Project work in mathematics is not a new idea, although it has different names and different forms, including investigations, rich tasks, or inquiry-based learning. However, despite the idea being around for a while, it is still seen by some as an optional extra, to be added on if there’s time at the end of a topic, and perhaps only suitable for the highest-attaining students. We believe project work should be part of every student’s experience of learning mathematics, so let’s take a look at the case for including project work in every mathematics classroom.
One powerful image that has informed the thinking of the NRICH team over the years is the rope model proposed by the National Research Council in 2001. This model includes five strands of mathematical proficiency woven together:
• Conceptual understanding
• Procedural fluency
• Adaptive reasoning
• Strategic competence
• Productive disposition
Much traditional mathematics teaching focuses mainly on the first two of these. Students need to have a good understanding of concepts and be able to use mathematical procedures and algorithms fluently. The National Research Council’s model draws attention to the need for students to also be able to solve problems and explain their reasoning, as well as having a positive outlook about mathematics. Project-based learning offers a great opportunity to develop all five of these proficiencies.
In particular, mathematics is often a subject that students feel ambivalent or even negative about, and if our aim is to develop students’ productive disposition (being positive about mathematics and their own capacity to think mathematically), working on rich tasks can be a great way to achieve this.
But what does it mean to think mathematically? Good mathematicians notice patterns, make conjectures and generalisations, explain their ideas, and justify or prove their results to others. Project work gives students a flavour of what it is like to work in this way. Take, for example, the project ‘Odds and Evens’ (Cambridge IGCSE™ Core and Extended Mathematics Coursebook, page 252). This starts with the simple game of drawing out two numbers from a bag and adding them together to find their total. If the total is even, the students win, and if it’s odd, they lose. The teacher invites students to calculate the probability that they will win. Students can approach this in a variety of ways – listing all possibilities, a sample space diagram, and a tree diagram – and this offers a great opportunity for some class discussion about the benefits of different methods.
To find out more about how to organise your classroom during project work lessons, see the accompanying blog post.
Then the task moves on to looking at other bags of numbers before opening up to the big question: does a set of numbers exist that makes it a fair game, where the probability of winning is 50%? This question opens up the possibility of in-depth exploration, collaborative working, and the potential discovery that there is a relationship between the number of odds and evens in the games which are fair and that there are infinitely many solutions, which, with the insights about patterns, can be generated very easily.
This experience of working mathematically, and generalising to big ideas that can be proved, is invaluable for developing students’ concept of themselves as capable and confident mathematicians. There’s something exciting about walking out of a lesson having discovered a big idea for yourself, and the projects in Cambridge IGCSE™ Core and Extended Mathematics and Cambridge IGCSE™ and O Level Additional Mathematics series give students the opportunity to feel this excitement and get a real sense of achievement. And once they are bitten by the mathematical exploration bug, there are plenty more activities freely available on the NRICH website for them to continue their mathematical journeys.
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.
Have you enjoyed reading this blog post? Listen to this podcast with Charlie and Alison to learn how you can facilitate a successful project-based lesson in maths.
Reference:
National Research Council. 2001. Adding It Up: Helping Children Learn Mathematics. Washington, DC: The National Academies Press.
