If you’ve read our previous blog post ‘Why is it important for students to undertake project work in mathematics?’, hopefully you are now very keen to try some of the projects from Cambridge IGCSE™ Core and Extended Mathematics and Cambridge IGCSE™ and O Level Additional Mathematics series in your classroom. The projects have been carefully chosen to complement the material within the surrounding chapters of the textbooks, giving students an opportunity to show off the mathematical skills they have learned as they tackle thought-provoking tasks which they can really get their teeth into. We want students to discover some big ideas so that they leave lessons with a real sense of achievement.
Each project is accompanied by guidance to help you make the most of the activities, and this blog post offers some more general advice on what to say and do, to help you to support your students when they are working on rich tasks.
It is tempting to rush through the early stages of an investigation, because as teachers we know what’s coming and we are in a hurry to get our students to the punchline. However, it’s really beneficial to start a rich task slowly, and give students plenty of time to explore and get to know the problem. In ‘Marbles in a box’ (Cambridge IGCSE™ Core and Extended Mathematics Coursebook, page 550), the original problem invites students to work out the number of winning lines in a three-dimensional game of noughts and crosses by placing marbles in a 3 by 3 by 3 box with layers. By immersing themselves in this task, exploring a variety of representations and strategies, and learning from the mistakes they make along the way, students do the necessary groundwork that prepares them to go on to consider larger grids, and eventually find a formula for the n by n by n case. This is at the heart of the ‘low threshold, high ceiling’ philosophy of these projects, as well as many others which appear on the NRICH website. ‘Low threshold’ means that everyone can get started on the activity – there is no barrier to entry, and students can immediately begin exploring mathematically. ‘High ceiling’ means that the task includes natural extensions that offer all students the opportunity to be challenged, get stuck, and develop their resilience by getting themselves unstuck.
The role of the teacher in the early stages of the project is to create an environment in which students feel confident to explore and conjecture, and where they feel safe testing out ideas which may be wrong but can be modified. Working on projects is a perfect opportunity to show students that mathematics does not always come neatly packaged, but mathematicians learn from their mistakes and develop elegant ideas from messy starting points.
When deciding how to organise your classroom for project work, you may wish to arrange students around tables in groups of three or four, so that they can collaborate more easily. Some teachers like to provide a single, large piece of paper in the middle of the table to encourage messy and chaotic collaboration at first, which can then be refined into a neater presentation. Other teachers make good use of individual student whiteboards, or even writing on the desks themselves, to allow for mistakes to be easily corrected.
Throughout the project, the teacher’s role is to act in a way that will develop students’ higher-order thinking skills. This might mean being quite ‘hands-off’ at first – circulating round each group and listening to conversations without interrupting, and building up a picture of what each group is working on. This allows you to decide on the best time to intervene, perhaps bringing the class together and inviting groups to share their insights. As work progresses, invite students to move from particular examples to more general cases – in ‘Marbles in a box’, for example, ask groups who have a method and an answer for the 3 by 3 by 3 case to consider winning lines on a 4 by 4 by 4 grid, or even the n by n by n case. Some groups will reach the ‘high ceiling’ of justifying and proving their findings, and even if not every group gets to that stage, students can then see what other groups have been working on and get a flavour of the mathematical excitement that comes from being able to assert a big idea with certainty.
One barrier to using more project work in lessons is the time pressure which faces all teachers. The projects we have chosen for the Cambridge IGCSE™ Mathematics series can be completed in a single lesson, but as with so much in life, what you get out depends on what you put in, so it might be worth allowing more than one lesson for students to fully appreciate the richness of the tasks. One way to mitigate the time pressure is to encourage students to continue their work on the task between lessons, or to set the initial task as a homework activity before the first lesson of project work. Inviting students to work in groups also ensures that students aren’t getting stuck on their own, and frees up your time as a teacher to work with those students who perhaps need a little more support.
We hope after reading this blog post you have a clearer idea about how to get the best out of the projects. If you want to read more about using rich tasks, including video material of us working in this way with a class, take a look at ‘Tilted Squares – Teaching Using Rich Tasks’, and to read more about the research and beliefs that inform our thinking about teaching mathematics, visit the NRICH website.
This blog post was written by Alison Kiddle, a freelance mathematics education consultant, who has previously worked for NRICH as a Key Stage 4 coordinator, and Charlie Gilderdale, the Secondary Coordinator at NRICH.
