Teachers Thinking about Mathematical Thinking
This is an account of a Work Group being run by the East Midlands East Maths Hub, but this is one of 35, running in every Maths Hub across the country as part of a Maths Hubs Network Collaborative Project. To get involved, contact your local Maths Hub.
“What could be on the hidden labels?”
The maths teachers present, all very familiar with graphs of this type, think for a while, and some lively discussion ensues.
“I’m not going to show you all the labels. Why not? Because getting it right or wrong is not the point here – it’s the discussion we are having that’s important.”
So says James Thomas, Work Group Lead for East Midlands East Maths Hub, leading the first session of the Work Group: ‘Mathematical Thinking and the new GCSE’, with a small group of teachers from local schools (you can find out more about the Work Group here).
What would your GCSE students make of this hide/reveal task?
James admits that if he was as evasive about the ‘reveal’ part of this task, with his Y10 class, “there would be a riot”, but his point is made – that learners of mathematics often get very tied up with whether they are ‘right’ or ‘wrong’. This can be a crutch for those that are normally right, and equally powerfully, something that makes maths terrifying for others (and not only those that often get it wrong). In a task like this, James has taken a graph that you might find in any secondary maths textbook (usually with a set of closed questions alongside) and created a rich and meaningful discussion, where students have to engage with the data to argue why it could, or could not represent the data suggested by their classmates.
Hide/reveal is just one of the powerful techniques that the Work Group is discussing, for promoting reasoning and problem solving in secondary classrooms. They form a toolkit with which teachers, even inexperienced or non-maths specialist teachers, can adapt standard textbook questions to discourage students from adopting a ‘learn the algorithm’ approach. The techniques are designed to adapt standard, and readily available material in a way that promotes deeper thinking and therefore more connected understanding. Teachers in the Work Group acknowledge that the content demands at GCSE means that reasoning and problem-solving can easily get ‘tacked on’ to an already full curriculum. There is a danger that it becomes the part of the curriculum that gets left out when time is short, or only accessed by higher-attaining students, rather than being integrated throughout learning. The toolkit provides a way to bring reasoning into every lesson.
Below, are a couple of examples of ‘generic’ approaches that can open up thinking on any topic, question or context:
Example 1: Sequencing Lines of Working Out / Here’s the Calculation, What’s the Question?
A shop sells flour from large sacks at a rate of £1.20 for 750g. As an introductory offer the shop is currently offering a discount of ___% on all sales of flour.
The working out a student used to answer a problem related to this information is shown mixed up, below.
Can you construct what the original problem might have been, and can you rearrange the 'working out' into the order you think the student might have had it?
Write a reason explaining the meaning of each line of working out in relation to the problem.
What was the discount on offer?
Example 2: Developing chains of reasoning and deepening understanding
Below is an Edexcel question from the June 2017 GCSE exam to which a number of the Work Group activities could be applied to open up opportunities for reasoning and deepening understanding. Where pupils may already be confident answering the question, we look here at opportunities to use the structure of the problem to develop further chains of reasoning.
On another day Daniel bakes a different number of vanilla, banana, lemon and chocolate cakes but the quantities remain in the same proportion.
- If you knew how many vanilla cakes were baked could you work out how many banana cakes were baked? Give a chain of reasoning (What about in reverse?).
- Try something more challenging: if you knew the number of chocolate cakes baked could you work out the number of banana cakes?
Choose any two of the items below, how could you work one of them out if you know the other? Explain your chain of reasoning. What other discussion issues might arise out of the answers?
In the session we visited, the Work Group also spent some time considering ‘what is reasoning?’, and how does it differ from problem-solving? Using the NRICH article Reasoning: the Journey from Novice to Expert, that describes five stages of reasoning (Describing, Explaining, Convincing, Justifying and Proving), they considered some pupil work and tried to determine which of the NRICH stages best describes each piece. Participants in the Work Group concluded - perhaps unsurprisingly - that the pupils were able to reason better verbally than in written form. This highlights a challenge for teachers: to teach students to express what they can tell someone verbally, in effective written form for examiners.
The problem is fairly standard:
What presents a challenge to pupils is expressing their reasoning: verbally, but also in written form.
Try assessing their reasoning yourself. Have a look at these two pieces of work. Consider first, the written argument– how complete is the reasoning? (You could use the NRICH scale). Then play the short video to hear the verbal argument – is it any more complete?
The Work Group that we visited is part of a Maths Hubs Network Collaborative Project, ‘Mathematical Thinking and the new GCSE’, being run by Maths Hubs throughout England. James is one of the 35 Work Group Leads facilitating the project in his own locality. Based on best practice researched and refined by the Multiplicative Reasoning at KS3 project run in Maths Hubs over the past two years, this national project aims to support long term development of skills throughout the secondary curriculum, as well as the immediate needs of GCSE students. The approaches are practical, accessible and classroom-based.
This model of professional development is designed to engender long-term, sustainable change in classroom practice, throughout the departments of participating teachers. When schools sign up a teacher, or preferably, a pair of teachers, they are committing to four days of release time, plus the time that the teachers need to lead their departments in introducing more reasoning into their classrooms. Between each Work Group day, there is a ‘gap task’, typically involving trying out one of the activities so as to be able to reflect deeply on the process with other teachers in the group, or using Lesson Study to observe a colleague using one of the activities. It’s a commitment for departments that are looking for genuine, long-term improvement in how they teach reasoning and problem-solving.
If you are interested in taking part in this kind of professional development, contact your local Maths Hub now – there may still be opportunities this year, and certainly Hubs are keen to hear from teachers wanting to get involved in 2018/19.