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:
- I hadn’t assessed the prior knowledge of the students.
- I was teaching topics as though they were seeing them for the first time.
- 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.
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:
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.
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:
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.
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.