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#MathsConf23 Reflections: What before why

Posted by Mr Alhassan

MathsConf23 has now been and gone, and as my first ever maths teacher conference I have to say it was an absolute blast – the team at La Salle education should be feeling very proud right now. I’ve been interested in the past, but a young family and the demands of normal teaching always made going difficult so to have something so high quality presented virtually presented a wonderful opportunity. Although I would love to go to a physical event so I can fanboy over various maths teacher rockstars so I can socialise with other wonderful educators, I do hope that a solid online offering is now a routine part of the calendar year.

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.

General Reflections

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.

An example of one of Mathigon’s amazing interactive demos.

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.

Jo Morgan’s proposed journey of teaching trigonometry at Y11

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.

Kris Boulton-gate: How before why

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’.

My own notes on the early part of Kris’ session

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!

All the rest…

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

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