A couple of months ago, Jo Morgan published a blogpost in which she discussed the level of autonomy maths teachers are allowed (and which she claims is, on the whole, quite high. My experience would agree with this). For a general read and a background to what I’m going to write here, I recommend reading Jo’s post first.
It’s a dilemma for any head of department or MAT lead – to what extent should I tell people what to teach and how to teach it? – but these discussions take a particular flavour in mathematics. In future blogposts I’m going to talk specifically about mathematics curriculum, assessment and more, but first I wanted to articulate a few thoughts about my general perspective.
I think there are times when autonomy is appropriate and times when it is not. It’s not simply a matter of saying “do it how you choose” or “you will teach everything this way” because, as always, there’s more nuance to it than that. Jo considers a spectrum of autonomy which encompasses a teacher free-for-all, through ‘loose’ schemes of work with associated learning objectives, through shared departmental methods or shared lesson resources, to scripted lessons. We might call this spectrum content autonomy as it, to a greater or lesser degree, dictates the substance of the content to be taught.
Some schools also dictate how long must be spent on certain objectives. This spectrum can go from ‘here’s the national curriculum for KS3, cover it by the end of Y9’ (does anyone give this much autonomy? I don’t expect so, but you never know), through ‘here’s what you must cover each year/term/half term/unit’, to lesson-by-lesson plans. Let’s call this spectrum time autonomy.
The third area to consider is to do with lesson delivery itself. Some teachers are allowed to decide exactly how each lesson will proceed, others have to include a certain type of starter or plenary (e.g. a ‘Do Now’ or ‘retrieval starter’), others have to include many more features (e.g. the old three-part lessons, ‘non-negotiables’ or the more recent Rosenshine checklist, where people have to use every element of the list in each lesson) and others are given scripts. Let’s call this spectrum instructional autonomy.
There may be more spectra, but these are the ones I want to consider for now. It can be helpful to think about where you/your department/school fall on each spectrum. Does your school dictate generic non-negotiables but leave what you teach up to you? Does your department have shared methods but leave the length of time spent on a topic up to each teacher? Thinking about where you fall, and the pros and cons of each approach, can help you to find a position that works best for your situation.
In my next few paragraphs I will set out my current position on these spectra, which can sometimes fluctuate depending on context and the school I am looking at. I have mathematics in mind, but I think the ideas would transfer relatively easily to other subjects.
I believe that a huge source of pupils’ difficulties in grasping mathematics comes from the piecemeal nature of its presentation and the lack of coherence this produces. As teachers, we can see the connections between topics and our own schema of mathematics is probably a nicely-connected web, where one idea feeds seamlessly to many others. We know that we didn’t always see it like that, and we know that many (most?) of our pupils don’t see it like that. To them, it can be more like a random collection of stuff they have to memorise. We need to accept that we exacerbate this disconnectedness when we don’t explicitly reveal the connections. In other words, we can take pupils over the bridges, or we can put up roadblocks.
I would argue that finding consistency in the we teach maths in a department is an important way to cross the bridges. I don’t believe there is one correct or best way, but I do believe that having things taught differently between teachers is more likely to make a roadblock.
Let’s take a classic example: expanding and factorising brackets. There are lots of viable approaches, but if Mrs Sherwood teaches expanding a double bracket using the parrot’s beak arcs – the way she herself was taught – in one year, and then the class moves to Mr Barton, who decides to teach it using FOIL (not to suggest Mr Barton would ever consider such a travesty) then, to pupils who are still getting to grips with this and everything else, it can feel like a completely different process. However, if pupils worked on multiplication grids with Mr Hall in Y7, then went to Ms Quinn in Y8 who expanded and factorised single brackets with the same multiplication grid, then onto Mrs Sherwood in Y9, who extended the presentation for double brackets, then onto Mr Barton in Y10 and Y11, who also uses it for factorising when the leading coefficient is greater than 1, then onto Mr Gokarakonda in Y12 and Y13, who uses it for polynomial division, then the method becomes subordinate to the concept – multiplication of numbers and expressions – because the method remains invariant as the concept gains complexity.
This allows pupils to make meaning of the concept, and its links to others, more readily, without our teacher-made, artificial obstructions in the way. I like to call method consistency a curriculum thread – something which weaves its way through our curriculum, sewing all the parts together.
There are other threads which achieve similar aims of making connections and weaving strands of the curriculum together. One of these is the metaphor thread. A mathematical metaphor is an image that helps us to conceive of something abstract. The number line is one such metaphor, as is the Cartesian grid, as are manipulatives such as algebra tiles or Cuisenaire rods, or images like the bar model. If we pick our metaphors and weave them throughout the curriculum we create coherence through building more connections. Algebra tiles, for instance, can be used to think about the four operations with numbers and expressions, and particularly to grid multiplication, which in turn links to area. They help us to make greater meaning of something like completing the square and add to our understanding of the abstraction in algebra.
Metaphors, like methods, work best when they are used consistently across topics and teachers, so that regular exposure creates familiarity and they become a tool with which to think.
A third thread is concepts themselves. If we are clear where we expect concepts to be seen and re-seen, we are more likely to get consistency in pupils’ exposure to them between teachers, and coherence in the curriculum in the sense of a purposeful design where concepts deliberately grow in their complexity, detail and use. For an example, I have taken the concept of ‘solving equations’ and mapped out where I expect it to be deliberately taught, used and built in throughout my curriculum.
The idea of doing things deliberately, rather than leaving them to chance, is one that permeates all teacher activity and I’ll return to it in another post.
It’s this first content autonomy spectrum where I see the greatest benefits from less individual teacher autonomy (although it’s essential to work as a team to ensure that everyone understands and is party to decisions made). Less autonomy on this spectrum ought to result in better outcomes for pupils and understands that pupil results are a function of everyone who’s ever taught them. A silo mentality helps no one. Done well, the benefits are clear: an improved experience for pupils through a coherent curriculum, time saved for teachers through reducing the need to search for resources online, and a curriculum that is well understood by teachers at all stages in their career. However, autonomy allows professionals to thrive and that’s where I see benefit in looking at the other spectra separately.
A head of department should know what’s going on in their classrooms, but does every class need to be doing the exact same lesson at the exact same time? Categorically, unequivocally, no.
Lesson-by-lesson plans are the antithesis of responsive (read ‘good’) teaching. A teacher needs to be able to adapt what happens to the class in front of them. Some students may need more practice with an idea than others. You won’t know until you’ve taught them, done some formative assessment, and responded accordingly. A lesson-by-lesson plan assumes all students learn all things at the same rate and in so doing it confuses consistency with homogeneity.
A homogeneous system is one that has the same properties at every point, one without variance. Classes are not homogeneous. Pupils in ability groupings of any kind are most definitely not homogeneous, but neither are mixed-attainment groups. To treat them as such and have a lesson-by-lesson plan that every class must follow is nonsensical.
A better way (I don’t say ‘right’ way, because there isn’t one right way) is messy and difficult, because we’re dealing with humans who are, well, messy and difficult. There are options, and each has its own pros and cons. We have to choose which suits our context – students and teachers. Moving pupils on is one decision that we have to consider in this way. If your pupils are setted and every set moves at its own pace, then pupils ought to be experiencing a well-tailored curriculum. It does, however, make set changes difficult, or even impossible, placing an unnecessary ceiling on progress over and above what we might ordinarily expect. There is a risk that less experienced or skilled teachers will move too slowly or too quickly through the material, each of which carries its own problems. If you teach in mixed attainment groupings or you have sets that all move through units at the same pace, you can pull some students along who might otherwise have stagnated and you enable movements very easily. You do, however, risk going too slowly for some while too quickly for others. These are perennial conundrums in teaching, and we all have our own opinions on better or worse options. Whichever option your department chooses to take (moving on together or at different times), you need to be aware of the pitfalls and work to reduce them. The road to any solutions involves communication and professional development – being aware of which teachers need support (and giving them that support) but also opening channels of communication that enable you to understand, as a team, what is happening in classrooms and what needs to be changed at a departmental and classroom level as time goes on. It’s part of the ongoing, essential development and iteration of curriculum that any good department will put at its core.
My personal preference (at this moment in time) is for a department to develop units of work together that everyone uses (linking to my position on content autonomy) but which include tasks and activities that are deemed essential for all pupils, those that can be used with pupils who need more practice or support, and those that can be used with pupils who are ready for deepening or extending. Sequencing is important, and should be thrashed out in discussion, but timing through the sequence is up to the professional judgment of teachers.
If such an undertaking is done collectively, we can help those who might not yet realise just how far they can stretch their pupils, we pool our collective knowledge and experience for the benefit of all, and we free up lesson planning time for professional discussion and upskilling of all staff.
This third spectrum is where I am least concerned about consistency and where I believe autonomy can be most powerful. There should be a high-level baseline, where every teacher understands the importance of key factors that make good teaching. Something like Rosenshine’s principles can be very helpful here, provided they don’t turn into a reductive checklist, and there ought to be an expectation that every teacher develops their personal knowledge and skills over their career, just as all professionals should. However, I’m not sure there’s anything to be gained from dictating how long a period of teacher exposition should last for, or what specific type of retrieval practice should take place, or that it should be at the start of every lesson.
As HOD, it really doesn’t bother me that Mr Barton chooses to use a multiple-choice question each lesson to formatively assess his pupils while Ms Quinn goes for mini-whiteboards and a weekly low-stakes quiz. I’ll have spoken to both and will know why they’ve chosen their approach and will spend time with them thinking about whether one seems to be working better than another in certain scenarios. I’d quite like us to develop a department-wide bank of formative assessment skills that form part of our collective craft.
If we, as a department, decide to begin every Y7 lesson with Numeracy Ninjas, because it’s only five minutes long, builds routine, helps students to settle quickly and without fuss, and provides lots of well-planned distributed practice on essentials that students need to practise and fits with our Y7 curriculum plan, then that’s probably a decent decision. If we carry on when a group of students no longer needs that particular practice, then that’s not such a good decision. If SLT says that there should be ten non-negotiable minutes of “Do Now” at the start of every lesson, because retrieval practice, then that’s not necessarily a good decision. Why should it be then? Why that long? How do you know what’s being done is useful? These kinds of decisions should be made at a departmental level, and a head of department might justifiably leave it to their teachers to decide what’s done when. Discussion between staff and HOD, and then between HOD and SLT, are key to this. These discussions, where we question and probe and help each other to get better are essential. Top-down diktat is far less likely to achieve what you want it to, while creating the illusion to those in charge that they are making things happen (see Chapter 5 on autonomy here). Provided decisions can be justified and you’re happy they’re good ones, they don’t all have to be the same ones.
Allowing autonomy to develop
There are other places where autonomy is absolutely fine. Teachers’ record-keeping is one such (only petty micromanagement would dictate that). There’s also an argument for allowing teachers increasing autonomy commensurate with experience. CPD is a good example, where early career teachers deserve set development which will allow them to flourish, but where more experienced teachers ought to be able to choose what interests them and what paths they want to pursue (much like what happens in school, through to GCSEs, A Levels and degrees). I expect that instructional autonomy fits this pattern, where more experienced teachers are better able to adapt their instruction to groups of students and should be allowed to do so.
Let’s open up discussions in school around autonomy and consistency. Where do we draw our lines and why? How can we strike a balance between improving everyone’s teaching, allowing space for innovation, not suppressing excellence, and creating the best possible conditions for learning within and across classrooms? These are our conundrums, let’s not pretend we have easy answers and let’s not shy away from them.
Your entire post was a essay on how to teach us maths. That is what I thought to myself after reading it. I teach functional skills maths to people who have dyscalculia etc. And I always use basic Year One methods in addition to help them. Maths is all about teaching them skills through technique and methods. For example I like to use stuffed toys to teach numbers of and shapes. Good luck.