I will be preaching to the converted when I say that mathematics is full of beauty and wonder. Even the ‘simplest’ of concepts never fails to surprise me. Only the other day, Alex from VicMathsNotes pointed out something I’d never thought about before:
The proof, using nothing more than the principles of division and multiplication of polynomials, is quite straightforward and yet through being taught about tangent lines in the context of calculus – and then spending years teaching the same – I found this surprise beautiful.
I expect every one of us has had moments where something we thought we knew all about has revealed a secret that made us go, ‘wow!’ (It’s one of the reasons I wrote my book – even an experienced maths teacher with a maths degree can learn more about school-level maths). I’m going to call these ‘bridge moments’ – moments where a connection is made in our mind that makes our learning a little more sophisticated.
These bridge moments can’t always be predicted, but we should try to create opportunities for them to happen throughout our curriculum. They can be mind-blowing (the many ways of proving that 0.99999… is in fact 1 blew my Year 11s’ minds before half term) or they can be a moment of ‘Ah, I see!’, something that creates a connection in a quiet but profound way. However they occur, we should think explicitly about what they might be and where to bring them in: the moment where last month’s learning on linear graphs is linked to this month’s learning on linear sequences and students get the chance to notice that it’s the same maths; the moment where they see that the process of factorising to find the roots of a quadratic also works to give the x-intercept of a straight line; the moment where they understand that finding the mean from a histogram and from a frequency table is the same process; the moment where they see that the terms in a geometric sequence are just points with integer x-coordinate that lie on an exponential curve. I suggest to you that it’s a good way to spend some time thinking about other such bridges.
School maths can be so compartmentalised. Partly this is because of the lack of maturity in our students’ mathematical knowledge: the more advanced their knowledge becomes, the more likely they will see and make sense of the links between ideas. However, we can’t ignore our role in exacerbating or mitigating this compartmentalisation. I argue that one of the most important jobs in creating and delivering the curriculum is to build in the links that make it coherent. In fact, I’ll go a step further and say that our curriculum, from start to finish, should tell a story.
In many of the best novels the master storyteller starts with seemingly separate threads of character and place and gradually reveals how these threads interlock and interact. There are times of quiet, where the reader is getting to grips with what’s going on; there are times of climax, where suddenly you realise that two characters are brought together in the most unbelievable way; there are times where you’re shocked by what you’re reading and times where you can see it coming and anticipate the reveal with glee. These stories are careful in their construction, giving you the right information at just the right time to maximise its reception.
So should our curriculum be: a story of carefully crafted experiences, such that the right information appears at just the right time to maximise its reception. Ask yourself this today: how can I craft experiences into my lessons, into my curriculum, that make the learning coherent, so that my students see the story unfold in the best way?
I took a entire Year Four class to a supermarket fairly recently. They were given a budget in order to control their spending on the trip. At the supermarket they were asked to count the numbers of eggs. And they had to tell me about the total number of cake mixes. This, they found on the shelves in one of the aisles. Good luck. I made them spend as little as possible. They were there in order to work hard on their math skills not to waste money on any unwanted items after all. Every single purchase was recorded on my teaching pad. We had a full day of fun. They carried their exercise books in case they needed them.
Other tasks included calculating costs of the books in question. They had fun doing the maths for me. They also made a colour coded tally chart on the number of magazines afterwards at school. They even designed a pie chart based on the number of sandwiches they saw. They even made a nice bar chart on types of the wine they looked at. They then prepared a analysis report on their findings. They learnt a lot during the math investigation. We compared the item prices. They even analysed all the data.