I am a relatively new convert to the maths CPD community, and my gateway drug was Craig Barton’s seminal “Reflect, Expect, Check, Explain” – a sprawling, yet tightly focussed, work that deftly balances the weighty tide of education academia and the stochastic fever dream that is a typical year 9 classroom. One of the interesting features of the book is that it regularly pauses and explicitly asks the reader to reflect on the preceding nugget of wisdom, asking them to evaluate it and identify whether they agree with the general idea and if it would fly in their local context. Of course, because of the dual hammers of well-evidenced research and a pool of experimental evidence, I found myself nodding furiously at pretty much every juncture, taking the text as a holy manna from heaven, immaculate in its conception. Who knew God was a northern maths teacher?
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 polygon (/ˈpɒlɪɡɒn/) is a plane figure that is described by a finite number of straight line segments connected to form a closed polygonal chain or polygonal circuit. The solid plane region, the bounding circuit, or the two together, may be called a polygon.From Wikipedia (yeah, I know)
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”
“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.