Final ones! I’m so glad that lots of you have found these useful.
These are based on the advanced information from Edexcel for Paper 3 – there are four different levels of difficulty and an editable version.
Coming later than expected, but hopefully still in time to be useful! The versions for Paper 3 will be created over the weekend and shared after Paper 2 has been sat (on Tuesday). These are based on the advanced information from Edexcel.
Please do share if you find useful.
The 2022 versions can be downloaded below – either as a PDF or an editable version. There are four different levels of difficulty – two Higher, and two Foundation.
The plan will be to create similar sets of questions for Paper 2 and Paper 3 – so watch this space!
I’ve been using OneNote for various things over the past 8 months or so – one feature I have recently discovered is that it can be used to generate self-marking Maths quizzes. These are automatically created in Forms and can be shared with students directly, providing a super quick way of gaining some insight into how students are getting on. Usual disclaimers – the questions (and incorrect answers generated) are not going to be as well-designed as they could be, and for that reason, I’d probably veer away from this when introducing classes to a topic for the first time. That said, as another tool to take advantage of – I think this certainly has its place!
2. Select everything you’ve just written – I would use the ‘Lasso Select’ tool here.
3. Select your Maths, then click the ‘Maths’ button on the Draw bar.
4. This screen should appear. Clicking Ink to Maths will transform your handwriting into a beautifully typed equation! Then click ‘Select an Action.’ Depending on what you’ve typed, you’ll have a number of options appear. (e.g. Differentiate w.r.t x, Evaluate, Factor, Solve for x).
5. Select ‘Solve for x‘.
6. This where it gets exciting – you’ll have an option to ‘Generate Practice Quiz.’
7. Select that button – this screen will appear. Choose how many questions you’d like.
8. Generate the quiz! This will automatically embed itself within your OneNote page, as below.
At this point, the OneNote page can be distributed to students using Class Notebook. Alternatively, you can copy the link to the quiz and send to students directly. To view results, visit forms.office.com.
Hope this is helpful!
For various (non-exciting, just being busy!) reasons, Twitter and blogging and MathsConfs have taken a bit of a back seat for me over the last 18 months or so – but today I attended the virtual #MathsConf23 and I feel inspired all over again. It was a busy day (6 sessions) and I have so many thoughts and questions and strategies which I cannot get to wait to try out in the classroom.
So, my first workshop of the day was ‘Why Learning Maths is Hard…. And What We can do to Help’, by Stuart Welsh (@maths180). I’ve never seen Stuart speak before but I took so much away from his session. The title of this workshop really resonated with me – I think during my first couple of years of teaching I would frequently tell my students ‘this is easy – you’re going to understand’. This was a mistaken approach at motivating them (‘if they think it’s going to be easy, they’ll be more prepared to persevere’), but typically it had the opposite effect (‘if it’s so easy, and I still don’t understand, the problem must be me ‘).
Stuart began by drawing upon Geary’s distinction between biologically primary and secondary knowledge. Biologically primary knowledge is that which we have evolved to learn (walking, talking etc.), whereas we need to be instructed to acquire biologically secondary knowledge (how to solve a quadratic equation). This is one reason why learning Maths is difficult – acquiring any biologically secondary knowledge is hard and requires effort. What else makes learning Maths difficult? The inherently hierarchical structure of the subject means that to learn new maths, many prerequisites need to be secure already; if we try to instruct our pupils in new content when they are not secure, the new learning is likely to fail.
One of the points Stuart made that resonated most with me was that many mathematical concepts are ‘more than the sum of their constituent parts’. For example, a student could recall and use a procedure to solve an equation but there is still something beyond just the procedure that we would expect a student to possess if they really understood what it meant to solve an equation. Much of this comes back to what students are thinking about and what their attention is on – while we cannot control what our students think about, successful task design allows us to influence what they might be attending to at any moment. This is an area of my practice that I am keen to develop; although this is something that I have improved at, I am aware that I don’t expose my students as frequently to rich tasks as I could.
Another real take-away for me from this session was on problem-solving; problem-solving is something that students find hard. Students often feel overwhelmed when faced with an unfamiliar problem, and will not know where to start – something that I need to make more of an effort to do is to model problem-solving as though I am a novice. This is difficult! It’s hard to imagine being faced with a GCSE type question where I wouldn’t immediately be able to identify what area of Maths was required and select an appropriate strategy to solve. I need to do more to put myself into the position of my learners and make my thinking process more explicit.
Finally – the other distinction that Stuart drew upon what between necessary and arbitrary knowledge. Arbitrary knowledge is that which could be differently as it is often a matter of convention; names of shapes and function notation could be examples of this. This knowledge cannot be ‘discovered’/deduced – it can just be told to our students. Necessary knowledge is different; it can be deduced and reasoned about – such as the idea of why were using inverse operations to solve an equation. The problem, then, is where necessary knowledge is taught as though it is arbitrary. In doing so, maths just becomes a list of unconnected and random procedures to be memorised. This really rang true. More of my students than I would care to admit would view Maths as just lots of different things – and don’t have the bigger picture understanding of how concepts and ideas all link together and make it such a joy! I need to make this a priority.
Next, I attended Behaving Mathematically by Jonathan Hall (@StudyMaths), who is also the creator of www.mathsbot.com. I was really looking forward to this session – as I’ve alluded to earlier, a definite tendency in my teaching is to prioritise procedural competency/fluency, and I find it easy to neglect providing my students with sufficient opportunities to practice thinking and behaving mathematically.
Jonny opened with this quote:
Now, I couldn’t agree with this more – I would always articulate my aims for my students in a similar way, and yet I know I don’t think this is what is happening in my classroom (at least, to the extent which I would like it to).
The session focused upon prime factorisation; this is an area of Maths that I do really enjoy teaching and I think it’s something I’ve improved my delivery of. I am aware that prime factorisation can be viewed by students as ‘just another procedure to memorise’, and so I was keen to explore ways to prevent this.
Jonny began by showing us prime factorisation tiles – I’ve been aware of these for a while but had no idea of how to effectively use them in the classroom. It would be hard to do justice to the session in this post, but I was really struck by how fun and joyful the entire session was. We looked at a sequence of questions (express 300 as a product of prime factors, express 600 as a product of prime factors etc.) which wouldn’t be dissimilar to a sequence I would use, but the addition of the physical manipulatives made the experience just feel so different.
Moreover, the tiles led much more intuitively to the idea of using the prime factorisation of two numbers to find their Highest Common Factor and Lowest Common Multiple. I have generally opted for the Venn Diagram method in the past; while this generally works well and most students can be successful with it, I am not convinced that enough of my students understand the underlying structure of the mathematics. I particularly loved how straightforward the tiles made it to find all possible factors of a number, in a systematic way.
The session closed with a couple of real ‘wow’ moments! I don’t want to say too much more as I think it would be far more worthwhile to catch up with the session yourself. Honestly – I was so amazed and I am buzzing about the possibility of using some of the tasks Jonny suggested in my classroom. The tasks provide students with the opportunity to specialise, generalise and conjecture – which are the constituent parts of what it means to behave mathematically. Ultimately, being in this session for me was the way I would want students to feel when they are in my lessons – I cannot recommend catching up with it more highly.
For session 3 I was meant to be attending ‘Pi vs. Tau and other Mathematical Arguments’ with Ayliean McDonald (@Ayliean). Disappointingly, this session was cancelled due to technical issues and so I am hopeful that I will still get to attend this workshop on a future date. However, I went along to ‘Working with Low Attaining Pupils’ with Gary Lamb (@garyl2). I did miss the beginning of the session and so I am glad I will get to catch up with it!
Gary made the point that a mastery approach to learning does not mean that we should solely focus down on tiny aspects of procedures – to do so is reductionist. A really interesting quote from Guskey (1997) was shared: ‘some students are good at guessing what they are expected to learn, many others are not. For those who are not, learning soon becomes a frustrating experience.’ This really resonated with my experience within the classroom.
Similarly to session one, the idea of ‘what is it like to be a novice?’ kept recurring throughout this session. In many ways, I am a terrible novice learner. I don’t come from a particularly Maths-y background, and my degree isn’t in Maths, and so something I do intermittently is to try and learn some Further Maths A Level. Although I am enjoying the experience, it’s still true that I find it hard and frustrating and I really struggle to ask for help. I don’t like people looking at my work unless I am certain that it is right: it is something I just find quite stressful. Yet, when I am in the classroom and in my comfort zone I forget that this is what it can be like to be a student. I expect them to be as excited as me about solving equations or using Pythagoras’ Theorem – which is an unfair expectation. I have so much to think about here.
The workshop also looked at the relationship between motivation and achievement. Gary made the point that motivation needs to be built through achievement; I think this is incredibly important, and that trying to develop motivation where achievement isn’t there is never likely to yield positive results. The importance of sincere praise should not be underestimated here; by praising the specific mathematical behaviours we want to see (‘well done, I know you found that problem challenging but you looked back in your work to find a similar example which you used to help get started – that’s fantastic!’), we are more likely to see a repetition of those behaviours in the future. This is particularly important for our lowest attaining students.
Gary also highlighted the importance of alternating examples and problems within our teaching, and shared some research which suggested that this leads to the greatest long-term gains. He made the point that with a typical example-problem pair, a student can follow the steps during an example, but forget those steps when faced with a similar problem. By introducing additional scaffolding (in the form of naming the steps or allowing pupils to ‘trace the mathematics’) we increase the probability of all students being successful. Over time (and this is why patience is key!), these scaffolds can gradually be removed.
The final takeaway for me from this session was Gary’s point about overlearning. I think it is perhaps easier with our lowest attaining students to provide them with more repetitive practice, with the hope that this will means they can be successful, leading to greater confidence and (hopefully) long-term retention. However, there is little evidence to suggest that this approach is effective – ‘practice shouldn’t just be more of the same’. Gary shared a structure of ‘Do, Think, Fix’ which has been really useful for me to think about. I know that I am guilty of prioritising the ‘Do’ stage when working with lower attaining students; this is mistaken as all students need the opportunity to Think and to Fix if they are to be truly successful in Mathematics.
For my fourth session, I went along to ‘Always Teach ‘What’ before ‘Why’’ by Kris Boulton (@kris_boulton). I’ve seen Kris speak quite a few times before (and was lucky enough to spend a week at King Solomon Academy when he was teaching there); I know how deeply he thinks, and so I was looking forward to this workshop.
Kris shared this model of ‘understanding’. To improve students’ understanding of a given area of mathematics, we would want to increase both the quantity of knowledge they have access to and the connections between them. The top right quadrant, then, is what we are aiming for. So far, this felt relatively uncontroversial!
Kris shared his own experience of his first year or two of teaching: in his efforts to expose his students to the wonderful connections in maths, he would often try to teach everything in one go. This felt very similar to my own experience. Now, though, he would recommend teaching one thing at a time; subsequent topics should be taught on topic of that existing knowledge and connections should be built that way.
A specific example Kris gave of this was teaching students how to simplify with index notation. The approach outlined is one that I have used myself:
After running through several similar examples, he would introduce students to the general form (a^m * a^n= a^m+n), before students would complete some practice. Now, while specifics may vary, I think the approach here is almost certainly the most standard approach used in classrooms today. As I say, I’ve used it myself and it feels like a reasonable approach to take. What could the problems be, then?
Kris highlighted three main problems with this type of approach. Firstly – for at least some students this is likely to induce cognitive overload. This resonated. I can think of particular students who after (what I thought was) an excellent and clear explanation would be completely overwhelmed; upon pointing out that that we just need to ‘add the powers’, everything fell into place. It also has the problem of how we ask students to practise the why – do they need to practise they ‘why’ stage in order to have understood? Moreover – when we return to this topic (as part of revision or in some other context), we do not practise the why. Finally, the journey from the start of teaching to the point where students have ‘understood’ is often long and meandering – and above all, only really interesting if you know the destination.
This final point made a lot of sense to me. Several years ago, when teaching a sequence of lessons on how to find the area of 2D shapes, I arrived at ‘area of trapeziums’. I was very keen for my students to understand where comes from, and so came prepared with several different methods of how to derive this formula. For various reasons, it was very unsuccessful and it created unnecessary confusion. Students found the derivations of the formula challenging – when they actually went on to practise finding the area, they were confused: ‘why is this bit of easy when everything earlier didn’t make any sense?’. However, I think there was something going on beyond that. I can’t especially remember when I first learnt the area of a trapezium formula. I feel reasonably confident that one of my maths teachers would have shown us why it worked at some stage, but I have no recollection of this. As such, when I started preparing to teach area of a trapezium I was so excited about all of the different ways that I could show why the formula worked. I can’t help thinking that I was only excited because I already knew what the formula was – once the ‘what’ is established as fact, the ‘why’ becomes more interesting.
For that reason, there are a couple of topics that I now teach ‘back to front’ – area of a trapezium is one of them, as is the quadratic formula. Honestly – one of my favourite all-time lessons was with my year 10 class earlier this year. I made a really big thing that we wouldn’t be looking at why this formula worked just yet – we were just going to trust that it did and get on with it. I did promise them that I would show them why, but only when I thought they were ready. This intrigued them, and they kept asking and they kept asking. Finally, I ‘gave in’ one Friday and we went through the proof of the quadratic formula and it was just magical. I’m not prone to being overly sentimental but to see it all come together for them was such a wonderful moment. Anyway, I say all this to make the point that in many ways I am onboard with teaching ‘what’ first and ‘why’ second.
Back to Kris’ workshop: Kris then went through a sequence of instruction he would now use to introduce simplifying with indices. This sequence involved starting by showing students a true statement (e.g. 7^3 * 7^7 = 7^10 ) and then changing just one thing at a time. Pupils will work through questions using mini-whiteboards. He made the point that this is a rapid process – by changing (for example) just the base number, pupils only have to change one thing on their whiteboards. It is a process that all pupils can be successful with.
I don’t want to go through the entire sequence here (honestly, catch up with the workshop!) but there were a couple of moments I particularly liked. One was the use of the question ‘193^40 * 193^7’ – Kris made the point that to students this will look difficult – and so guaranteeing their success with difficult questions will lead to increased self-efficacy. I also really liked the introduction of multiplying three terms (e.g. 2^5 * 2^8 * 2^3). With careful narration (‘I know I haven’t shown you how to do this yet, but I’m pretty confident that if you have a guess then that guess is likely to be correct’), pupils are able to make correct inferences for themselves – again, I think this can only be a good thing.
Up to this point, I was fully onboard with Kris’ ideas. However, title of the workshop was ‘Always Teach ‘What’ before ‘Why’’. This idea of ‘Always’ seemed dubious.
Kris then shared how he would now introduce students to solving quadratic equations. Rather than beginning with a quadratic equation in expanded form and asking pupils to factorise, he would begin with a quadratic equation in its factorised form. This seemed perfectly reasonable. At this point, Kris shared the example below, with the commentary ‘flip the sign, put it over this number’.
I cannot explain just how taken aback I was by this: ‘you CAN’T just say ‘flip the sign’, pupils won’t understand what they’re doing at all, this is unbelievable.’ This felt very much like a trick to show students – they might be able to do the trick but they won’t understand at all, so the ‘trick’ must be wrong.
The above slide has been shared on Twitter quite a lot (on this, I am genuinely always so impressed by the way in which Kris engages in meaningful debates), and I think without context it is easy to respond in horror (as I did!) and discount the whole thing. However – I really want to stress that Kris was not saying that we should teach pupils this and then consider that ‘solving quadratic equations’ can be ticked off the list. Once pupils are secure with this stage, they will then return to why it is working and what those missing steps are etc.
I am still very unconvinced – but I can’t really articulate why. I think much of my discomfort can be attributed to biases and currently held convictions about the ‘best’ ways of teaching Maths. I am not sure that it is best to always teach ‘what’ before ‘why’ – I think for many things the two go hand-in-hand. Above all, though, it was so valuable to be challenged and to think hard and to confront existing beliefs. I may not be about to revolutionise my teaching of solving quadratic equations but I will continue to puzzle over things and try some new things out and see what works.
After lunch, it was time for ‘Teaching Exact Trig Values’ with Jo Morgan (@mathsjem), who is creator of the incredible www.resourceaholic.com. Jo’s sessions are always fantastic – they are filled with resources and practical strategies of things to try out. Jo began by providing some background to the topic. With the ‘old GCSE’ in Maths, trigonometry was only assessed at Higher and Intermediate tier, whereas the introduction of the ‘new GCSE’ (which is no longer that new!) meant that all students could be assessed on trigonometry, including at Foundation tier. Jo made the point that the specification says that students must know the exact trig values, but not necessarily ‘use’ or ‘apply’ them. It is worth being aware of this distinction!
There is, of course, an argument about whether exact trigonometric values should be included as part of the GCSE. For the most part, I am glad they are. I think they offer a really nice opportunity to ensure that students’ understanding of trigonometry goes beyond a set of procedures to be memorised. Moreover, for students who are going onto study Maths and Further Maths at A Level, it makes sense for them to already be familiar with these exact values. However, I am still spectacularly unconvinced for the need for exact trig values to appear at Foundation Tier. I think it is very much an example of the ‘wrong maths’ to be teaching Foundation students – and it becomes an exercise in pure memorisation as opposed to anything deeper.
Jo shared the above as a proposed structure for how to order the teaching of ‘advanced trigonometry’. This has given me a lot to think about! Typically, I would teach sine and cosine rule before introducing exact values/non-calculator trigonometry. A more effective, approach, however, could be to introduce exact trig values earlier in the learning journey; thus, creating more opportunities for interleaving and (hopefully) attaching more meaning to these exact values. The importance of ensuring that students are fluent with surds before delving into exact values is something that Jo emphasized. This is very true; I can remember one occasion when I tried to introduce this to a class who should have been confident with manipulating surds. They were not. As such, I had to spend the lesson dealing with any number of surds misconceptions which meant that the exact trig values were lost in the surrounding noise.
I really enjoyed thinking about how to teach this topic, and also about how to ensure that students have memorised the values! Like many Maths teachers, I would always refer back to the ‘special triangles’ and encourage students to derive them as opposed to simply memorising. However, if students can gain 1 mark for recalling that , then perhaps it is worth spending more time on memorising a list. I feel deeply uncomfortable with promoting the finger trick or similar, but I need to consider if I have grounds to feel this uncomfortable. I am not sure!
My main takeaway from this session was to use exact trig values throughout all my teaching of trigonometry as much as possible. Jo shared all sorts of excellent resources to help with this (particularly from Don Steward, MathsPad and the Level 2 Further Maths qualification from AQA).
Final session of the day now – ‘Misconceptions in Mathematics: Angles’ with Craig Barton (@MrBartonMaths). I’ll be honest: angles is not one of my favourite topics to teach. One of the reasons for this is that I think practice of basic angle properties tends either to be routine (and thus just become arithmetic practice) or increases significantly in difficulty. At this stage, some students will have become overwhelmed by the increased level of demand and I’m not as successful at helping them to bridge that gap as I would like to be.
Craig shared several questions from his Diagnostic Questions website (www.diagnosticquestions.com) – we were invited to guess what the most popular wrong answer would be. Before we even delve into any specifics, this was fascinating and something I’d like to try with our department! I would have imagined that a (virtual) roomful of Maths teachers would be extremely successful with identifying the most common misconceptions, but the results would show otherwise. It’s really interesting to think about why that might be.
One question I particularly enjoyed thinking about is this:
Students tend to opt for B, C and D in roughly equal proportions, with C just being the most popular misconception. Why might this be? There’s various reasons, but I was particularly interested in this student’s response.
This has really made me wonder: to what extent do my students do view Maths questions as needing to reach a numerical answer? I’d like to think that my students would be able to reason about the remaining angles being acute and obtuse, but then I also think that needing to ‘get to the answer’ would lead some drawn to C. I mightbe wrong, and I will need to assess this more carefully in fiuture, but if my intuitions are correct then I need to more actively cultivate a culture of reasoning, not answers, in my classroom.
I would really recommend catching up with this session and see how you do. I’m not going to go through each question here but rather discuss some of the implications of these misconceptions for how we might approach the teaching of angles.
The most important thing that I took away from this workshop was ‘include the unusual’ from the start. There are several ‘standard’ triangles in terms of orientation etc. that we tend model to our students – I rarely actively include triangles with obtuse angles, for example. This is a problem, and something that I need to actively address. Craig also highlighted the use of Intelligent Practice when introducing students to angle properties. I’m excited to explore using Geogebra to show students more explicitly the effects that changing just one thing has on the remaining angles. Finally – we need to go deep from the start. I’ve been exploring Don Steward’s Median blog (www.donsteward.blogspot.com) to find some nice resources for this:
Look at how fun this question is! It is a lovely opportunity for students to go deep and to be challenged.
Finally, I will be thinking a lot about my choice of language when teaching angles next time. The most obvious example which springs to mind is ‘angles on a straight line add to 180°’. While this is not untrue, it is full of vagueness and ambiguity, which is unhelpful to novice learners. Is at as unhelpful as ‘two negatives make a positive’? I don’t know. Maybe. Craig suggested several possible alternatives: maybe ‘angles that form a straight line add to 180°’ or ‘adjacent angles on a straight line add to 180°’. I can’t imagine that this change in language alone will have a significant impact on students’ understanding, but I think it is likely to positively affect at least some students. This can only be a good thing.
As ever, #MathsConf23 was the most incredible day of CPD. Every single session left me thinking hard about my own practice; I have so much to reflect on. I enjoyed the virtual nature of the conference more than I thought I might, and while I am excited to see MathsConf friends at the next in-person conference, I am so impressed by how well the whole of the La Salle Education teams put together such a well-run day for so many attendees in such a short time. I am extremely glad that the conference presentations will be available to watch after. The six sessions that I attended are only a small part of the entire conference, and it’s often incredible difficult to select sessions as many similarly amazing workshops will be happening simultaneously. I would strongly recommend each of the workshops above, but I have a list to catch up with now. A huge thank-you to all of the presenters, Mark McCourt (@EMathsUK) and La Salle Education (@LaSalleEd) for the most fantastic day.
I’ve been thinking a lot recently about the possible tension between having high expectations of students and teaching them the ‘wrong’ maths.
When I first started teaching, differentiated learning outcomes were still very much in vogue at the school where I worked; it was required that lesson objectives were structured as All, Most, Some. This always felt uncomfortable; after all, when I was planning lessons/units, I was thinking about what I wanted all of the students to be able to do (e.g. ‘add fractions with unlike denominators.’). To then make this fit with what was required, I would end up with something like:
Firstly, this was logistically extremely frustrating. It typically required multiple worksheets and activities for pupils to complete; resourcing would take far more time than it needed to. It also meant that when I was modelling a procedure, I was acutely aware that it wouldn’t be directly relevant to the work that each student was completing; this led to teacher exposition being rushed through and then most lessons being spent rushing around trying to correct misconceptions which had been formed. It also meant that it became incredibly difficult to plan the next lesson (I am aware that the lesson is the wrong unit of time to be thinking about!). If lesson 1 of a sequence had the outcomes detailed above, then lesson 2 would end up being something such as:
Logistical issues aside, however, there was something more that bothered me about the need to differentiate learning outcomes in this way. Some pupils would never be engaged in achieving the ‘most/some’ outcomes; after all, they had completed the work that was compulsory! ‘Most/some’ were seen very much as optional extras. The greatest issue was that it resulted in me lowering my expectations of what my students were capable of.
That’s a really difficult thing to admit; I certainly wouldn’t have recognised this at the time. Whenever I was planning around which students were likely to end up completing which activities, the vast majority of the time my groupings would have been extremely similar. The same pupils, lesson after lesson after lesson would work on the ‘all’ objectives, without being provided with the necessary materials to develop beyond this. It also meant that my planning focused almost entirely on the middle ‘most’ group; ‘all’ and ‘some’ were added on at the end (in primarily what felt like a box ticking exercise) and I had failed to think enough about my questioning and the support for those two groups.
A couple of years later, and differentiated learning outcomes are nowhere to be seen in my lessons! This is hardly surprising – they appear to have fallen out of fashion pretty much everywhere. My planning process has improved (with far more of a focus on ensuring long-term learning over short-term performance, ensuring regularly opportunities for revisiting, giving pupils sufficient time to secure new concepts/procedures before expecting them to problem solve etc.) and so my ‘lesson objectives’ are much a thing of the past. While I always have an idea of what I would expect to achieve in a particular lesson (keeping in mind how this fits into the bigger picture), I know that learning isn’t linear – if it takes more lessons than I had previously spent to ensure all pupils can add fractions, then so be it! All of this has very much proved to be a change for the better.
I’m not quite sure I’m there yet. I’ve written before about the mixed attainment year 7 class I teach – how I spent a year differentiating by task in every possible way, compared to now, where I spend more time focusing on ‘low floor high ceiling tasks’ and differentiating in terms of teacher support and time given. The thought that keeps coming back to me is this: am I teaching them the ‘wrong maths’?
Mark McCourt (@EMathsUK) has written/spoken about this idea of the ‘wrong maths’ – see here. Mark makes the point that we only learn things which are ‘just beyond our current understanding’ – new things that are only just beyond can be assimilated into what we already know. New things, however, which are way beyond our current knowledge, cannot be assimilated. This seems to be a perfectly reasonable claim!
(Anecdotally: I attend a weekly pub quiz. Each week, we are given the themes of two topics in advance, to enable revision if one is so inclined. When the topic is something such as ‘Eurovision’ – that is, something I already have a reasonable amount of knowledge of – I am able to read and assimilate new information fairly quickly, as I can ‘hook’ new facts onto those I already know. When the topic is something such as the Sengoku period (a period of Japanese History from 1467 – 1600), it proves significantly harder to learn or retain any relevant facts, as I am missing so much of the bigger picture knowledge into which new facts can be incorporated).)
Thinking ahead to next term, I’ll be doing some work with my Year 7 class on algebraic notation and solving equations. One of the things we will be covering is expanding brackets, and I’ve a fairly good idea on how those lessons might go. I’m aware of some of the resources I’ll want to use, I’ll use some diagnostic questions as formative assessment to gain a more accurate idea of pupil’s starting points, I’ll be interleaving previously taught content (e.g. calculations with negative numbers) and I’m reasonably confident that I can ensure, through this, that all pupils are successful.
The nature of this success, though, is questionable. As you would expect from a mixed attainment year 7 class, the pupils have a significant range in their starting points. Let’s consider Pupil A: Pupil A cannot confidently/consistently recall their times table facts to 10*10. Pupil A’s knowledge of number bonds to 100 is weak, and Pupil A can currently only divide numbers with the aid of drawing dots and sharing them into piles. I am still fairly sure that with careful scaffolding and support, I can ensure that Pupil A achieves success when expanding brackets – except – I’m not really sure what the point is. The short term knowledge of ‘multiply the number outside by everything inside’ is not going to be assimilated successfully, as Pupil A doesn’t have the required knowledge to assimilate it into – quite simply, it is too far removed from their current mathematical frameworks.
I want to be clear that this is not simply a case for not teaching mixed attainment classes; I think it is more than possible to teach the wrong maths to pupils in more traditionally setted classes. I teach a Year 11 Higher class, in which they will (hopefully!) go on to achieve grades of between 7 and 9. One thing I’ve taught them is how to solve quadratic inequalities – for some pupils in this class; this was absolutely the right maths for them to have learnt when they first encountered it (back in Year 10). For pupils who can factorise quadratics where a>1, can solve any quadratic equation and any linear inequality, solving quadratic inequalities is a lovely topic! It is something that is just beyond their current frame of reference but that can easily be incorporated. For other students in that class, it was just too much – and even though it is perfectly possible to differentiate and to put scaffolding in place to ensure their initial success, it simply wasn’t an appropriate area of maths for them the encounter when they first did. However, at the time, I approached it with the view that by teaching such a challenging topic to all students, I was ensuring that I had the highest possible outcomes of what all pupils can achieve.
The reason this happens so frequently is obvious – teachers open up Schemes of Work, check what they are meant to be teaching and go from there. Of course – we will consider if things are appropriate for the class we are teaching and adapt and differentiate as appropriate, but it still feels that there is a need to follow the Scheme of Work even when it doesn’t fit.
I’m thinking about our Key Stage 4 Scheme of Work at the moment – it needs reworking to ensure a more logical sequence of content and various other things. I need to ensure a way in which it can ensure that as many pupils as possible are being exposed to the right maths for them. I’m not sure what I can make this look like, practically. I’m also not sure of the best ways of assessing to determine exactly where pupils are in their journey of learning mathematics. I do know that I am going to make more of an effort to move away from teaching the ‘wrong maths’ (that is: mathematics that is too outside the realm of students’ current knowledge) and that doing so does not result in my expectations of students’ potential being lowered.
Teaching topics that students are not ready for typically results in long-term learning not taking place; it also leads to students who encounter the same topics time and time again, who continue to fail to be successful. While I don’t yet have a clear set of actionable steps of exactly how I want to improve this in my own classroom moving forward, I know that I need to stop equating ‘challenge’ with ‘high expectations’. Indeed, by ensuring that the level of challenge is appropriate (rather than just challenging!) pupils’ long term success will improve – the best possible outcome!
Yesterday (March 9th 2019) I attended #MathsConf18 – my fourth Maths Conference! This time around, I wasn’t presenting which meant I could attend five workshops, alongside mathematical speed-dating and making hexaflexagons in the Tweet Up at lunchtime. As ever, the whole day was packed with content and I came away (as always) more determined to improve my teaching and with strategies of how to do so! In this post, I will share some initial reflections on the workshops I attended.
My first workshop was on Managing Workload from @MrEdWatson – a Head of Maths from Bristol. While I would generally describe my workload as being ‘manageable’, this is something that I have to work hard to manage and typically involves working longer term-time hours than is probably optimal. As such, I was keen to leave with some strategies to put into place to help here.
Ed encouraged us to think about a typical teaching day, and place the tasks that we complete into the below table (taken from the work of Stephen Covey).
During a typical day, I would describe most of the tasks that I complete as being both urgent and important (quadrant 1). This, though, often leads to a feeling of ‘fire-fighting’ and higher stress levels about the tasks to be completed – it is better for productivity if we are working in quadrant 2 (important, not urgent) as much as possible.
Ed also spoke about how it is really easy to spend a lot of our time being busy, but not productive. This really resonated with me! It is quite easy to make it to the end of a long day at work and still feel as though you have failed to accomplish what you set out to do. A possible strategy to help rectify this is to write a ‘to-do’ list of tasks for the next day the next evening; while this sounds obvious, it isn’t something I do routinely which means I end up completing tasks ‘as and when’, rather than scheduling my time most effectively. We also looked at how all tasks expand to fill the time you have available – if you have 3 hours to plan a lesson, it will take you 3 hours. If you have 10 minutes, it will take 10 minutes. Again, this is something I want to work on! While I am certainly not a perfectionist, I am definitely guilty of spending hours on tasks which will not have a significant impact on student learning, simply because I have the time available. Ed really stressed that time is precious – to be as impactful as possible, we need to be ruthless with how we allocate our time.
There was so much food for thought in this session alongside some potentially controversial points raised (textbooks – yes or no?) and Ed was a fantastic speaker – I came away with things that I want to try immediately; we will see how that goes!
I finally managed to attend a workshop by Naveen Rizvi (@naveenfrizvi) – after the past two conferences where I was presenting at the same time as her, it was such a wonderful opportunity to see what she’s been working on!
The focus of Naveen’s presentation was on atomisation (that is, breaking down a task/skill/procedure into all of its sub-skills/tasks), with a specific focus on teaching angles in parallel lines. I’ve been working hard over the last year to break down content as much as possible when I am teaching, but I always find this much harder when trying to teach geometrical concepts, so I was really keen to find out how Naveen had done this.
One thing Naveen said was that most typical exercises for students to practice angles in parallel lines were either really easy, or really hard, without much for the ‘in-between’ stage. This is something that I have definitely found in my experience – most students can find pairs of alternate, corresponding or co-interior angles where that is the only skill required, but then struggle to access multi-step ‘exam-style’ questions. Naveen took us through a series of examples/questions that she had developed. It was fascinating to see how by focusing on changing just one thing (for example, including an extra transversal) each time, her students would be able to be successful at the most challenging mathematics. Throughout her presentation, this emphasis that Naveen placed on ‘success for all’ was apparent – she stressed how by spending time on planning carefully chosen examples and scripting explanations, it meant that all her students would achieve success, rather than just a handful of ‘high-attainers’.
The other thing Naveen said which resonated hugely for me was that by ‘breaking down a topic into its sub-tasks, a teacher can identify 100% of the domain of knowledge’. When teaching any topic, it is easy enough to look at a scheme of work, teach what you believe to be the required content, and yet later testing reveals that your students have significant gaps. For example, when teaching ‘angles in parallel lines’ previously, I may well have forgotten to include any instruction related to isosceles triangles and angles in parallel lines, or parallel lines involving multiple transversals. I almost certainly would have failed to include multiple orientations. The amount of thought that Naveen puts into how a topic should be taught to ensure everything followed in a logical order and didn’t lead to further misconceptions was remarkable. I’m not sure when I’m next teaching this topic but I already cannot wait!
For workshop 3, I attended Andrew Taylor’s (@AQAMaths) session on old exam questions and papers. Having only sat my Mathematics GCSE in 2009 and having started teaching in 2015, I wasn’t that aware of how the qualification had changed over time, so I was excited to find out.
Andrew started with a couple of non-calculator arithmetic questions for us to try (I cannot remember at all which year these were taken from!) – it was very apparent how the level of challenge for arithmetic in 2019 is certainly not what it once was. Several people commented that even their highest attaining year 11 students would struggle with what was required.
Andrew shared a number of examples from previous Mathematics exam papers with us; it was so interesting to see how certain types of question haven’t changed much at all in the last fifty years whereas others would be virtually unrecognisable (my favourite was was the one featuring Welsh towns and trigonometry!) to a student sitting the paper today. I also had no idea how much choice there used to be in Mathematics papers/qualifications. The University of London’s O Level in 1957 featured three compulsory 2.5 hour long papers on Arithmetic and Trigonometry, Algebra and Geometry, alongside an optional paper (which could be on the History of Mathematics!). It certainly put into perspective the three ninety minute papers that our students sit today.
Again – this was a really fascinating session! Andrew gave us an old O Level paper which I cannot wait to show my students next week.
After lunch and the mathematical tweet-up, I attended Jo Morgan’s (@mathsjem) session on Topics In Depth: Unit Conversions. Now, I cannot pretend to be particularly thrilled when I see that I have unit conversions coming up on a Scheme of Work, so I was hopeful that I would come away feeling at least slightly more inspired.
Jo started with showing us the KS2 National Curriculum requirements for working with measures and different unit conversions – she made the point that many of our Year 7s will be fairly adept at converting between different metric measures (and solving problems involving this skill) but that this often seems to be lost somewhere between Year 7 and Year 11. This is something that has definitely been true in my experience; I have taught numerous Year 11 students who cannot remember the number of centimetres in a metre or grams in a kilogram.
Jo shared with us some of her research about the etymology of unit prefixes (kilo-, deci- centi-, milli-, etc.). While this is something that I do mention to my classes, I think I tend to skip over it in favour of practice. I hadn’t known that bigger units (kilo, mega, giga etc.) all come from Greek, whereas smaller units (deci, centi, milli etc.) all come from Latin – while not necessarily ‘useful’ knowledge, definitely something interesting that I want to share with classes in future!
Jo shared with us a number of different ways that we can teach students to convert between different units; conversations with the people I was sat with suggested that we generally favour a ratio table or a common sense/reasoning approach. I’ve recently loved using proportionality diagrams for all sorts of ratio/proportion problems, so I was really pleased to see that this excellent post by Don Steward was mentioned.
The most fascinating part was Jo teaching us a new method for unit conversions; that of ‘last unit standing’. I have still not decided how I feel about this method or if it is something I will ever use in the classroom – but it was such a delight to see something completely new! Here is how it would work for a relatively simple problem:
Convert 250mm into metres.
It also works for compound measures! For more on this, I would recommend look at Jo’s slides which are available here. I cannot stress enough – even if I make the decision to never use this with students, I love knowing that it exists and that it gives me another tool at my disposal.
For my final workshop of the day, I went to see Kris Boulton (@kris_boulton) presenting on solving equations. It is always a real pleasure to see Kris present because it is so clear how much thought has gone into everything that he shares. Having spent a week at King Solomon Academy when Kris was teaching there in 2016, I know first-hand how really carefully sequenced instruction leads to remarkable outcomes, so I am always keen to learn more about how he approaches his planning.
Kris shared with us 17 processes that students need to be fluent with to be successful at solving one-step equations. Something I really noticed from this was how much important Kris would place on what the equals sign means, alongside looking at statements of the form:
before beginning the process of solving. While I have been developing this over the last few years, I am definitely guilty of choosing:
as the starting point, when to do so is presuming an overwhelming amount of prior knowledge that my students wouldn’t have access to.
The method Kris shared with us for solving equations is 4 steps:
Usefully, by identifying these four steps as exactly what needs to be done to solve any linear equations, we don’t have to expect students to complete all four steps from the very beginning. For example, it may well be beneficial to spend a period of time on just the ‘break’ stage, before moving to looking at ‘repairing’. I was also particularly taken by the use of the words ‘break’ and ‘repair’ – typically, I have stressed ‘having to do the same to both sides’ when teaching equations, but I am not sure the extent to which students actually understand what I mean or why they are doing it. There was something intuitive about how Kris introduced breaking and repairing, which I am really keen to try out.
Now, it is obvious that to teaching solving equations to a class using this method is going to take far more time than I would have spent on this previously. However – whenever I have taught this previously (to any class at any stage of their mathematical journey), they have had to be retaught various parts at various points. Ultimately, it will be better to invest more time from the beginning than to have to reteach and fix and correct many many times over. I left with so much to think about and ideas to try out.
Finally: I cannot thank Mark (@EmathsUK) and the whole La Salle Education (@lasalleed) team enough for all they do for Maths teachers alongside raising money for Macmillan Cancer Support. The whole #MathsConf movement has been such a massive support for me in terms of making connections with other teachers and improving my own practice. If you’ve not attended before, make sure you book for Sheffield in June – even if you are nervous about coming alone (as I was, the first time!) you will meet the best people and have the best CPD. See you there!
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.
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.
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.
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.
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:
All students were quite happy with the idea that this would be 2, once it had been shown diagrammatically. We moved onto related questions:
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:
They could replicate the changing of ¼ to 4, and the division sign to a multiplication sign, but I still think something was missing.
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:
This gave me a really useful starting point.
I began the lesson by giving them these three related statements:
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:
(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:
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:
After multiplying and simplifying, they noticed that when you have:
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:
I challenged them to write a related division statement. After a few moments of thinking, pupils shared the following ideas:
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.
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:
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:
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:
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.
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):
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.
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!
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):
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.
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.
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.
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.
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!.