While I was driving to a viewing with my wife, we were discussing our estate agent’s fee, which is 1.25% of the sale value of our house. She asked “would it ever not be worth it to get more money for our house?” This struck me as odd, but after a bit of back and forth I realised that she was effectively wondering whether or not increasing the sale price by an amount would lead to a net loss because the rise in sale price is less than the marginal increase in the estate agent’s fee from the house being sold. How would you respond? Have a think and write something down before reading on – I’d be interested in what you think!

My actual response wasn’t particularly inspiring – I could use the excuse that I was driving at the time and I very much had my summer holiday brain wired in. I stumbled over an explanation talking about the fact that a percentage retains its value in proportion to the given amount. If we go from 100 to 101 then that 1 rise can’t be swamped by the rise in 1% of 100 to 1% of 101. My wife seemed placated, or sufficiently bored, and we carried on to the minefield of “how many toilets do we need in a house?” (At least 2 is the answer).

I wasn’t happy with my answer, however. I feel that it didn’t really deal with the issue in sufficient depth and it would be easy for a student to then make any number of misconceptions: that the amount due to the estate agents never changes or that it would always be a really small change (which my wallet actively disagrees with). It also didn’t tackle head on the potential to explore the maths around the idea – what if it was 1% of the extra amount as opposed to taking 1% of the new total? Would that be better – why or why not?

Colin Foster recently released a video on the NCETM around armchair discussions on the mathematics classroom – the basic idea being that there are questions or events that come up during the classroom that open up a myriad of different responses and that in the moment we may choose any given response without fully appreciating the power of the different paths. Colin suggests that it would be a good practise to take a bit of time after a lesson to think about how we may have answered questions differently – he calls this an armchair reflection. Given the present issue, it struck me that it was a good opportunity to indulge in such a reflection. So, how else could I have responded to the prompt: “If you an increase an amount will 1% of the new amount ever be larger than the actual increase in the total?”

It might be a good idea to use specific examples at different extremes. This is what I tend to do myself when getting a feel for the generality of a concept, or finding out what a function seems to be doing (e.g. sketching graphs). What happens to X when Y is really big? What if it’s really small? In this case this might be a prompt to the student of “make some calculations when the price increases by a little bit and also by a large amount. I would be really interested in what you find out.”

Alternatively I may decide to use algebra to help shortcut to a general result. To do this you would need to do a bit of coding: call the current price X and the new price Y – how can we phrase our question? Ideally we need to identify what we are comparing – the absolute difference between X and Y and the the absolute difference in 1% of X and 1% of Y. The next issue is what sort of relationship are we positing? A student may be tempted to bang an equality sign in there without really thinking about it. However it is important to understand the implications of doing so. If we write Y-X = 0.01(Y-X) we are trying to find the break even point, the value of the variables where this must be true. Dividing through and getting 1=0.01 is a curious result – what does it tell us? How can we guide a student to interpret that answer? “Could this ever be true? Have you done anything incorrectly in your working? So what must that tell us about our starting point?”.

We could also bust out a bar model, but I’m not sure I could use it effectively here. My initial idea is to frame the argument as “what would it look like if the fee rose by more than the increase?”. Using diagrams it would look ‘wrong’ in the sense that the portion of the bar ascribed to the fee would look over 1%. I feel like there is definitely a better way of going about this, but not sure how, definitely something to give deeper thought to at a later date. I am sure that the bar modelling gurus of twitter will show me something that will blow my mind and make me go “WHY DIDN’T I THINK OF THAT?!”

Finally, I could use graphs. Writing that down, it feels like the quintessential mathematician’s phrase, but plotting a graph of y=x and y=0.01x is a great way to get a feel for the quantities involved and beginning a conjecture, and one could quickly look at gradients and consider them as the rate of change. Linking back to the original question, how might this system of graphs show us that the rise in the fee will always be less the rise in the absolute value?

Thinking on it, I would probably plump for the idea of using concrete examples at a range of magnitudes for most of my classes. The algebraic approach has a lot of baggage to it – you feel like to embark on it you need to get students to sign some sort of disclaimer – and is probably more suitable in a typical A level classroom. Not to say it is a bad approach, but it could well take up the vast majority of lesson time and distract from the main point you want to focus on. Although this may shock you, I’m still not proficient enough with bar modelling to be comfortable winging it in the moment. The graphical approach would potentially be my other go to, but it depends a lot on how comfortable students are with straight line graphs.

I hope you’ve found this interesting, and I wonder how you would approach such a question!

-Hisham

]]>Well, that was until I reached the section where Craig discusses his ‘rule’ variant of intelligent practice. The basic idea here is that it is simply not very helpful trying to describe what a concept is through text or even a static initial presentation of examples and non-examples alone. The obvious limitation of text is that, in attempting to define a mathematical concept with any degree of specificity, you end up with a beefy piece of text that basically requires one to know what you’re talking about in the first place to fully make sense. Let’s take ‘polygon’ for example, as it’s the definition used in Craig’s book.

In geometry, a

From Wikipedia (yeah, I know)polygon(/ˈpɒlɪɡɒn/) is a plane figure that is described by a finite number of straight line segments connected to form a closed polygonal chain orpolygonal circuit. The solid plane region, the bounding circuit, or the two together, may be called apolygon.

Case closed, everyone. Present that in front of your Y7s and watch the pennies drop, the lightbulbs spark on and maybe just a couple of the more emotional ones shed a tear for they now see the sheer beauty of geometry.

The idea is therefore that it is better to provide examples and non-examples to help highlight boundary cases in a much more efficient way than words alone could provide. Craig takes this idea further and argues that we can effectively get students to elicit what something is by carefully sequencing examples and non-examples in such a way as to draw students’ attention to specific aspects of a case. The process is presented below in one of the flowcharts from the book. To summarise, effectively the process is to present an example/non-example, have students meaningfully engage with it by discerning what is different from the prior example and make a judgement about whether or not the presented example fits the rule under consideration. There’s then a process of peer-discussion and whole-class explanation before repeating the cycle with the next example/non-example. For present purposes, the post-sequence activities (going deeper, recall) are less relevant so I’ll just point you to go and buy the book and read more about it yourself.

It makes a lot of sense and feels like it has the potential to squeeze ever more utility out of examples and non-examples to really hone in on specific features of a concept and create a more full boundary in students’ minds without having to resort to impenetrable tier 3 vocabulary to provide the definition that gives you the specificity you really need.

That said, I do wonder about the end point and in particular how able students are in articulating the definition of the concept they now understand. The point I’m making here isn’t that the worded definition is bad – it isn’t bad, it’s really good and it’s important as mathematicians that we have concepts codified to efficiently communicate them to one another. Even though it isn’t necessarily much help to a novice, it is not something that we should hide away from a student as they increase in their understanding, or presume beyond their grasp. It’s our job to get our students to the point where they are comfortable with the definition because it marries with their own internal understanding of the concept.

It’s with that question ringing in my mind that I became really interested in using the rule sequence for definitions. I think there is so much power there, but also a very difficult journey for the teacher to take – with each example/non-example they need to give students the space to form their own idea of the concept but also ensure that there’s very little variation in those ideas and that no-one is developing an incorrect picture in their minds.

So. Functions. This actually isn’t the first time I’ve experimented with a rule sequence in lockdown, so the way I ran it was actually quite different to the way that Craig suggests. It feels crass to suggest this, but a lot of the difference was actually because of practical reasons: I’ve got a real need to cover content efficiently with my Y12s due to the end of the year approaching and also it’s incredibly difficult to do any form of pair discussion remotely (although I did ask students to try and do it via whatsapp this time). I also altered the method of presenting examples and non-examples from what is presented in the book; given that I present such an unusual representation of a function, I felt it was helpful to constantly have it present as a reference point; the diagrams were too large to present all of them at the same time.

Here’s how I ran it: I began by asking students to think about what they understand a function is. Although it’s a thought for a different blog post, students have worked with functions before extensively and will have some sort of idea of what that word means. I thought it important to draw out that existing understanding so we can prime students to identify why that understanding may be incomplete. Below I’ve written some anonymised responses to that.

**“set of steps that returns a result given an input”“the f(x) stuff”“an equation that does something to a number”“f(x)”“a set of actions applied to a number”“(Computer Science) A subroutine that returns a value”**

So, looking at it, we have two main camps: students that have a rudimentary idea of a function as a process that transforms inputs into outputs (which is true although incomplete), and an idea that functions must be represented to f(x) notation, or that f(x) notation is its own discrete ‘thing’. Interestingly, we also appear to have a definition from a different domain. Let’s contrast this with the definition from Wikipedia:

“In mathematics, a function is a binary relation over two sets that associates to every element of the first set exactly one element of the second set.”

Source: Wikipedia

Skipping over the notion of a binary relation, I think some of the students have a general idea about the idea of associating one set of value (inputs) to another set of values (outputs). It is clear that their understanding is incomplete, and potentially highly tied to the surface structures that they formally learned at GCSE – the notation we typically use with functions.

This sets the scene for how I introduce the first example, where I use a pictoral representation of a function that they haven’t seen before, and I spend a decent chunk of time explaining the various features of the diagram, trying to tie it into their understanding of what a function is. I then draw their attention to what will happen over the next couple of slides, asking them to reflect, check, explain. Notice that I’ve omitted the expect part of the ‘Reflect, Expect, Check, Explain’ process – Craig acknowledges in his book that in certain cases where the ‘check’ portion is very quick, it can be difficult to form expectations of how the change in the question will affect the answer, but that it still important that we do so. I felt that in this context, trying to do it via remote learning, that there was little chance of enforcing an ‘expect before you check’ system, so I focused on the remaining three parts of the process. I also do not stop and discuss each case; this is partly for time reasons, and also because I wanted to see what the students could pick up in aggregate. These are A-level maths students, so I made the assumption that they could hold a fair amount in their working memory, and I made sure to give a decent chunk of time for self-explanation on each slide.

Below you can find my sequence – I’ve cropped it and stripped out any time when I mention individual students, so it should just be me doing the sequence.

Did my accent catch you off guard? I’d be interested in what accent you think that is!

At the end, I ask students to write down what they think a function is – I mangle the delivery of this a bit by trying to give them an initial phrase of ‘a function is…’ or to focus on the excluding factors ‘a function cannot…’ Here’s the interesting bit – their responses.

**“Functions map a set of values to another set of values. There is only one result per value, but the multiple values can have the same result…And every value must have a result.”“Each value in the function will have one output value. Some output values may be equal.”“a function must give one singular value for the range of x specified”“It must change any input value not equal to 0, one input cannot create 2 outputs, two inputs can have the same output”“A function cant have more than one outputs(Y) on a single input(X) and Every input needs a output”“each item in the first set must be mapped once to any value in the second set but can be mapped to the same value as another item in the first set”“all x values must be mapped to a y values. 2. can’t map value of x to two values of y but can map two values of x to one value of y.”“Every dot has to be used, but only once”“an expression involving in one or more variables.”**

It looks like the majority of pupils made the realisation about the two defining features of a function – that every value in the domain must be mapped, and that every value in the domain must be mapped to exactly one value in the codomain. However, there are a couple of misconceptions arising here – can you spot them?

- A value of zero does not need to be mapped. This is likely because the student is thinking each ‘dot’ in the diagram must be a non-zero value, potentially even an integer as that is the example I used.
- That we’re talking about dots, not values – too tied to the surface structure of the example
- The restrictions of a function go both ways (every dot has to be used, but only once – implying each value in the codomain must have only one value in the domain).
- (Not a misconception) a student clearly hasn’t really grasped much from the process and is also demonstrating a lack of understanding about basic mathematical vocabulary in the way they use ‘expression’.

Am I happy with the result? The answer is actually yes – I think that the misconceptions are quick to rectify and potentially smaller than if I had simply thrown a Frayer model at them and called it a day. Indeed, I spend a couple of minutes then clarifying the definition of a function to ensure a common understanding before then providing them some independent practise on identifying whether each of the below cases are a function – the results are that everyone is able to identify functions in this representation.

I then proceed to introduce other representations of functions and get students to identify whether they are functions or not using the definition that we have developed and practised – the representations I show are graphs, coordinates and xy tables.

Overall, I had my doubts about the use of rule – I wondered whether it would enable students to be able to articulate their sense of understanding of a concept. The answer is a little mixed and muddied by a couple of factors – the context in which this was run (particularly time pressure and the constraint of not being able to orchestrate pair discussion) and the fact that this was formalising a concept that had already been extensively worked with in the past. The way in which I had modified the sequence may also have hindered the development of a precise definition – undoubtedly talking about each example/non-example as they came would at the least clarify the identifying features. On the whole, however, even with the adjustments and adverse context, I felt as though it did add significant value and that, with a little further practise, students had a really good grasp of the idea.

I’m really looking forward to building this into my practice more and more during next year and seeing how different groups respond to it. So far, so positive; I used this via remote learning to identify appropriate lines of best fit with my middle set Y9s, and that worked really well and drew out a few misconceptions.

Another reason I’m particularly excited about this, is that it’s one of the first concrete modifications I’ve made to my teaching as a result of my wider reading. I feel like I’ve followed a process of reading, analysing + adapting, implementing and reflecting to make the most out of the research and advice given and it feels really refreshing. I’m looking forward to this being the start of a really long road.

-Hisham

]]>I wish I could remember where I read/heard this (I want to say it was in a video Dylan Wiliam shared) but someone said that “saying going on a course is great CPD is like saying going to Tesco was a great meal – it’s what you do with the information that counts”. In that spirit, I’ve written this post to give a general overview of some of my key reflections from the conference and to engage with one point in particular – Kristopher Boulton’s fantastic ‘what before why’ session which was perhaps the most controversial thing to hit the UK since Brexit. At one point, someone in the zoom chat said ‘I am done’ which I believe is the maths teacher equivalent of bottling someone on stage at the Reading festival.

I think I can chunk the general reflections into two discrete categories – implications for specific topics, and implications for more general pedagogical choices.

For specific topics there is a wealth of change. **Mr Mattock** (@mrmattock) presented a barnstorming session on conceptualising mathematics and proceeded to show how we need to consider the deep mathematics over surface techniques, and how that may help build understanding (and implicitly, build storage strength in long-term memory, as well as flexibility of the knowledge itself). A particularly nice example was transformations, which students often see delivered to them as a disparate collection of procedures to be delivered when the magical trigger words of ‘rotate’ ‘reflect’ ‘enlarge’ ‘translate’ are activated. Peter (Mattock) illustrated how, fundamentally, we could represent all of these transformations by considering the impact on vectors. I’ve used the vector method for enlargement for a while now and would never ever go back to the three separate-ish rules for positive, positive fractional and negative scale factors, but have never really considered rotations or reflections. Indeed, reflections fall into the category of maths which I feel is best labelled “*things that you need a good sense of spatial awareness to do because the method is really clunky and if you don’t have an intuitive feel for it you probably won’t be able to check it easily*“. Catchier name pending. The upshot of his presentation was that we can show really clearly how transformations could be tied together – thinking ahead to KS5 and beyond, this primes students really well for transformations represented by matrices.

Another topic that Peter discussed was using a visual representation to convey an understanding of the mean. As a side note, I found it very difficult to find the above images with a casual google search, so I’m pretty sure that this way of representing the mean is not yet widespread, but I have seen it before. I’m a little conflicted on this model – it does help with the concept of levelling out the data, and there are various interesting things you can do (what would be the effect if I had 100 squares instead of 14? Would that change the median as well? What would be the effect if I added another 9 squares to the set?). However, I don’t know if it easily transitions to the idea of the mean being a ‘central value’. In my own mind, I feel like there’s not a bridge between the ideas of ‘what if all the values were replaced by the same value” and “what number best represents the central point of the data” – the latter being the interpretation of the mean that I feel is more generally needed. What does that mean for my use of the model? I still want to bring it in as it’s a powerful visual and can nicely highlight some of the characteristics of the mean (particularly its vulnerability to extreme values), but it is not sufficient in of itself – it does not convey the actual point of the thing itself. It is important to note that Peter then went and used a second visual representation which looked really interesting (and I think linked closer with the idea of centrality), but I don’t 100% understand it yet – will need a repeat viewing for sure.

Moving on, the wonderful **Jonathan Hall** (@studymaths) led a lovely session on ‘behaving mathematically’ with a concrete focus on prime factors. It happens to be quite timely given my recent post on the process of prime factor decomposition , and was wonderful to get an insight into the types of activities to engage students with beyond ‘find the prime factorisation of … ‘ and HCF/LCM questions. Lots of lovely puzzle-esque activities that opened up nicely after a bit of poking and prodding (one of my favourites was making calculations a lot simpler using prime factorisation – some lovely interleaving of distributive and commutative properties as well). Something that came out quite strongly was the neatness of using physical prime factor tiles – something I had never seen before. While initially I was ready to hit ‘print’ on ten billion small pieces of coloured paper, I think I need to pause a little bit. The practicalities of having a lot of pieces of paper flying around in the classroom (“sir, James took my tiles!” “no I didn’t you bellend it’s HERE”) is an important factor and it may get in the way of working. After all, memory is the residue of thought and if students are thinking about teeny-tiny paper aeroplanes then that’s what will stick.

I will admit I was a bit surprised about **Phillip Legner’s** (@mathigonorg) highly engaging session on discovering mathematics. The surprise was simply because I thought it was going to be totally different in focus. As it was, it was a really nice travel through a fair few of Mathigon’s lovely interactives, which really made for great hooks into topics. The idea of an initial hook into a teaching topic is something that I’ve really decided to work on and make an integral part of my practice, and Mathigon is definitely on my go-to list for those now. Particularly nice is that it is both free and extends right up to A-level further maths, which is great when KS5 often gets slim pickings resource-wise.

**Jo Morgan **(@mathsjem) was another topic-specific session, this time on exact trig values. Let me just start by saying that it was a moment of pure catharsis when she, effectively, went on the attack arguing against the inclusion of exact trig values at foundation tier. Going forward, that will definitely be on a list of ‘if all students have mastered (properly mastered) everything else’. Jo did a great job of highlighting some really excellent resources as usual, but the real stand-out part of the session was her proposed progression through Y11 exact trig values, taking into account that ETVs require really solid trigonometry in the first place and very solid surd and fractional fluency. I managed to photograph part of the journey below, but it really helped to make explicit that ETVs shouldn’t be this standalone bit of maths, but integrated seamlessly into a larger journey, with plenty of interleaving through sine and cosine rules.

Lastly for this section is the first more general pedagogic session, which was hosted by the absolute ball of energy which was **Tom Manners** (@mannermatics). Dense with energy like a neutron star, Tom took us through a journey of examples where lack of specificity in vocabulary leads students through some very worrying misconceptions. I think the biggest thing that I took away from his amazing session was that, because we can’t be quite sure what our students are actually taking away from our lessons, we need to be as specific as possible in our language to limit the potential interpretations of what we say. In doing so, and in building consistency of vocabulary – and methods! – within a department, we can ensure that our students, too, communicate with purpose as mathematicians.

Before anyone takes this too seriously, I’m largely hamming this up for fun, although this session really did throw up quite a lot of controversy due to some of the apparent take-aways. I will also say that any and all of the below is taken as a professional discussion and that I truly respect Kris and his efforts to move the profession forward by sharing his expertise. It’s been heartening to see the vast majority of interactions on twitter about this being conducted in the same vein of polite professional discussion rather than personal attacks or unprofessional comments. The latter really have no place in our discourse and it is sad that they occasionally crop up – not least because they are incredibly distressing to deal with, but also because they act as a barrier to entry for people wanting to present their own thoughts and ideas in a polite and constructive manner, which is a real shame. One day, I hope to present at a MathsConf (when I have something of interest to say!) and I worry about getting nasty remarks.

I won’t go through Kris’ entire presentation, but the general gist of it can be summed up thusly: a typical model of teaching is to walk students through the derivation of a method (this may also be through an investigative task) before then isolating the method itself and using it over and over. Kris proposes turning this on its head – first present, in a fashion, the method and once students are proficient with the method then go back and explore why it works. At its heart, I think Kris is basically talking about the sequence of learning as opposed to adding new bits or losing bits.

A persuasive example was shown early on with index laws. I didn’t manage to get a picture during the time, but I’ve attached a clipping of my notes below (good luck with the handwriting!). Effectively, there’s the use of variation theory to get students to get a feel for how the rule works (Dylan Wiliam talks about ‘having a nose for quality’ in regards to formative assessment – I feel like a nice phrase for this is ‘having a nose for the method’), along with backwards fading in practice before what Kris terms ‘expansion questions’ – effectively varying the surface structure of the questions to begin to make that inflexible knowledge more flexible. Once that has been secured and fluency is assured, then the teacher must move onto the ‘where does this method come from?’ One point that Kris is at pains to make is that this is ‘what before why, not what without why’.

In the example presented here, I think the model is persuasive, and Kris’ arguments as to why he prefers the model (revolving around reduced cognitive load and the fact that the why is often not reinforced explicitly) make perfect sense. On that second reason – the why not being reinforced explicitly – I feel a sense of kinship. I’ve been varying up the way I teach fraction addition and I lately took my Y9s through a journey of using an area model of fraction arithmetic. It definitely made it clearer, but when planning I had to sit and really think ‘to what extent do I get kids to stick with this representation? Showing it a couple of times on the board doesn’t feel sufficient, and it feels like something I need to do across multiple lessons to have any chance of sticking’. In the end, I enforced students always drawing the area model of addition and subtraction for multiple lessons and writing the abstract underneath, but reflecting on it, I’m not really happy with it. I feel like – yes, students get why we need common denominators when adding now, but also it is an incredibly inefficient method of working out the answer, not to mention not generalisable (practically) when you have a large common denominator. In this case, I’ve been trying to make the ‘why’ stick by fusing it with the ‘what’ in the hope that I can kill two birds with one stone, but it didn’t quite deliver everything I wanted it to.

The particularly contentious point, I feel, came when Kris was exploring solutions of quadratic equations, particularly those which have already been factorised. I didn’t hear it but it’s purported that he either said or condoned the idea of ‘change the sign of the constant and divide it by the coefficient’. The idea was largely the same as his example using index laws – to get students really good at mechanically finding the solution, and then go into what that means later. Here, I’m more conflicted. I think I’m pretty happy to say that while I’m on board with getting students to infer a rule, I disagree with presenting them a ‘trick’ in the sense of a shortcut that is divorced from the conceptual maths. I think change the sign and divide is one of those tricks – yes, it comes from the maths (as all quick procedures do), but it feels like a totally stand-alone thing, and inorganic in a sense. The thing, I suppose, that feels particularly egregious here is that I don’t think the understanding – the ‘why’ – is particularly difficult for students at this level anyway. **In terms of cognitive load, I would strongly argue that it is more cognitive load to recall this rule that isn’t really used anywhere else in maths at GCSE than to write out ax + b = 0 and apply a procedure that students should already be fluent in**. In addition, I think this is a perfect case of where the ‘why’ is reinforced continually – I always get students to write out the factor = 0 and solve for the roots, because it is really only one extra line of working. It is far from clear cut that we gain efficiency here, and, in my view, it seems to muddy the water of ‘why’ as well.

I think I’m pretty happy to say that while I’m on board with getting students to infer a rule, I disagree with presenting them a ‘trick’ in the sense of a shortcut that is divorced from the conceptual maths.

I’m not convinced that it is as universal a solution as Kris purports it to be (he is, however, very upfront about being persuaded, which is very refreshing), but I do think it is a potentially powerful strategy to employ with certain topics. For example, I think I would readily apply this to quite a lot of geometry (area of a circle, area of a trapezium, Pythagoras – please don’t hurt me) and some algebraic concepts as well (potentially expanding brackets?). That said, I think it runs into direct competition with the idea of manipulatives – which I think are at their most powerful when initially introducing an idea. For example, using double-sided counters for negative number arithmetic – what would students really gain from them if they were already fluent with negative number arithmetic? However, they are such a brilliant and powerful visual when introducing the concept and are potentially less prone to building misconceptions than mis-applying an implied rule.

Fundamentally, I think this is an area where lived experience speaks volumes, and the only way to tell for sure is to try it out, record the results and reflect on the process. My gut is that this a technique to use with a subset of topics in maths, and potentially use less with stronger mathematicians (who have stronger retrieval strength) than weaker mathematicians. I’ve signed up to one of Kris’ other courses this week since my appetite has now been whetted!

It is fantastically exciting news that La Salle are placing the entire conference up on their site for free – there were plenty of panels I wish I had been able to attend, but had to make a few difficult decisions. I really think it is so important for the profession to have a bank of easily-accessible CPD such as this; I know I will be sharing with my department(s) and look forward to the discussion that ensues. If anything, it is energising getting the time and space to consider the teaching of mathematics when so much of our time is bogged down in admin and day-to-day. I truly enjoyed it, and can’t wait for the next.

-Hisham

]]>Clearly, we do not live in a wonderland, as recent news has shown us. It’s important to note that this isn’t new. There is nothing new about the systemic racism and bias that riddles a lot of institutions in the UK and abroad, and before you harrumph your way to a reply on that, then just do even the World’s tiniest bit of research on it. I will say that I think massive strides have been made, and that – functionally – there is the access to institutions and acceptance in today’s society that I feel there was sorely lacking even a generation ago. I am not called a nigger while walking down the street like my father (a doctor) was when he arrived in this country.

I am not an expert on race, nor am I particularly schooled in the issue of race differentials in schools in the UK. However, I am mixed race, I am a maths teacher going on to be a Director of Learning for maths, and I feel that it is important to say *something* at this juncture of my experience. I do not claim that I can provide solutions, nor do I claim that my experiences are universal. However, I suppose one thing that all of this has raised is that it is important to listen. Far too often, when BAME people, or people from lower SEGs speak, no one listens – although plenty of people are often on hand to patronise.

Far too often, when BAME people, or people from lower SEGs speak, no one listens – although plenty of people are often on hand to patronise.

**Explicit racism hurts, but what hurts more is the inaction of those around you.**This lesson I learned when on a night out with my now-wife and her teacher friends, in a pub in North-West London near her school. I didn’t really know any of them, and most of them were, fairly enough, a bit tipsy at this point. There was this absolute waster of a man (not in our group) who kept yelling loudly about how he HATED PAKIS, and FUCK PAKIS, while making purposeful, gloating looks in my direction. A smug grin on his face saying “go on mate, hit me and you’ll end up in jail”. Now, that was obviously a bit upsetting, but the biggest thing was that no one spoke up about it. I don’t really know what I expected – but at the least someone to come up to me and go ‘hey, you okay?’ Instead, it just seemed that he could do whatever he wanted and get away with it. Which he did. If you see explicit racism happening then don’t sit there and do nothing. Where safe to do so, try to intervene – or at least check in on the victim. In a school setting, if you see racism in the class or on the playground, act publicly, decisively and follow it up all the way to the top and all the way to the end. Do not pass it to HoY and wash your hands – like it or not, you are part of that now, see it through.**The subtle racism of lowered/raised expectations is real and we have to be constantly on guard for it.**I’ve heard staff members in various schools say sweeping generalisations about groups of pupils – “all the black students are in bottom sets”, “the chinese pupils work so hard!”, “what do you expect, she’s from a traveller family”. Look, the thing that I’ve noticed being a teacher is that you can’t lump all kids sharing a common characteristic together. “Like*BAME*, you say?!” you might smugly retort, before cocking your eyebrow and sitting down of your throne made of pure right-all-the-time. Sure, like BAME. The idea is that a group of pupils may share a common feature and that feature may be linked to adverse educational outcomes, so that’s a reason to be vigilant about the student. However, humans vary massively, and they vary massively even when given the exact same conditions. That is not the same of being shocked every time a black student solves an equation correctly. It works the other way – not every student with Asian heritage works incredibly hard or is interested in STEM. That’s fine, and it’s wrong to barrel in there and go ‘huh, I thought X would do better than that’ because they only got y% on the test.**Representation is important**. I have spent the majority of my life in incredibly white environments. Environments so white that you could imagine that, at any given point in my upbringing, I could have been in a boozy prosecco brunch talking about the sourdough starters. I grew up in Canterbury, went to university in Durham and then started a career in the civil service as an economist before moving to Oxford as a maths teacher. I basically have never really had a huge amount of interactions with people who look like me, or have similar ethnic backgrounds to me. At school, none of my teachers were BAME – they were all undulating white and middle class. Now this isn’t necessarily an issue – they were, by and large, good teachers and I am grateful that I got to receive their instruction. However, it made it difficult to imagine myself there. Trying to articulate it,*it felt as if I wouldn’t belong in that position*. One thing I can hope that I’m passively doing as I teach is just show BAME students that, look, you can be here too. There are mathematicians who look a little like you, and that you are welcome in this profession if you want to be here.**There’s always the ‘do you think it mattered that I’m not white?’ train of thought**. I recent was interviewing around for positions. I’m going to say that, on paper, I’m a pretty good candidate for any maths teacher job – I have very good qualifications and a good track-record. I’m relatively eloquent and my references are really good. There’s been a couple of times where I never heard back from a position that I was very well qualified for (particularly at private schools, although this was only my experience and I really do not wish to get into private vs state – that’s something I simply do not have much experience with) and you just can’t help but think … was it my face? Was it my name? Was it because I disclosed I’m muslim? Imposter syndrome is real and experienced by a great many people for a great many reasons and I think it really manifests itself in me. I would like to think that I simply handed in a rubbish application, but I just don’t think that’s true.**Please do not play on stereotypes**. I once saw a white person talk to a black person and their opening gambit was about how they loved Beyonce, so please just don’t do that ever. For the record, I love emo music (2000-2010 era) and video games, and I can’t dance to save my life.**Please do not say you are blind to race**. Look, I get it – it’s very well-meaning. I understand that you mean to say that you don’t think of me as different. But I am different – I have a different bank of lived experiences and that is partly because of my mixed race heritage. That is absolutely fine and it shouldn’t be played down. The World is a little richer because of our differences if only we talk about them.

I am different – I have a different bank of lived experiences and that is partly because of my mixed race heritage. That is absolutely fine and it shouldn’t be played down. The World is a little richer because of our differences if only we talk about them.

Again, these are just my own thoughts. Yours may differ, BAME or not, rich or poor, and it would be great to hear those thoughts. I hope that the vast majority of BAME students have really positive experiences at school and that race isn’t an issue directly. I think the conversation about income is very different, and is inextricably linked because the socio-economic ties between race and income, however I can’t really speak to that as a person who has not gone through poverty myself, thankfully. I don’t really have a pithy summation of this post, but just have your eyes open, be aware of implicit bias on a personal level and do your best to just be there for everyone. Have a lovely day!

-Hisham

]]>It’s important to keep in mind that, when talking about methods, it can feel very personal – teachers often choose a method largely on the basis of what they feel their students would do best with. I do feel that some methods have more longevity and are easier to generalise with than others, and therefore I prefer them, but at the end of the day only you know your students and context. That said, during a recent livestream maths chat Dani Quinn (who is amazing) asked (paraphrasing!) “why would it make sense that the best way to teach adding fractions changes depending on whether you’re in London or in Edinburgh?” I don’t think she actually said Edinburgh, but you get the gist.

I think I align with that argument to a great deal, although I think there is a lot of nuance in there that could be added. There isn’t really a single method or representation that teachers should be resigned to using it; Mark McCourt talks about instruction needing to be varied in metaphor and that teachers should be able to draw upon different representations depending on the students. The idea that if a student does not understand a method, or it is not ‘clicking’, then a teacher can draw upon their expertise and offer a different lens rather than doing what I’ve been known to do – repeat yourself, slower, and hope that at some point they’ll just give in to your pleading eyes and go ‘yup, got it now sir’. Maybe the idea is there needs to be a selection of battle-tested methods that are embedded into the maths canon, with a few being shown to be demonstrably better for deep mathematical understanding. Maybe that’s robbing teachers of autonomy and creativity. Such a debate is beyond the scope of this post, so I’ll just light the fuse and walk away innocently.

The overarching point here is that, please do not be offended if I am not favourable with your favourite method – these are just my thoughts based on my experiences and your mileage may well vary.

There are three methods that came to the fore here: the factor tree method, the ‘pure’ method and what I’ve referred to as the ‘ladder’ method. I’ve shown all of them below along with a ‘hybrid’ method that I’ll explain in a minute.

The factor tree method will be familiar to a lot of people. The way it works is you choose any factor pair of a number and write them underneath. If one is a prime number, then you circle it and leave that ‘branch’ as is. If it’s a composite number, then repeat the process of splitting it into a factor pair, circling any primes. Rinse and repeat until you only have prime factors. Your prime factorisation is the product of your prime factors.

The pure method (again, just naming it for convenience) is, inevitably, the same fundamental idea. It’s interesting to read Peter Mattock’s blog where he eloquently talks about the need to draw out the underlying mathematical principles explicitly, and that the method used to get to the maths is somewhat immaterial beyond that. He makes it very clear that students should be able to see that prime factor decomposition is a form of factorisation, rather than a discrete area of mathematics in of itself. It’s a very interesting point, and having students be able to draw links between appearances of factoring in the curriculum is not something I’ve dwelled on deeply before, so it’s worth exploring a bit further.

I absolutely agree with Peter’s argument, and I think it is important that prime factor decomposition is linked to the idea of factorisation and any factorisation that’s already been encountered by students (e.g. factors of numbers and factorising algebraic expressions). This will help long-term retention and also the development of a coherent and cogent understanding of the wider mathematical principle.

Peter is critical of the ‘tree’ method for a number of reasons, but one criticism that comes through quite strongly is that he feels that the method divorces the mathematical principle of prime factor decomposition with other forms of factorisation. We do not use the tree method when looking at other types of factorisation, so the danger is that students cannot see past the method and at the actual maths.

My own gut instinct is that I think it, by and large, depends on the actual teaching. What is the teacher drawing attention to as they go through their examples? What sort of questions being asked? What are the tasks that have been selected? All of these variables may have a larger effect in placing the concept within a student’s schema than the method itself. If one were to strip away the context of the classroom entirely and just provide a textbook (with an appropriately grim title, along the lines of “Mathematics for the youth”) with the algorithm, then I think the argument of the method being overly fussy is likely to be true. When we add in all the other factors of classroom teaching, however, it’s definitely not clear cut.

All that said, I will hold my hands up and say that I have lost count of the number of times that I have asked students to find the prime factor decomposition of a number and they’ve stared at me gormlessly for a little while before going “what, the *tree* thing?” Yes, the tree thing. So there is empirical evidence from my own practice that students place a high weight on the surface method – although, once they recall the method, they are generally able to do relatively sophisticated tasks involving factors, so it’s still not clear if the method is posing too much of a problem.

“

I have lost count of the number of times that I have asked students to find the prime factor decomposition of a number and they’ve stared at me gormlessly for a little while before going “what, the“treething?”

The pure method is delightfully no-frills. It almost screams “MATHS” in its absolute single-minded intensity and there are no distracting features for students to get hung up on, potentially reducing the level of extraneous cognitive load. That said, a potential fault is that it could be difficult to keep track of which products have been reduced, and where those products came from. A mistake lower down in the working might be difficult for students to follow back up and spot, and I can see students overlooking factors that are not yet reduced to primes. A nice facet of the tree method is that it does organise thinking in a structured and clear manner, which makes systematic working and checking a lot easier in my opinion.

What about the next method – the ladder method? I’m sure there are a lot of variations of layout and specifics, but the way that I’m thinking of basically involves repeated division by prime factors, starting with the smallest prime factor and going up until you get a result of a prime factor. Conceptually, it is clear, and there is little fuss getting in the way.

I wouldn’t choose to use it personally for a couple of reasons, however. The first is that I wonder if some students will find it difficult to associate the fact that because an integer is divisible by 2 then 2 must be a factor of the integer. Of course, this should be solid understanding by the time students come to this process, but I am sure that I am not alone in understanding that this is not always the case. This is clearly not insurmountable, nor is it necessarily a bad thing to find (it is important that students are comfortable with the relationship), but it does drastically alter the sequence of lessons leading up to the idea. The second is that there is potentially the risk of it being too procedural – again, taking my caveat of needing to understand the delivery of the material into account. I could see students churning through the method without pausing to really understand it. There’s also the question that it may foster the misconception that you have to always divide by the smallest prime first, or that you couldn’t get the prime factorisation by dividing by any factor. Of course, a similar argument could be made for the factor tree method, although I think it is slightly more flexible.

So what’s left? The factor tree method might have students thinking about oak dieback, the pure method might leave students with an incomplete decomposition and the ladder method might be too procedural. That said, all these methods, wielded by someone who has given it very careful thought, will undoubtedly be effective and they all have something to add. What’s my opinion? Get off the fence!

I like the down-the-line approach of the ‘pure’ method the best. I think it’s nice that it never deviates from explicitly referring to the factors of an integer as a product, and so I think I would find it easier to build connections with other areas of maths, or develop understanding for deeper connections (this is one of the reasons I shifted away from the Venn diagram method for HCF/LCM). That said, I vastly prefer the tree diagram method for the clarity and scaffolding it gives students.

My hybrid method isn’t really particularly revolutionary and you may just go it’s the pure method with circles in it, which is exactly what it is. I think it’s important that students record the prime factors as they come upon them, so they are still actively thinking about primes, and recalling that this is the motivation in the first place. It will also help students visually decode potentially long strings of numbers.

Layout is important and I have neglected it somewhat in my example. Perhaps there is an iteration where you take a great deal of space and clearly write the factor pairs underneath each integer in successive lines of working – taking the general idea of the tree method but altering it somewhat.

Of course, this is all theoretical. One could write all day about it, but it’s not worth much until we get the chance to get it in front of a good many classes and see where it falls down and where it works. Nonetheless, it has been fun thinking about such things, in a topic that I had previously assumed to be done and dusted and not really given a second thought. If you’ve enjoyed this post, I would strongly recommend picking up a copy of Jo Morgan’s book *A compendium of Mathematical methods*, which I have enjoyed reading immensely.

-Hisham

]]>**I couldn’t*

Well, even if it’s just me, the point is that I really never used to take much stock of prior knowledge when planning for lessons. I pretty much just assumed that most of my students came to me as tiny baby mathematicians and then planned to ramp them up when I got going with them. Of course, I changed my baseline depending on their set (egregiously, I effectively used to assume that bottom sets had made little to no practice in their secondary career to date, and thus planned very conservatively), but it was generally sweeping statements with relatively little evidence to back them up.

On the topic of evidence, one could argue that actually, especially for Y8 onwards, there is plenty of evidence about prior competence. It may be through internal test data or Hegarty Maths-like systems, but students are incredibly likely to have some sort of paper trail a few months into Y7. Once you get going with a class, it may be you just know your class very well. You don’t need to spend time grappling with old spreadsheets because you know Jamal and Hurain are at the top end and will smash through everything and that Jasmine and Bob are likely to look at you with sad confused eyes at every turn.

However, there are problems with both these lines of logic. Regarding *prior data*, it is likely correct that there is data but there is a massive challenge in making any sense of it: data collection amongst classes or years may be fragmented and students are likely to have moved sets between years making collating data a nightmare. Even if you have all the data, actually analysing it (assuming its broken down in the detail you would want) is also hugely time consuming and likely to be beyond a reasonable ask of a teacher’s preparation. As for *knowing your class*, you may well know that James’ favourite band is Enter Shikari and that he can find the perimeter of any rectilinear shape between here and the moon, but it doesn’t necessarily tell you much about James’ relationship with algebra. Even the best teachers aren’t psychic.

All that said, let’s assume you’re in the position where you do, in fact know what the students know about the upcoming topic. The problem is that data is likely rather outdated – you don’t know if students crammed for that test or if they had just broken up with their boy/girlfriend that morning. Further, you don’t know if that knowledge has any degree of retrieval strength – have they been tested on it recently? Was the last time they saw it a full year ago? This doesn’t mean the information is useless – far from it – but you still need to assess whether or not it needs to be covered.

When I started out teaching, my department used to give all students a test at the start and end of every topic. The tests would be identical and it would help us see progress across a unit and also plan to take account of their weakness. I’m not sure if the pre-testing effect was also a rationale for these tests as well, but it is at least a happy coincidence that it may have spurred that sort of thinking in a selection of students.

As with any approach, this came with its negatives as well. First, we would give this test the first lesson into a new topic, and given the time involved in marking, analysing and then planning this meant that it was often difficult to use the information to its full extent. Secondly, and this is a sentiment I’ve seen on twitter recently, it’s no fun for any student to take an extended test (maybe 10-20 minutes) where you are, by definition, unable to do most of the questions. Lastly, it doesn’t tell you about relevant knowledge recall outside of the topic (for example, if you’re pre-testing percentages but don’t include any decimal or fractional questions, then you’re still going to run into significant difficulties).

During my reading, I stumbled upon the concept of Atomisation and, at first I really did not like it. The general idea is that you break down a new topic into its individual ‘atoms’ of knowledge – for example, Craig Barton identifies being able to read decimal scales as an atom of histograms (you could ask me for a hundred years and I don’t think I would have made that link!). You then focus attention and assess each atom in turn, and if the class are mixed or totally unable to do it, then you teach that atom explicitly before re-assessing and potentially moving onto the next atom.

Why did I not like it? A gut instinct screamed out that it would take too long, it would take far too long to plan, that it seemed patronising and you just need to get to the good stuff.

However, it actually makes a lot of sense. I believe Craig Barton says something along the lines of ‘you’ll have to deal with these atoms anyway’, which makes total sense. How many times have you been modelling or conducting some process questioning and discovered that a student/class can’t do a procedure that is just a single step in the new idea? And then you quickly tell the class what to do and carry on? As if that 20 second half-mumbled and rushed explanation suddenly kicks open the doors of knowledge, rips away the shutters of misconceptions and leaves them truly enlightened?

“How many times have you been modelling or conducting some process questioning and discovered that a student/class can’t do a procedure that is just a single step in the new idea? And then you quickly tell the class what to do and carry on? As if that 20 second half-mumbled and rushed explanation suddenly kicks open the doors of knowledge, rips away the shutters of misconceptions and leaves them truly enlightened?”

The idea of atomisation makes sense. One of my big reflections from my reading is the idea that we need to make sure prior knowledge is secure and if not, we take the time to secure it before building on it. Building prior knowledge alongside novel knowledge will rarely end with the success rate that you want to see, and will only get worse over time, like mathematical subsidence. Taking the time to assess prior knowledge is an investment we must make in our teaching.

I really like the model that Craig Barton uses for assessing prior knowledge via Diagnostic Questions. It seems quick to plan, efficient in its use and it also lends itself nicely to relatively well-defined (and therefore plan-able) actions. If the class basically do not understand it at all, then you can simply reteach it (at least using example problem pairs). If the class are split, then you could reteach it while giving out an extension problem for those already secure, or use a peer-teaching system where students pair up – although this is dependent on class dynamics and your comfort zone as a teacher.

That said, it definitely does take time. In his book *Reflect, Expect, Check, Explain*, Craig relates an anecdote whereby he took around 2 lessons just assessing and dealing with atoms ahead of a histograms topic and the main class teacher was unimpressed with the pace. I think a degree of pragmatism is needed – as much as Mark McCourt hates it, the reality is that most schools in the UK operate some form of conveyor-belt system in which there is a constraint in terms of time spent on a topic area. While it is possible to be flexible in that system, too much slack risks substantive divergence that the general assessment framework in the school may not be able to withstand (for example, if one class fall significantly behind, two different tests may need to be written, which minimises chances for comparison). In reality, we need to keep some level of pace, which means we may need to make a sub-optimal compromise on something – maybe the number of atoms we assess, or the depth in which we reteach.

Regardless of the issues, assessing understanding and using that information is something I will definitely be doing going forward with my classes. I’ve always known it was important, in a theoretical way, but never sure how to actually do it. Thanks to the efforts of a number of maths teachers, I now have that practical roadmap.

]]>Armed with Teacher Tapp and unspent petrol money, I’ve invested in a steady trickle of books from authors who I’ve long looked up to, and set about reading in every spare second I can prise in between childcare, job and general life admin. Honestly, it has reignited my love of teaching in a way that I simply had not anticipated. I knew I enjoyed teaching, but I simply had forgotten what it felt like to have the space and time to think about teaching. In the day-to-day of life, you work so hard just to keep going; with the lockdown the space opened up to step back and critically look at what exactly I was doing. As it turns out, there was a lot to critique: page after page, my self esteem and practice took heavy blows as I came to realise that, while my teaching was generally decent, there was just a vast amount of improvement to make.

This blog is an attempt to chronicle and make sense of that reading.

So, the first of my revelations is the confluence of Mark McCourt’s* Teaching for Mastery* and Craig Barton’s *How I Wish I’d Taught Maths* and *Reflect, Expect, Check, Explain*. To get to it, first we need to be familiar with the idea of a progression of a learning episode (distinct from a lesson – because, of course, it takes different amounts of time to tackle different concepts). The progression begins with some sort of *knowledge acquisition* phase where students acquire inflexible knowledge – knowledge that is tied to the surface structure of how it is presented (all Pythagoras questions look like an explicit right angled triangle with just two sides marked, for example). That inflexible knowledge is then carefully worked with and increases in its flexibility – the ability of the student to identify its application to an ever wider set of problems, which may look very different to the practice in the knowledge acquisition phase.

The below, taken from Gary Lamb’s post on the Complete Maths website, follows Mark McCourt’s proposed progression – showing the knowledge acquisition phase (roughly speaking, the ‘Teach’ and ‘Do’ phases) and the knowledge development phase (again, roughly speaking, the ‘practise’ and ‘behave’ phases). The bar model illustrates the general principle that around 80% of a learning episode should be going through the Teach, Do and Practice phases, with 20% on the Behave phase (which is effectively an opportunity to interleave ideas taught some years prior into the current episode).

In *Teaching for Mastery* Mark suggests that the typical teaching follows a pattern of a chunk of ‘Teach’ then a sizable chunk of ‘Do’ followed by a little bit of ‘Practice’.

Reflecting on my own practice, I realise that I had fallen into a cycle of heavy dependence on the ‘Do’ phase and tricked myself into thinking that performance (correct answers on mini-whiteboards or in books) was the same as my students properly understanding the concept. While that false equivalence is a topic for a different post, the general idea was that I wanted my students to become fluent in a topic, and that held so much weight in my mind that the ‘Practice’ part was often relegated to extension material alone. After the lesson, or set of lessons, if a student hadn’t gotten onto the extension we simply had to move on. The wheels of education keep on turning, after all.

Obviously, this is not right. Part of what makes maths engaging and fun are, in the parlance of Craig Barton, *purposeful practice* activities. It’s the weird, obtuse tasks one can find on Don Steward’s website, or some of the wonderfully crafted resources on mathspad that encourage you to dig a little deeper and wade out of the comforting embrace of umpteen correct answers in a row.

This is not to say that I never used these resources – in fact, I made heavy use of mathspad – but quite often they seemed to only shine for the higher sets, and if I engaged lower sets with them then I did an eye-watering amount of scaffolding and support. Scaffolding purposeful practice tasks isn’t necessarily a bad thing, but too much and it somewhat robs the task of its desirable difficulty (to potentially misuse the phrase).

Why did I fall into this? I can trace it back to my PGCE and NQT years, which, at the time, felt as though I was cartwheeling through a firework factory while on fire. In my view, I would explain how to do a procedure, do a little bit of fluency work and then whoosh, onto this really fun activity! To my surprise, reactions tended to fall into three camps:

- “Huh, this is pretty cool”: a glowing endorsement by teenage standards, but also relatively rare.
- “I don’t get it, I can’t do it, I’m not doing it because I don’t get it and I can’t do it so I’m not-“: this was one of the more common responses in the classroom.
- “Can’t be bothered”: this was the other dominant reception.

There’s obviously a fair bit of disguised ‘2’ in reaction ‘3’, but the cumulative effect was that the classroom rapidly would become unmanageable. A sea of hands, a particularly anxious pupil rapidly asking a tonne of questions to ring out as much support as they could and a good deal of discussion about the week’s ‘birthday beats’ rota all combined to make the environment approximately as stressful as defusing a nuclear bomb.

“…the environment [was] approximately as stressful as defusing a nuclear bomb.”

After attempting these sort of tasks repeatedly and watching them fall apart in front of me was demoralising for both myself and the students. As I progressed in my teaching, I saw how fluency practice (the Teach-Do cycle) allowed me to build confidence and (perceived) competence quicker as well as ensure a more controlled, calm and productive environment. Indeed, Craig refers to the idea that success is motivating, and I saw how students switched on for fluency practice and switched off for purposeful practice. Naturally, being an inexperienced teacher in desperate need of a win, my lessons shifted more and more to securing fluency above all else. As the years rolled on, I slowly began doing more purposeful practice, but it definitely had an ‘optional extra’ quality to it, on the whole.

Now, securing fluency is incredibly important – without secure inflexible knowledge, there’s little chance of developing robust flexible knowledge – but it is only the beginning of the journey. Therefore, I can’t keep pushing purposeful practice out to the fringes of my students’ experiences, and I have to make it an integral part of the journey because it holds real intrinsic value for *all* learners, not just the top end.

Given all of that, then how can I embed purposeful practice in such a way that doesn’t give everyone in a 5 classroom radius mild PTSD? What was I doing wrong all those years ago?

- The first thing is that I’m simply a better teacher now than I was then. I have hundreds more teaching hours under my belt, and I can manage a classroom much better than I could do before. I shouldn’t be scared of the lesson going off the rails because I have the tools and skill to bring a lesson back from the brink in a way that I didn’t then.
- I also need to select the activity to be appropriate. What is it doing, and what do I want the students to get from it? Is it of an appropriate difficulty, or does it include concepts that we haven’t seen yet?
- Scaffolding the task is important, but a much better way of ensuring access (and therefore that motivating success) is to use formative assessment before the task to ensure that students can do all of the skills needed for the task. Such tasks often require knowledge outside of the novel concept that is the focus of the lesson, and it doesn’t matter how fluent they are in whatever we’ve just done if they then come apart when adding some fractions together.
- Finally, I need to actually do the task. Shamefully, I often did not fully do the tasks I set for students – snowed under as an NQT, I cut corners where I could assuming my maths was strong enough to wing it on the fly. I can disclose to you, that I definitely did not wing it on any sort of fly. But more importantly, doing the task in advance tells me where students are likely to struggle, what knowledge they really need to do it and also the questions and prompts I could give during the task itself to stretch or support students as needed.

Who knows what will happen when I first try it with my classes (now likely to be in a new school). Maybe it will be a disaster. But I can’t back down – I have to make this a meaningful and robust part of my practice. In time, with active reflection and adjustment, being consistent about my expectations and also making sure I plan very carefully, I’m sure I can bolster my teaching and enrich all my students’ learning.

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