable. Having done some thinking, I’m not convinced that is a problem with the design of the task itself, but more in terms of how it can be implemented and used.

Two things that I now think about more carefully when using similar problems:

How can I provide a way in such that all pupils can access the task?

How can I model what success here would look like?

In terms of answering that first point: here is a task that I have used with classes since. This is what I initially give out:

mcvenn1 Now, the first time I tried using a Venn diagram, it did not go well. I think I had been too vague about what was expected of them, and so they didn’t really know what I was getting them to aim at – my suggestions of ‘make up your own example!’ was not received well, as they had such a focus on being ‘right’ or ‘wrong’ that the message was lost. As such, I adapted this task in the smallest way: I gave them three questions to place into the Venn diagram.

mcvenn2

Without meaning to be too dramatic about this, it was a revelation. All of a sudden there was significantly more of a focus – they were challenged to complete it as they understood what being successful looked like – once those three had been placed, the challenge of ‘can you create your own example?’ followed much more naturally. From that, it was easy enough to progress by considering three way Venn Diagrams with impossible regions – what makes it impossible? How do you know?

In conclusion, then: whenever you are teaching a topic to a class who are already familiar with all or some of the content: consider what their perceptions might be, and consider task designs which prevents students from feeling ‘familiar’ with what is being covered.

 

 

 

 

 

#MathsConf17

Yesterday, I attended my third MathsConf in Birmingham – #MathsConf17. As ever, the day was so much fun and is the most excellent CPD  – I cannot thank Mark (@EmathsUK) and the whole team at La Salle Education (@LaSalleEd) for the work that they put into organising these events. If you’ve not been to one yet, take a look at the upcoming list of locations and dates and come along! You won’t regret it.

I arrived in Birmingham on Friday night with one of my colleagues, which was an excellent opportunity to catch up with some Maths education Twitter friends. As you may expect, the conversations quickly turned into sharing and solving interesting maths problems and sharing ideas for enrichment lessons. My offering was this excellent area maze puzzle, which certainly caused a few headaches!

areamaze

On the day itself, after some opening remarks from Mark McCourt and Andrew Taylor (@AQAMaths), including a huge thank-you to the amazing Rob (@RJS2212) for all that he does by selling raffle tickets and organising tuck shop, it was time for speed-dating.

From speed-dating, I took away the following ideas (apologies for lack of credits – completely forgot to write down names):

  • When teaching factorising quadratics, atomise the procedures/steps, before bringing it all together. I still don’t do this as effectively as I would like, so it was really interesting to see how someone else had split the process up, and hear their reflections on how it had worked in the classroom.
  • I chatted to Ashton (@ashtonC94) about how is now teaching finding Lowest Common Multiple – effectively by finding the product of the numbers in question, and then dividing by the Highest Common Factor. I’ve been aware of this method but never used it before – I’ve generally gone for using Venn diagrams, but given how frequently my students make mistakes with these, I really like this approach. Definitely food for thought!
  • I spoke to someone else who shared the retrieval practice grids that he is using with his classes at the moment as starters. I really like the layout of these, and the way that they allow questions to be targeted effectively to the topics that your classes have studied.

 

Session 1:

After speed-dating, session 1 began! I attended Literacy in Mathematics by Jo Locke (@JoLocke1). I’ve met Jo several times before (and had dinner with her on Friday!), but it was my first time seeing her deliver a session, so I was really looking forward to it. Jo made the point that we are all teachers of literacy, and that students with low levels of literacy will go on to underperform in Maths, if they are not able to access what is being asked of them. This really resonated with me.

It’s also worth pointing out that I teach lots of students who I would describe as having excellent literacy – their spelling and grammar is always impeccable, and they can speak and write eloquently. Even then, I’ve seen students like this fall apart in exams if the word ‘integer’ is used, and I’ve seen similar students look astonished when asked to write down ‘the range of solutions which satisfy an inequality’. Mathematical language needs to be taught clearly and explicitly, and this is something that I have definitely neglected in the past.

Jo shared lots of different resources and strategies to ensure that our students develop the literacy skills that they need. A key takeaway for me was discussing etymology with our students; she made the point that students are frequently really excited to hear about the origins of interesting mathematical words, and so I am going to make more of an effort to do that when teaching. Even aside from student interest, I think it can help to develop their long-term recall of key ideas. If, for example, we explain how algebra is taken from the Arabic meaning ‘the reunion of broken parts’, then this leads into the process of what it actually means to solve an equation.

The other idea that Jo shared that I loved and will definitely be using was Frayer models. This is an idea which I feel sure I must have seen at some point before, but have definitely need used in the classroom. The model looks like this, and can be used in various different ways.

Screen-Shot-2016-02-16-at-10.33.31-AM

I really love the opportunity for students to create their own examples/non-examples: all too frequently, we show our students lots of examples of what a probability is, but never what a probability is not. I think non-examples are incredibly powerful for students to build a fuller understanding of a given concept, and so I am excited to try this out.

 

Session 2

For session 2, I attended Bernie Westacott’s (@berniewestacott) session on ‘Making Maths Memorable – Progression in Fractions’. Again, this was my first time seeing Bernie speak and I was not disappointed!

Bernie focused on the use of different representations of fractions and how we can use these to support students to acquire a conceptual understanding. We had the opportunity to explore Cusinaire rods, Numicon and Multilink cubes, and build various models to represent various problems. Throughout, Bernie discussed the influence of ‘Singapore Maths’, and how that approach can be used in British classrooms to ensure our pupils can become as successful as possible.

Now, to be completely honest: with the exception of the odd box of multilink cubes, I don’t make use of manipulatives in my classroom. This is partly because we don’t have other equipment, but primarily because I don’t know how to use them effectively! The reason that I found Bernie’s session so useful, in spite of this, is that it gave me a much better insight into what is going on in primary Maths classrooms at the moment, and I think a better understanding of what our students will have been previously exposed to can only improve the quality of teaching and learning.

The other reason that I found it so fascinated was how careful and precise Bernie was with all of his language used throughout. If students are not 100% fluent with the idea that the parts of a fraction must all be equal, then any subsequent teaching is being built on shaky foundations. As such, rather than using the word one half to describe ‘1/2’, Bernie described it consistently as ‘one out of two equal parts of the whole’. ‘2/3’ became ‘two out of three equal parts of the whole’, and so on. This was fascinating, and it has certainly made me think about adding and subtracting fractions and the language that I am using then. We also looked briefly at division, and how it can be interpreted in a number of different ways. Again, this is something that I typically gloss over in the classroom: I am going to make more of an effort to slow down to allow my pupils to explore the nuances which are lost when going at speed.

Session 3

I led a workshop for Session 3 (my solo #MathsConf debut!), entitled ‘Tackling Re-Teaching’. I am not going to speak about it here, because I want to write something about it separately and this could otherwise become unreasonably long. I will just say the hugest of thank-yous to Pete Mattock (@MrMattock) for stepping in and lending me his laptop at the last minute – the crisis was averted thanks to him! Similarly, I cannot thank enough everyone who attended and has spoken to me about it since; I really hope that you all found something in it useful (and thank-you for carrying on listening as my microphone was replaced mid-sentence!). As I say, there will be something written about it which will explain everything in full.

Session 4

I found it incredibly difficult to choose which workshop to attend for session 4! Luckily, I had two colleagues who were also at #MathsConf with me, so the decision was made to split up and to feedback to each other later. I went to Pete Mattock’s session (@MrMattock) on ‘Time to Revisit… Teaching For Mastery’ and I am so glad that I did.

There was so much to take away from Pete’s session that it is hard to know where to begin! He made the point that students who achieve the best outcomes have typically had a mixture of ‘teacher-directed’ instruction and ‘inquiry-based’ methods. Specifically, teacher-directed approaches appear in most/all lessons or learning episodes, and inquiry-based approaches appear in some. Too frequently, it is easy to see Maths education as a series of dichotomies which can be really unhelpful – it was refreshing to see the fact that both approaches have a valuable role to play.

Pete took us through a number of different ways of modelling different mathematical procedures –  and made the point that just teaching ‘rules’ is unhelpful. An excellent example he gave was that of BIDMAS: students who are often exposed to BIDMAS are taught it as an arbitrary set of rules that must be followed, but Pete made the point that the correct priority of operations is a mathematical necessity. He supported this with visual representations, which I still need to explore more for myself before showing to students, but which were incredibly powerful.

I also loved the way in which Pete showed the addition and subtraction of negative numbers. I’ll be honest: I have tried teaching adding and subtracting negative numbers in multiple different ways to multiple different classes, but never particularly successfully. I have tried pattern-spotting and analogy and rule-following and number-lines and everything has seemed to collapse at some point, and students revert back to ‘two minuses make a plus’. The model that Pete showed involved modelling negative numbers as vectors: I cannot do justice here to how effective it was, but I will definitely be using it in future. (I will also point out that Pete has a book being published soon – you can order it at: https://www.crownhouse.co.uk/publications/visible-maths), which includes these ideas.

We also looked at the division of fractions using a Cuisinaire rods approach – this is another topic which frequently comes back to rule following (‘keep, flip, change’) and it was so useful to see a model that actually works and provides meaning to the process. Finally, I am so pleased that Pete spoke about ‘exit strategies’. He made the point that our students won’t have access to manipulatives in exams, and that often, drawing diagrams is a more time-consuming approach than being able to answer a question without. Pete modelled how, over time, he will move his students onto a more ‘abstract’ approach, but this has acquired meaning through the multiple representations that students have used on their journey.

 

Overall, I had the most fantastic day! MathsConfs are such well-run events and there are so many different workshops being offered that you can really tailor the whole day to suit your own personal CPD needs. Thanks so much to everyone who makes the day what it is – see you all in Bristol in March for #MathsConf18!.

Teaching Mixed Attainment Year 7: A year of Mistakes!

At my current school, Year 7 Maths lessons are taught in their tutor groups; they are not placed into sets until Year 8. If you’d asked me six months ago how I felt about this, I’d have responded with something along the lines of ‘it is just so hard’. It really was.

Whenever I taught my Year 7 class last year, I had an overwhelming feeling that many were not progressing as well as they should be. In terms of the assessment system in place at my school, they were generally performing as expected, but I was still not convinced. I felt that strategies that I was using to stretch and challenge the highest attaining pupils were not sufficiently stretching and challenging, and I felt deeply guilty that I wasn’t supporting the lowest attaining pupils to master fundamental concepts. Above all, I was knackered! Despite being a lovely class, it felt that I had so much more to think about compared to teaching other setted classes, and so planning and preparation took significantly more time.

This year, however, I’m feeling quite differently about the whole experience! Obviously, we are only three weeks into term right now, but so far, it seems to be going significantly more smoothly. I will also say this with the caveat that I am not necessarily advocating for teaching mixed attainment classes. I am aware that the research is fairly inconclusive as to which is ‘better’ (especially when considering better for whom and the extent to which this is the case). Nonetheless, I am currently loving teaching Year 7, and so in this post I am going to share three main mistakes I was making last year, as well as the strategies that I’ve put into place to improve things.

Mistake 1: ‘Teacher explanation and modelling must be done quickly.’

The reason I continually made this mistake last year was that I didn’t want my highest attaining pupils to be bored. If, for example, I was teaching pupils how to find the nth term of a linear sequence, I knew that the highest attaining pupils may well be able to do this after seeing one example. As such, I would often talk quickly and at a more superficial level than I would with other classes. Regularly, I would say things along these lines:

Okay, so today we’re going to be adding fractions. I know some of you in here will already be quite confident with this, so we’re just going to go through a couple of super quick examples and then you can be moving onto some more challenging questions’.

In short, I was consistently apologetic for having to explain or model or demonstrate.

The consequence of this was typically as follows: while my highest attaining pupils would have understood a given procedure at a superficial level, several would have over-generalised or under-generalised and formed misconceptions, which I often didn’t detect until later. The lowest attaining pupils were often completely overwhelmed by these highly rushed demonstrations, and I would find myself immediately going over to the same few pupils and re-explaining at a more appropriate pace. Over time, this resulted in some pupils tuning out during whole class exposition, as they were aware they were only going to hear it again from me. For all pupils who fell somewhere in the middle of these two groups (a significant proportion), their success would have depended pretty much entirely on how much prior knowledge they already had. I was failing to use effective formative assessment strategies (multiple choice questions, mini whiteboards) because I was so keen to move them on to their differentiated work, which meant I was never really sure until the end of the lesson if they had been successful.

 

Mistake 2: ‘All independent work must be differentiated into multiple levels.’

In these Year 7 lessons, there would always be ‘bronze, silver and gold’ tasks. Sometimes, in my planning, I would worry that gold wouldn’t be sufficiently challenging, and so would have to introduce the platinum super challenge level. At other times, I would worry that some pupils wouldn’t be able to access the bronze task, so I would spend time creating support sheets to be used alongside.

First of all, this was a nightmare to facilitate. I would spend forever at the photocopier before teaching them, and because I tried not to allocate in advance which level pupils should be working at, I would print multiple spare copies which would end up wasted. It also meant that simple things such as reading out the answers to the first five questions to assess that everyone was on the right track took far longer than necessary. More problematic than this, though, was that pupils often did not understand the task they needed to be doing.

If, for example, the lesson was on adding fractions, the gold or platinum task would often be in the form of a puzzle or investigation from Nrich, or some UKMT problems. Due to the superficial exposition that had happened, pupils didn’t have the core knowledge in place to be able to access these. Equally frustratingly, as it would have taken too much time to detail exactly what was required for each task, pupils may have had the required knowledge but remained unclear about exactly what they were being expected to do.

At the other end of the spectrum, if the lesson was on adding fractions, the bronze or support task may have been something related to adding fractions with the same denominator, and in this case, the examples that I had rushed through to the whole class wouldn’t have helped pupils to complete it. Now, I absolutely still differentiate work for all my classes, but the way that I do this now is quite different (and easier to implement): this will be discussed later.

Mistake 3: ‘Pupils will be working on different skills within the same lesson.’

This mistake, on the surface, appears very similar to the second mistake outlined above. However, this feels that it was an even bigger barrier to being successful when teaching Year 7.

As an example of this: in our Year 7 Scheme of Work, we teach/review the process of multiplying and dividing numbers. Obviously, we are aware that all pupils will have learnt this at primary school and most pupils are reasonably proficient at it in Year 7, but we are also aware that without complete fluency in these topic areas, pupils will struggle with more complex procedures. Now, within my Year 7 class last year, I had pupils who could not recall their times tables, alongside pupils who could demonstrate multiplying three and four digit numbers. Neither point seemed quite appropriate to start, so I began by quickly modelling multiplying two digit numbers before pupils began practising.

For the strongest pupils, I began by giving them some worded problems which required them to multiply large numbers. This was a terrible idea: they didn’t need to think about the context in the problems (they knew) it was just going to be multiplication, so they rushed through and became bored, as is understandable. Quickly, I improvised, and moved them on to practising multiplying and dividing decimals, as this seemed like a fairly natural progression.

However, later in the year, I was required to teach the whole class how to multiply decimals. At this stage, I thought a good next step for the highest attaining pupils who were already fairly fluent would be to look at dividing decimals, and then decided to introduce them to standard form. In effect, throughout the year, I helped the strongest pupils to access lots of topics and replicate lots of procedures, but they were not being given the opportunity to think more deeply about the maths and the concepts involved: it was a case of procedure, procedure, procedure. Meanwhile, I had other pupils struggling to access the primary content of the lesson, and it felt as though pupils were developing the mindset of ‘oh, well, those are the people who are good at maths’: ability was being viewed as fixed.

What next?

At the start of this academic year, I became determined to improve my Year 7 teaching. While I am aware that it wasn’t completely terrible last year (honestly, believe it or not, there were moments of excellence in the midst of all the mistakes), I knew I could be doing a better job.

The first decision I made was to slow down when explaining and modelling. This has not been easy. I have to keep reminding myself that even the very brightest pupils will not be unduly negatively affected if they have to wait an extra five minutes before they begin their own work. Obviously, I am still the person in the classroom who knows the most maths, and so I’m trying to feel less guilty about making sure my explanations are clear and my examples are well chosen. This still does not need to take that long (indeed, by making use of silent teacher and example problem pairs), this is often very efficient. Moreover, when I am talking, I have realised that the prior knowledge my pupils have isn’t what I had presumed it would be.

Just yesterday, for example, I discovered that none of my current class had been shown the divisibility rules for 3 and 9 – it was so joyous when they discovered this was true. If I had rushed through as before (‘right, this is how we divide numbers, I know some of you can do this already, this will be quick’), then that point would have been lost.

I am also developing my AfL strategies with this class so they are far more effective than before. This is so important with mixed classes; all too frequently you will find a student who excels at a particular topic but struggles elsewhere, or you’ll find that one of your typically highest attaining students has a blind spot when it comes to nets of 3D shapes. I will now use multiple choice questions every lesson (pupils can vote with their fingers), which means I can detect misconceptions earlier and put steps into place to remedy them.

In terms of differentiation: this will always, always be so important. It is really important that all pupils are supported and challenged in terms of the demands placed upon them, but this does not have to mean five different worksheets.

Recently, when students needed to practise multiplying numbers, I gave them this task (by the ever excellent Don Steward).

https://donsteward.blogspot.com/2017/01/consecutive-digits-in-multiplication-sum.html

Picture2At a surface level, this doesn’t look particularly interesting. Using it in a classroom, however, was absolutely fascinating. For pupils who are unsure where to begin, you can just tell them to write the numbers in any order in the gaps, multiply them, and see what the product is. You can chat to other pupils about what the units column of what the multiplier and the multiplicand must be: why? How do you know? Is there more than one option? Quite simply, by giving pupils the same task but which will be accessible to most (if not all), you can plan your questions and conversations and prompts much more easily than if all your energy is going into distributing the next worksheet.

https://donsteward.blogspot.com/2013/12/six-digits.html

Picture1

A similar activity is this, on adding decimals. Again, this is something that looks almost uninteresting on the surface, and it is certainly something that I wouldn’t have given to my highest attaining pupils in the past, as I would have reasoned that they could already fluent add decimals. When I did use this task, I discovered that even some of the pupils who could add decimals really struggled with reasoning about where various digits must be – but the nature of the task meant that they were forced to think about this while also practising the procedure.

I think probably the biggest cause of my Year 7 lessons being so much more successful than before is this change of approach to resourcing. Don Steward’s blog (https://donsteward.blogspot.com/) is my go to place, as are www.openmiddle.com and www.mathsvenns.com This is not always easy; some topics lend themselves to this style of activity much better than others do. As I say, this is only an overview of what is working for me in the classroom at the moment. I know that there are other people teaching mixed attainment classes with a very different approach to the one that I am using, and so I am really not suggesting that this is the ‘best’ way. I’ll hopefully write something in a few months time about whether this year’s experience is continuing to be quite so positive!

Teaching Surds – Part 3

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.

is this a surd

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.

surd your turn

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.

Slide3-6

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.

Slide9-4

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.

rational irrational roots

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.

Teaching Surds – Part 2

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).

Teaching Surds – Part 1

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!