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!