Conceptualising Maths Pre-Session Blog

The most common complaint levied against learning mathematics is that it is parcelled up into discrete methods that have to be memorised and wheeled out when required. What is worse, that also means one has to memorise when every method is required!

Instead, what if we taught what unifies these methods and when it is required? I am talking, of course, about the mathematical concept from which the method arises. I have long advocated that maths teachers should switch their focus from “teaching to do” to “teaching about”. If we teach about mathematical ideas, the structures that underpin them, how these structures interact with each other and how the structure allows us to make sense of how associated procedures do what they do, then learners of mathematics have the ability to recognise when certain procedures are useful as well as how to carry them out. What is more, they are in a position to adapt and improvise for new problems and to select methods appropriate to the situation.

So, what does this look like. Let us take the concept of “perpendicular” as an example. Consider this line segment. 

What does a line segment that is perpendicular look like? We might choose to have pupils use a protractor to begin with to draw a perpendicular from either end (or not) but we would want pupils to recognise and understand that a perpendicular would have to go two squares right and three squares down (or conversely two squares left and 3 squares). But more importantly we would want pupils to see that this is because the distances that are horizontal become vertical for a perpendicular, and vice versa. Understanding this structure allows pupils much more flexibility to work with perpendiculars. For example, we should now confidently be able to ask pupils about which of these are squares:

Of course, this understanding will eventually be built upon to include perpendicular gradients and perpendicular vectors. But importantly, those are both clearly part of the same structure as this idea. Once 2-D vectors are introduced using the  notation, it is exactly the same thought process that allows us to see that a perpendicular vector is , and that another is . When we understand gradient as the vertical change for a unit horizontal change, we can see that that the perpendicular gradient will become a unit vertical change for a horizontal change that is the negative of the previous vertical change (or, more succinctly, the perpendicular to a gradient of m is ). There is no new mathematical structure in the concept of “perpendicular”; the new mathematics is simply how this concept ‘fits’ when we combine it with the concept of “vector” or the concept of “gradient”.

If I could give one piece of advice to any maths teachers (new or experienced) it would be to think about what the mathematical concept is that they want their pupils to learn about, think about all the different concepts they might have to use it with, and then teach it in a way that means it is recognisably the same concept every time pupils encounter it.

The rise of the "method"

As I write, I have been engaged in a discussion on twitter around "methods" for decomposing a number into the product of its prime factors. The thread is interesting enough and can be found from this tweet.

Now I have no problem with discussions around methods. Each one is an interesting mathematical process, and there are often pros and cons between one "method" and another for doing whatever it is you want to do. I love @mathsjem's book that examines different methods for doing different things.

However, I feel there is a real danger arising in maths teaching whereby teachers think that in teaching a collection of methods, they are teaching mathematics. Even those that purport to teach "mutliple methods" are, I feel, missing the point. The point being that the method is not the mathematics. No matter how much conceptual understanding runs alongside it, the method itself cannot be the mathematics. And this is because the mathematics is that structure, that idea, from which the method arises.

Take, for example, the above mentioned "prime factorisation". There are two mathematical ideas at play in this - namely "prime" and "factorisation". What are these ideas? What is the structure that underpins them. Let us deal with the second one first, because that is required to understand the nature of "prime". What is factorisation? What do I understand by this idea?

Simply put, factorisation is the manipulation of an expression to write it as the product of two or more factors. Of course, this leads us further to ask what a factor is, and from there further down. There are lots of representations and manipulatives that can support in developing these ideas that are explored heavily in Visible Maths, so I am not going to re-hash them here. The key is, that this is the underlying structure, and it is true no matter when the word "factorisation" is used:

3 × 6 is a factorisation of 18.

2 × 3 × 3 is a factorisation of 18.

3x × 6 is a factorisation of 18x.

6(3x + 5) is a factorisation of 18x + 30.

It is this key understanding of what factorisation is that pupils need to understand, so that they can apply it no matter what the context. If I understand this, then any "method" I use for factorisation has to be taken in the context that this is what it is accomplishing - the decomposition into the product of two factors. Personally I would choose methods that appear the same/similar across as many contexts as possible, which is why I have moved away from things like "the factor tree" as it is typically only used to accomplish prime factor decomposition.

Once we have a clear idea of what it means to "factorise", we are in a position to understand primes as those numbers greater than 1 that cannot be factorised using numbers greater than 1 (or however you want to define primes that results in the same outcome). Again, there are lots of ways of exploring this idea with pupils - the Sieve of Eratosthenes being one of my favourites and the idea of factor towers being another:

With a thorough understanding of these two concepts, then the idea of "prime factorisation" becomes self-evident (or if it doesn't, easily linked to the two concepts already covered). A prime factorisation must be a factorisation that only involves prime numbers. Nothing else makes sense. so 3 × 6 is not a prime factorisation of 18 as 6 is not prime, but 2 × 3 × 3 is. At this point, the "method" for actually carrying out the decomposition is almost immaterial - as long as it is clear how it does the job it is is supposed to be doing. On this I stand firmly with @jemmaths, that the method should not be designed to insulate from the mathematics, but rather to highlight it. This, again is where I feel factor trees fall down - they not only obscure the links between prime factorisation and other factorisations, but they also obscure that it is factorisation that is happening at all!

For another example, take the simple method of "column addition". Again two ideas are a play here, namely the idea of addition, and the idea of place value. If, in isolation from each other, I am helped to understand that a way of making sense of addition is the collection of like objects, and that place value uses numerals in different columns to represent the size of different parts of a number (as well as the relationships between these columns), then addition in columns for larger numbers follows naturally. Not only does it follow naturally, but it is adaptable. I can see why this calculation can be done in columns in any order:

 

whereas it might be beneficial to do this one from right to left:

but that if I do complete it right to left, I can see what has gone on:

and also why I might not use column addition at all:

If you are teaching maths, I implore you to actually teach maths. Teach kids about what is happening when you add, or factorise. Teach them what it means to be prime, or how place value works. Then teach them one or more methods that arise from these understandings.