I don’t profess to knowing the absolute ins and outs of the history of maths but one thing I do is that at one point, we didn’t know that the curve f(x) = x² has gradient 2x everywhere. I also know that nobody on the planet knew that (-1) multiplied by (-1) = 1 at some point. Likewise, there was a time in the world when we could not just put an arithmetic series into a formula to find the sum of the first 1000 terms. What has happened?
Maths is still evolving; we still don’t know everything there is to know. We are still refining techniques. We are still disproving things and we are still itching to prove something new. Why then are we spending time as educators teaching people how to solve a problem? “This is how you find the length of a side if you have an angle and its opposite side.” That’s not learning the sine rule – that’s being told the sine rule. That’s learning a technique to reproduce the sine rule. This is fantastic if the exact conditions are replicated but what if the pressure ramps up or if someone, God forbid, forgets the technique. This style is akin to coaching a rugby team to play in one way and telling them there is only one way to succeed. There are infinite ways to outscore an opponent in sport and sport coaches have worked out now that the players will work out the best method for them to do that. Sure, you probably know of an easier route to score a goal or a basket or whatever but the players will see someone else achieving easier success and work out what the differences are. They will not however feel unsuccessful. I wrote a piece earlier in the week about why I teach and it was to do with making people feel success. Everyone will find success if you let them wander.
Imagine being dropped in a forest and being walked to your house by someone every day for a year. By the end of the year, you’ll probably be pretty good at walking that path and I’d say you’ve got a good chance of getting to the house along that path. What happens, then, if you mistakenly turn right? What happens if you aren’t dropped in the same place every time? What happens if someone asks you to get to someone else’s house?
This is analogous to learning mathematics. Yes, I can tell you how to use the factor theorem in this context. I can even show you 100 different ways the factor theorem can be tested but someone will develop the 101st way, what do we do then? You don’t understand the factor theorem, you “know” the factor theorem. This year, I have been teaching A-Level single mathematicians using a teaching style called Harkness. This was developed by Phillips Exeter Academy in New Hampshire. You can read more about it here https://en.wikipedia.org/wiki/Harkness_table
The main philosophies behind it are as follows:
- Pupils come to class with work done
- Pupils share their answers to the worksheet (first writing up solutions on the board and second talking about their solutions)
- The teacher sits at a circular table with the students and facilitates discussion
- Maths is taught. We don’t need to split topics (eg. series then functions then differentiation then probability).
- Students help students.
Firstly, I must say I was one of the most sceptical when I first heard of this teaching theory. That being said, I am now totally on board. It seems absurd I said to someone just six months ago that “it is stupid that you think we shouldn’t teach in lesson time”.
Every maths teacher and whoever else is reading this should sit and think for a second about how much work each student does in each lesson. We are wasting lesson time which is the most valuable time for the pupils. We don’t need to spend 10 minutes explaining the sine rule and the rules of differentiation then do 10 minutes of examples then do 20 minutes of “individual” work before talking to the group again. Lesson time, within the Harkness system, is a place to embed learning. The “learning” happens outside of lessons. The students go away and play! Wow, what a thought! Imagine children actually solving problems and enjoying getting to an answer. It doesn’t matter if it’s a super quick method or even whether it is all correct. What is important is that they are trying to win the game! The first time I played rugby, I threw the ball forwards. Another player who was better than me said I couldn’t do that and I never did it again. Chances are if I had just read or been “taught” the rule book (100s of pages), I would forget.
Children want to win; indeed everyone wants to win. Children are better at working out the best way to win than we are. They learn off the ones who are winning more often and they try to emulate them. They try to pick things up from “the better kids”. Why not just do the same in the maths classroom?
In a group of twelve children trying to factorise a quartic that they have never seen before, chances are one person has done it wholly correct, a few have done some parts correct and others had no idea but tried some stuff that didn’t work. Once an attempt is up on the board and the children are talking about it, all 12 are invested in the process. They can all relate because they all have been there and done it. Think about whether this is the same when you stop the whole class to discuss a question that one pupil asked. How many of the other pupils are invested in that process? I agree that you can ask the answer of another student but that is now two students plus maybe the half of the class that are listening, what’s going through the rest of the class’ heads?
I absolutely love the idea that these pupils are doing 45 minutes worth of maths work pre-lesson every lesson. They don’t see it as maths work because it isn’t A-level questions but when they see A-level questions they are super easy! “Sir, these are so much easier than the worksheets!”. Through directed questioning and some clever leading of the discussion a teacher reinforces so many ideas through a lesson.
For example, we have just come through 8 worksheets where the pupils had to rearrange anagrams followed by picking 2 red balls and 5 green balls from 7 boxes and then 2 x’s and 2 1’s from 4 boxes. Yes they will be seeing a binomial expansion soon but they don’t know what that term means. Whilst this has been a constant theme, we also have them working out the average speed of a car moving according to distance = 2t² between intervals of 5 seconds, 1 second, half a second, very small h seconds. They don’t know that they are setting up for calculus but they know they are finding an average speed function. They have worked out that if they graphed what they are doing that they are getting pretty close to finding the gradient of the quadratic curve. Isn’t that fantastic?! We can find the gradient of something that isn’t straight. They are not confused about that fact. They can clearly see that the average speed over 5 seconds is a rubbish measure but as we make the interval small, we are making a pretty good estimate of the average speed at that time. They haven’t seen a single f'(x) yet and although that will come, it is not the important part! They will get there when they get there. What is true is that when they get there, the notation will just be a short way of writing “the gradient function is given by”. No confusion.
The key thing is we put our utmost faith in the pupils. We trust them to give things a go. All teachers are trying to foster a culture where being wrong is acceptable. It shouldn’t be embarrassing to make a mistake – this goes some way towards that target. Every footballer has miskicked a ball, every tennis player has hit the net, every mathematician has written 1 + 1 = 1 at some point! What a brilliant world we will live in when all maths classrooms are full of pupils giving stuff a go because they know that one step in any direction cannot be wrong. Even if you went North West from Birmingham, you can still get to London! It isn’t a race. When you did eventually get to London, you would find others who had done the same journey who had been waiting a while and you’d ask “how did you get here so fast?” and they’d explain their route. You’d listen and you’d take on board what they are saying because you want to win. You want to be as successful. You achieved the end goal but you want to be better. That is human nature. Next week when asked to get from Exeter to London, you’d look at a map and see where is Exeter and work out which direction to go in because you want to be the first one in London this time.
This is what we are trying to achieve in our classrooms. I think it is brilliant and I think the children will see this. It is not all rosy during the process. The pupils do find it difficult and do find it very different to what they have done before. This makes for lots of discomfort amongst them but slowly and surely, we are seeing them feel successful. Every child has something to contribute every single lesson. They look at others who have less homework or who have hundreds of formulae that they “know” and think this is unfair or we are behind. As educators, we have to believe in the long term goal and the pupils must trust us in this. I am excited by the challenge and I look forward to seeing down the line the level of mathematician and whether we can actually have a few who come out and say they enjoyed this subject a lot more!