Mathematics education has had something of a renaissance in recent years. The curriculum has undergone an overhaul, with more advanced topics included and an increased onus on developing a broad mathematical understanding that spans different concepts. However, most topics previously taught still feature. Therefore it is tempting to not alter the manner in which we teach a topic. Most teachers’ pedagogy and resources have been honed over one or two previous specifications, and are still incredibly valuable now. But a few years on from the introduction of the new curriculum, schools now have a sense of what does need to be modified. Perhaps part of the solution is through fully embracing curriculum changes through the prism of Stem. The amount of investment, technology and resourcing that now exists means pupils can have content delivered to them in engaging and exciting ways.
Certainly one aspect of the new curriculum that does necessitate a change is the increased focus on problem solving. This needs to be a core skill that pupils have been practicing throughout their schooling, not just tacked on in Year 11. Something potentially seen as an obstacle should be considered an opportunity. Facing this head on and meaningfully attempting to develop this skill during secondary encourages teachers to deliver rich and engaging lessons. They also make the subject more relevant and expose the wider applications of maths. In turn this could help to make mathematics education more of a positive experience for pupils.
So how can problem solving be taught? Being a skill it cannot simply be presented or derived in the way you might with the quadratic formula. It has to be drawn out and developed. It should come from the pupil, with the teacher utilising tasks that enable them to develop. Through support and feedback the pupil hones this skill over time. There are a number of fantastic resources available now that does this. These resources present mathematics in unfamiliar contexts. It is not as simple as solve this or calculate that. The mathematics is hidden, often in a wider, practical problem. As a result pupils are immediately engaged with the task – they understand what they are trying to work out, such as the maximum volume created with a fixed card. The difficulty comes with translating their mathematics to the situation, which is where the role of the teacher comes in. We need to manage the learning environment, allowing pupils time to try out ideas and explore the problem, but resisting the urge to show them how to do it. Give them a nudge in the right direction: “Have you thought about this? Why didn’t that work?”
An activity I have used a number of times is based around these principles. The title is: How to beat Usain Bolt. There is always at least one pupil who glibly answers ‘run faster’. But the task is a bit more complicated than that. It works particularly well with the use of Excel or graphing software. Pupils are presented with raw data and have to guess what it represents. The final time and distance is the giveaway here. Then through a series of prompts, the task will take them through various areas of maths. Pupils will draw Usain Bolt’s distance time graph. They will have to make inferences about what happened at the start or the end. They uncover the idea of a gradient being a rate of change and its connection to speed in this context. That leads to a velocity time graph – where it gets really interesting. What does the area represent? How could it be worked out? Is the gradient significant? What about the peak? Is it better if this comes earlier or later? How could this period of tapering off at the end be minimised? If we mapped the Berlin record against his result in Beijing what comparisons could we make? Is it true to say he was faster in Berlin?
All of this mathematics is on the GCSE syllabus, but the way it is presented is more meaningful, tangible and related to something they already grasp. They are forced to think deeply about the concepts instead of mechanically applying a method. In that lesson the mathematical principles were the stars of the show and the problem solving supported it in the background. But as a skill it can and should be tackled head on as well. A great way to do this is to use a so-called back of an envelope problem. The Italian physicist Enrico Fermi was famed for these. They are problems often quite abstract in nature, which encourage pupils to make a series of assumptions, estimations and calculations, all of which could supposedly be done on the back of an envelope. The classic Fermi problem is “How many piano tuners are there in Chicago?” This is a great question that could be easily translated to the classroom and used as a lesson starter.
The emphasis is not on the validity of the answer but more on the sequence of steps that takes someone there. You do not want to hear a guess of the final number, but more how the problem is broken down into steps where it is easier to make assumptions and then how these are combined in calculations. These are quick. They take 5 minutes. The first time a pupil does this it is alien. But repeatedly challenging them with these problems will shift their focus from the answer to the method.
At school I was fortunate enough to have a teacher who constantly pushed us out of our comfort zone like this, and anchored abstract mathematics in the real world. In our first ever lesson he said: “It is now 9.25. What is the angle between the hour and minute hands?” When most said 120 degrees – we were forced to re-evaluate our answers. I had never before experienced mathematics presented in such a way. Like a magician he was able to draw it from the most unexpected of sources and made us appreciate why mathematics is truly a core subject. Problem solving – this is the true transferrable skill in maths. Some pupils will use maths the rest of their lives in a very explicit way. They will become engineers, astrophysicists, economists and coders. But for most the main transferrable skill will be problem solving - that practical ability to take some given propositions and logically deduce an outcome. Mathematics is a so-called gateway subject. It teaches people how to think and analyse, to see an error as an opportunity and become more resilient as a result.
I was also a member of “the maths club”. Enjoyable though it was, it was an extra-curricular activity squarely aimed at pupils who already enjoy maths. We would be given riddles, logic problems and open-ended tasks. This still has a place now. Many schools still have similar groups and these are invaluable in enriching pupils’ learning and maintaining engagement. However, with the technology available nowadays, it should be possible to inspire even more pupils, not just those with a pre-existing talent or enjoyment of the subject.
We recently bought a 3D printer, and are currently creating a project that will allow pupils to knit together their mathematics with that of design and engineering. The idea behind the project is for pupils to work towards creating a 3D object with the printer, uncovering a great deal of mathematics along the way that assists them. It will be problem orientated – they are presented with a problem at the beginning and their object must be a solution. It will link up areas of mathematics often taught separately, such as algebra, geometry and statistics. Every discrete skill is seen as an important part of the whole. Pupils will use mathematics previously learnt and so have opportunity to consolidate it, extend further and also apply it in an applied context.
The goal is for pupils to create a boat, for a specific purpose. Within certain fixed parameters their boat needs to be able to carry the maximum possible volume of sand across a water tank. The parameters set would be things like a specific motor used in all models or that the entire boat can only be printed in two parts. It is likely this list will be added to as the task is developed. Perhaps I would want pupils to only consider certain shapes for the hull. Or considerations made relating to buoyancy.
This is ambitious. There is a lot going on here and there are some particularly sophisticated areas of mathematics involved. But whilst I hope to simplify the task enough for pupils to access it, I believe there is also plenty of scope for them to take it quite far. I hope by the end that all pupils involved would have designed and produced something. They will see first-hand the role of mathematics – not just in modelling a hull with equations or calculating densities, but also in how breaking a big task into logical steps, making assumptions and evaluating them is the heart of the subject. It is tangible – they will produce a physical object. And there is something exciting about seeing an object you designed and created in action.
Codebreaking also lends itself particularly well as a STEM enrichment, where activities could be built around mathematics and technology concepts. We have previously held codebreaking days where pupils were able to experience this exciting and stimulating subject from different perspectives. In one task pupils received a paragraph of encoded text and used frequency analysis to decode it, before guessing the book and finding it on a bookshelf. In another, they had to decipher a blinking message using Morse code and open a lock. There are applications across many areas of maths and the tasks themselves could be linked to lessons that explore concepts further. In the examples above I went on to explore the history of codebreaking and its uses in our technological world with the class. There could be follow up with computing lessons where pupils experience the importance of encryption in the digital age.
If we are going to turn around the decline in pupils pursuing mathematics or related disciplines we need to make it as relevant as possible at an earlier age. We need to get them on their feet, link up these abstract processes with the physical. We can explore the subject’s wider applications in our curriculum. That classic question of “Why am I learning this?”, though sometimes asked facetiously, deserves to be answered. And better yet, if possible, let them answer it for themselves by building a boat or coding a program. Mathematics enrichment shouldn’t just be for those that are naturally drawn to the subject – all could have opportunity to explore it.
Two of the biggest challenges I see facing mathematics teaching and STEM as a whole are: pupils’ understanding of the subject’s importance in the wider world and a culture of fear or apprehension that often surrounds it. One of the core principles of STEM is to see each strand less as a discrete area, but in its wider context. The two challenges are obviously linked, one feeds into the other, so that by engaging them more with maths they should become more confident. Also, by better understanding why they need to study the subject irrespective of where their interests lie after school, they may be less hesitant to engage with it. All pupils should leave school as mathematicians in the sense that they can solve problems with a logical and rigorous approach. Whether it be an explicitly algebraic problem, or simply the manner in which they break down a complex task into manageable components, all could benefit. The opportunities we have in terms of technology, curriculum and investment are so different from even 5 years ago that it seems like we are better equipped than ever to meet these challenges.