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!