MathsConf23

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.

Session 1

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.

Session 2

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:

image 1

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.

image 2

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.

 

Session 3

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.

Session 4

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.

image 3

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:

image 7

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’.

image 4

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.

Session 5

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.

image 5

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).

Session 6

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:

IMAGE 9

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.

image 8

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:

image 5

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.

Final Thoughts

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.

High Expectations vs. The Wrong Maths

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:

  • All students to add fractions with common denominators.
  • Most students to add fractions with unlike denominators.
  • Some students to add mixed numbers

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:

  • All students to add fractions with unlike denominators
  • Most students to add mixed numbers
  • Some students to problem solve (read: answer some questions with words in) using addition of fractions

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.

However.

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!

#MathsConf18

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.

Workshop 1

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).

covey-time-management-matrix.001.001

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!

Workshop 2

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!

Workshop 3

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.

Workshop 4

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.

conversion1conversion2

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.

Workshop 5

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:

7+3=6+4

before beginning the process of solving. While I have been developing this over the last few years, I am definitely guilty of choosing:

x+4=9

 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:

  • Decision-make
  • Break
  • Repair
  • Simplify

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!