Dear Reader, Although the content of this post is more specifically for the study of Mathematics, the useful ideas can easily be adapted to suit a variety of studied subjects.
Someone once told me that to be truly good at teaching mathematics, it was better to be just slightly rubbish at it yourself. And while I advocate for subject specialists within Maths, I get where they are coming from. Teaching Maths is almost as much about understanding students than it is about understanding numbers. Part of this comes from the stigma around the subject: If I had a pound for each time a parent has told me “Well, I can’t help because I’m terrible at Maths” then I’d be perhaps slightly wealthier than I currently am. (Let’s be clear: as a Maths teacher I know exactly how much wealthier I’d be, even counting for inflation). So it’s important for us to understand how people work, how the world works and how we can tie that in with how Maths works. And this is problem solving. In this blog I will have a look at problem solving, why it’s important and how we can move forwards on it. I am a Mathematician so a lot of my advice will link back to this, but I am also a teacher so I hope I’ve provided something here for everyone.
So What’s the Problem?
I love Maths. To me, Maths is art, music and therapy. I love learning it; I love teaching it and I would say I’m relatively good (not capital G good) at both. But even with my passion for the subject I am left scratching my head about the best way to go about Problem Solving. I am baffled as to how I can successfully engage a mind to solve a range of complex mathematical problems (we’re talking all the big hitters – trigonometry, quadratic equations, inverse proportion) independently in a stressful situation, but it seems a different story when it comes to mastering numbers mixed in with some context, or posed in a non-standard question format.
Why is it that students believe the average of some single digit number might be 42, or that they deem it reasonable to declare that when making 60 biscuits rather than 30 Louisa might need 78,000kg of butter rather than the original 120g. Faced with a problem-solving question why do even my best students start multiplying every number on the page with increasing panic and ever larger writing? Perhaps I’ll just pop a zero on the end? Maybe it’s a good idea to multiply them all and then halve it? One student even just wrote: “Beijing” when asked to compare temperatures in the northern and southern hemisphere. Sheer panic.
When did the Problem start?
With the introduction of a new syllabus we started to see the rise of problem solving. For anyone unclear on what I mean by this I am referring to any question that is presented with a context (however realistic or otherwise it may be) or one that requires students to combine methods or reverse methods in what I’ll refer to as a "non-standard question format". This is a fancy way of saying a question that isn’t presented in exactly the same way that the textbook presents it. Essentially – you haven’t explicitly told them how to solve this question before.
The new syllabus seemed keen to boast a focus on “real life” Maths, a chance to teach outside of textbooks and the classroom, for students to see the point, the application, the joy in the subject rather than simple number crunching. And I get it. But it seemed to imply that making the paper harder would make students somehow better. Unfortunately, what actually happened is the pass rate dropped, in some cases by as much as 15%. What we saw next was hugely demoralised students, bemused parents and an added wariness that the marmite subject of Mathematics scarcely needed.
Things have improved and we’re now building walls with 10 men rather than trying to prove the cosine rule with an algebraic term, a surd and a melon balanced on our heads. It’s better, but the problem of 'problem solving' has lingered, and with it a real need to find a solution.
So how do we solve a Problem like problem solving?
Going back to my earlier premise that to teach mathematics, you must have empathy. It is understanding how the jumble of letters and numbers looks to someone who does not (like you) worship at the alter of Pythagoras. And the answer is, in many cases, scary. So I have set upon myself (after much rambling) to investigate some approaches to problem solving that may help.
My first approach was likely similar to many other teachers in the face of this increased wave of problem solving. Reach for the highlighters, colour keywords on the Powerpoint and focus on making links between words and numbers. Having Studied French, as well as Maths, up to degree level, I undertook this approach much like a French translation. This has helped, discussing how words and numbers interact together is at the core of the solution. Here are a few of my initial strategies that I have found work.
Use colour well.
I like to have black writing on a pale-yellow background and use no other colours on my slides until I come along in the lesson and start furiously highlighting keywords. I like to do each step in a different colour and encourage my students to invest in a pack of highlighters and a glorious 4 colour pen. I’ve never been a fan of PowerPoints where the answer magically appears, these answers need to develop together. They need to be a conversation. Note that there is an emphasis on using colour well – students should read once and highlight the second time. In the timeless words of Sheldon Cooper: “If you highlight everything, you highlight nothing.”
Ask a lot of questions.
And don’t stop with your first answer. Why do you believe that you should multiply these numbers? What in the question made you think of this? What’s the keyword? Is the answer reasonable? Do you think Sainsburys will have 78,000kg of butter for poor Louisa? What would that even look like? Make sure every child has been asked at least 2-3 questions every (problem solving) lesson... ideally more. Volley questioning can work especially well here, getting students to build up the answers together. Ensure you praise each student before asking for other answers, they aren’t wrong – they are pieces of the whole puzzle. The more we train ourselves to build up answers together the more our brains will get better at doing it alone.
Pepper them in and gather them up.
By which I mean make sure you do them little and often, but also a load of them every now and again. Put them in starters, in plenaries, in exit tickets, mini tests, question of the days, but also do a whole run of lessons just on problem solving. Remind them to bring their coloured pens, highlighters and at least half a brain of logical thinking. Pick your moment – your class are unlikely to want to engage Friday period 6 with a page of writing that requires an algebraic proof at the end of it, but after break on a Tuesday? It might just work.
Rely on literacy.
As Mathematicians we balk at the idea of grammar and spelling and mutter “nothing to do with us” whenever the English department leave the room. But it is no longer the case that we can simply do numbers and they letters, we must teach students to read, comprehend and process words in the same way (if not with the same elegance) that they must for Shakespeare. Recently I challenged my Year 11 class to come up with 20 words that all meant they needed to do an addition. We started with the easy ones like total, altogether and sum, but pretty soon were debating whether increase was just for addition or whether it could be a multiplication in a percentages question and would we be allowed average because you did have to add them all up first.
Build Up the Skill and then add the Knowledge.
For a higher tier GCSE class, I often start off Problem Solving using Foundation Tier to get them going. It’s not important that the Maths isn’t hard enough yet. Jumping straight in to Problem Solving PLUS a level 8 topic might be overload – plus working on the lower tier questions builds confidence which is important. Knowledge and Skills can of course co-exist but it’s OK to focus on them separately at first, and mix them later.
Praise Mistakes and Questions.
When a student quietly responds they believe the answer is 42, instead of cringing inside and finding your kindest voice to say “wrong answer but good try” turn it around – ask them to explain why they thought it was 42. They won’t be the only one. Foster a situation where any answer is a good answer. Try “Who else got 42?” “Is 42 reasonable?” “Let’s take a look at your method” “Can someone explain where we’ve gone wrong here” It can take a lot of work to create an environment where students feel safe to make errors but being positive and celebrating them will not only promote trying, it will also reduce the anxiety that many students feel in the subject. Praise mistakes, praise responses. A colleague of mine thanks each student when they ask a question and I am trying to do the same with a mistake.
Don’t worry about the final answer.
This is a bold claim from a mathematician. I am sure there are those reading, shivering in their seats in horror… “what does she mean? We live for the final answer.” And Yes, it’s the goal but bearing in mind the grade boundaries getting the students halfway to the answer means success. Instead of saying “how do we get the answer?” we can ask “How do we get the first mark?” and sometimes, the first mark will lead to the second and before you know it: An answer.
Have we solved Problem Solving?
Not just yet. The strategies above have the ability to support students to navigate their way through problem solving. They are a toolkit that gives the students the confidence to try. I see fewer empty pages with a big question mark and more highlighter pens, attempts at questions and almost-answers than ever before. Every mark is a crucial step towards a target grade and aside from good grades, the fewer students who are afraid of maths means fewer times I have to hear “Well, I can’t help because I’m terrible at Maths” in 25 years’ time – even if this does mean being considerably less wealthy – especially accounting for inflation.
Next time:
I hope to look beyond the individual lesson and consider some different approaches to problem solving that encompass the whole scheme of work. I’ll be investigating Big Questions and if they make a Big Difference, Problem Solving booklets and whether they are more than just worksheets stapled together, and also whether Fun lessons are Fun for anyone.
So there you have it, 7 ways to work on Problem Solving skills with your classes. For more useful ideas, check out our range of blogs.
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