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The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here they explain the thinking behind their latest creation - maths trails. You can find out how to obtain the trails here.

What do we see as the purposes of teaching mathematics? What exactly are we trying to encourage in our mathematics classrooms? These are "big" questions which we are not going attempt to answer fully in this article, but we would like to suggest that one aim is to give pupils the experience of feeling like a mathematician and therefore being mathematical. Being mathematical has many different interpretations, but some descriptions of what it means might include:
  • thinking about and communicating ideas;
  • engaging in problem-solving activities;
  • creating and identifying mathematical problems within given contexts.
It is clear that some knowledge of mathematics is involved in the above. Whilst the regurgitation of facts or the use of a certain skill in a particular context (for example, being able to do long multiplication) are useful, they are only valuable when they help us to solve problems. So, if much of mathematics is not about learning facts and practising skills, how can we support learners in being mathematical? They need opportunities to explore problems and to pose problems of their own. We need to guide and structure our pupils' experiences if we are to support problem solving effectively. This is where we hope the NRICH Maths Trails come in. The purpose of the Trails is to be more explicit about the skills learners need to develop in order to become better problem solvers.

Problem solving and mathematical thinking


What is problem solving?

When we are presented with a mathematical problem, it is only a problem if we do not immediately know how to solve it. The process of problem solving is like a journey from a state of not knowing what to do, towards a destination which we hope will be the solution. The key is to have some strategies at our fingertips which will help us to identify a possible route through to a solution. Our mathematical journey is often full of twists and turns where we revisit ideas or need to step back and look for alternatives. Often a mistake or dead-end gives vital clues to the mathematics of the problem and is therefore crucial in the solution process.

Polya (1945) was certain that teachers can help pupils to become better problem solvers by making the methods of problem solving transparent. The framework he devised for problem solving consisted of 'four phases': understanding the problem, devising a plan, carrying out the plan and looking back. Other authors have slightly different takes on the stages occurring in the problem-solving process. Mason et al. (1982) define three steps which they call 'Entry, Attack and Review'. Burton (1984) supplements these with a fourth which she labels 'Extension'. Piggott (2004) also proposes four 'elements' in what she terms the CAPE model: 'Comprehension, Analysis and synthesis, Planning and execution, and Evaluation', which are outlined in more detail in the Trails books. These frameworks give us a general feel for the processes we go through when solving problems, but on their own they lack sufficient detail. They are useful in order to identify where we are in the problem-solving process but what particular strategies will help us begin to understand the problem or interpret our findings?

What is mathematical thinking?

The particular mathematical skills we need to use when problem solving are more than numeric, geometric and algebraic manipulation. They include ideas such as:
  • modelling;
  • visualising;
  • being systematic;
  • generalising.
We would class these skills as elements of mathematical thinking that are needed to engage in mathematical problem solving. To date, we have published two Trails books, one focusing on generalising and the other on working systematically.

A Trail can be used as a "course" for pupils over a short or long period of time, working as a whole class, or in small groups or individually. Each Trail indicates an ordering of the materials to support each pupil's developing skills over time. However, it is also possible to dip in and out of the materials. The ordering of problems is not intended to be a straitjacket. The problems offer opportunities for pupils of a wide age and ability range, and do not imply a particular view of classroom organisation. However, there is an underlying message concerning classroom practice and the learning of mathematics as a collaborative experience, valuing the journey through a problem rather than the answer. While there is no need to offer group-work opportunities there is an underpinning expectation that pupils will be given opportunities to talk about their mathematical experiences en-route as well as at the "conclusion" of their studies. Models of using a Trail with the whole class, small groups or individual pupils are described in the books.

And here's a taste...

So, let's look at a couple of examples from the Trails. The Generalising Trail begins with a section on generalising from patterns and the first problem is Colour Wheels .
Colour wheels
In the notes for teachers in the book, we suggest the value of this activity as a demonstration of the power of using mathematical knowledge to predict and explain patterns. This power is perhaps more apparent in Colour Wheels than in other problems as the visual context means that it is not immediately obvious as mathematical. Ideas for how to introduce this activity are outlined in the book. This might involve modelling the creation of colours by using a circular disc with colours marked on its circumference. Pupils could be asked to close their eyes and imagine a wheel which produces a similar pattern and then describe or draw on whiteboards what they see. After considering the first sequence as a whole group, we have suggested encouraging pupils to work in pairs or small groups on the rest of the problem. In the notes, we offer guidance in the form of questions and prompts which will help students to:
  • build up ideas by describing what they see;
  • model the wheel either physically or numerically;
  • look for patterns and note where the repeats occur;
  • build upon their notion of multiples and divisibility.
For each problem in the trail, we have also recommended ideas for a plenary. In the case of Colour Wheels this could be an opportunity to discuss methods and findings from different groups regarding divisibility and remainders. In addition, we suggest that there is a chance here to comment on a method you might have chosen as a "teacher problem solver".

Pupils are often asked to "go away and be systematic", but we are often not clear about what this means. We cannot assume that someone can go away and, as if by magic, begin acting more systematically. There are skills we can identify that are about working systematically which can be discussed and shared with our pupils and the Working Systematically Trail aims to support this dialogue. In this Trail we identify three main strands, or ways, in which you can work systematically:
  1. Interpretation: where data or information within a problem accessed in a systematic way enables entry into the solution.
  2. Framing: taking meaning from playing with the problem -recording outcomes which lead to identifying patterns that need to be validated. In this case, the system adopted helps identify a property which needs explaining.
  3. Deduction: part of the explanation and underlying mathematics comes from the ability to plan and organise inputs to one or more algorithms.
Of course some problems can be solved using a mixture of the above. We feel that it is important to highlight the different features of working systematically to pupils so that they can begin to identify for themselves the sorts of strategies they are employing, or could employ, when solving problems.

The starting point of the Working Systematically Trail is A Mixed-up Clock .
Clock

In the teachers' notes, we highlight that the solution to this problem can only be approached by systematically interpreting information given in the clues and so this of course fits into the category of Interpretation identified above. We suggest that a "warm up" which serves to remind pupils of the meanings of number properties (such as odd, even and consecutive) might be a good way in to this problem. Students could be invited to:
  • suggest two numbers whose product is an odd number less than 100;
  • find out how many different sets of three consecutive odd numbersthere are between 50 and 70;
  • find the largest and smallest totals they can make using three consecutive odd numbers between 50 and 70.
We then go on to offer guidance on introducing the problem as it stands to the children. Initially looking at clues altogether, but revealing them one by one, provides a good opportunity to talk about which clues are immediately helpful and which must be put aside at first. This encourages the pupils to prioritise the information and to realise that it may not always be appropriate to start at the beginning and work downwards! We suggest that in a plenary it would be interesting to discuss the point at which there is an element of choice in the order the clues are utilised.

In conclusion

All the questions in the Trails books can be used in an appropriate curriculum content context rather than taught in isolation, but the important thing is that there is a journey to make and problem-solving skills with which we need to become familiar. We hope that, after tackling some of these problems, learners will recognise appropriate opportunities to generalise or work systematically and talk about generalising or working systematically when those occasions arise in the future.

The books will not necessarily make those who use them expert problem solvers, but we hope that they will help unravel some of the mysteries we encounter on problem-solving journeys.

You can find out how to obtain the trails here.

Bibliography

Burton, L. (1984) Thinking Things Through . Oxford: Basil Blackwell Limited.
Mason, J. with Burton, L. and Stacey, K. (1982) Thinking Mathematically . Wokingham: Addison-Wesley Publishing Company.
Piggott, J. (2004) Mathematics enrichment: what is it and who is it for? Paper presented at the British Educational Research Association Annual Conference, University of Manchester, 16-18 September 2004.
Polya, G. (1945) How to Solve It . London: Penguin Books Limited.

This article also appears in Primary Mathematics, a journal published by The Mathematical Association .