It is a time of great change for the teaching of mathematics. The Government has given high priority to mathematics, including an objective that virtually every student in post-16 education should continue to study the subject. This could be re-sitting GCSEs to achieve a C, studying AS/A-Level maths or taking a new type of course called Core Maths.

The introduction of the new National Curriculum in all key stages aims to address the formulaic maths teaching and teaching to the test that has resulted from an increased accountability and emphasis on results in English and maths. Emphasis is now on creating learners that ‘can reason mathematically and solve problems’, making content comparable to that in mathematically high performing countries.

Looking at international practice, high performing countries create generations of maths learners who are not only extremely adept, but also demonstrate a genuine passion for the subject. Lessons from these systems may help to develop our approach to mathematics teaching, taking it from a subject often loathed by students to one that is correctly viewed as presenting endless opportunities and challenges.

Problem solving approaches combined with lesson study for professional development has been shown to be an effective way of innovating and sustaining good practice in the classroom; so what can we learn from these methods?

### Problem solving

Mathematically high performing countries, such as New Zealand, Japan, Singapore and Finland, all have an emphasis on problem solving in their maths curricula. Stigler and Hiebert in The Teaching Gap (1999) report that in these countries, problem solving is seen as an essential part of mathematics education; problem solving is used to actually drive learning, as opposed to merely testing it.

Taking the example of Japan, Stigler and Hiebert found that teachers present “a problem to the students without first demonstrating how to solve the problem”. In this way, learning begins with a problem to be solved, and the problem is posed in such a way that the students need to gain new knowledge before being able to solve it correctly. Rather than seeking a single correct answer, students are able to interpret the problem, gather relevant and necessary information, identify possible solutions, evaluate options and present conclusions.

The aim of this approach is to develop simultaneously both the creative activities of the students and their mathematical thinking in problem solving. It is based on the belief that students’ perceptions of mathematics are formed by the work they are asked to do. For example, if they are mainly asked to carry out pre-taught procedures in a set of exercises, they will think that mathematics is purely about following a set of rules, largely irrelevant to ‘real life’. If students are to understand that mathematics is about solving problems that are applicable to real world experiences, they need to spend most of their time solving such problems.

Problems can be open in three different ways:

1. Process is open – where the learning comes from studying the different ways of solving a problem;

2. End product is open – where the learning comes from studying the different answers;

3. Ways to develop are open – where the learning comes from the students using the initial problem to generate new problems of their own.

The Japanese mathematic educationalist Sawada (1997) suggests that the success of the open-ended approach depends very much on the choice of problem. For example, he describes how Japanese teachers firstly determine if the problem is appropriate by asking three questions:

1. Is the problem rich in mathematical content and valuable mathematically?

2. Is the mathematical level of the problem appropriate for the students?

3. Does the problem include some mathematical features that lead to further mathematical development?

They then develop their lesson plan by:

1. Listing the students’ expected responses to the problem;

2. Making the purpose of using the problem clear;

3. Devising a method of posing the problem so that the students can easily understand the meaning of the problem, and/or what is expected of them;

4. Making the problem as attractive as possible;

5. Allowing enough time to explore the problem fully.

Ensuring that the problem is accessible to all students is the key here, and as this approach places ‘special emphasis on the mathematical thinking of individual students’, it is important that the teacher remains neutral, acting as a facilitator for individual methods.

This problem solving-centric methodology is one that is being championed by the Core Maths courses, whose objectives are based on DfE technical guidance and objectives, which are as follows:

1. Objective 1: Deepen competence in the selection and use of mathematical methods and techniques.

2. Objective 2: Develop confidence in representing and analysing authentic situations mathematically and in applying mathematics to address related questions and issues.

3. Objective 3: Build skills in mathematical thinking, reasoning and communication.

The unique aspect of these courses comes in the form of focus on the process skills, defined by the above objectives, taking precedence.

### Lesson study

Innovation in the classroom is difficult enough, but sustaining such innovation is even more problematic. Experience from the mathematically high performing countries previously discussed shows that lesson study, where groups of professionals work together to plan, observe and reflect on ‘research lessons’ is a really positive way forward to encourage greater classroom creativity. There is a four-step process underpinning lesson study, namely:

#### Overarching aim

You need an overarching aim for this professional development; for example, something along the lines of: Our learners will become independent thinkers who enjoy working together to produce creative solutions in mathematics in unfamiliar situations.

#### Specific objectives

The overarching aim can then be translated into specific objectives for the research lessons, for example:

Enjoy doing mathematics – to help learners to enjoy and sense personal reward in the process of thinking, modelling situations, exploring data and solving problems.

Be willing to take risks and to persevere – to improve learners’ willingness to attempt unfamiliar problems and to develop perseverance in solving problems without being discouraged by initial setbacks.

#### Design, deliver and review research lessons

Teachers work in groups of three or four, designing and planning as a group activity research lessons that meet some of the specific objectives above. The cycle design lesson plan

-deliver observed lesson -review lesson plan

-revise plan is repeated, with each teacher in the group taking a turn to deliver the research lesson (one or two cycles each half term is a good benchmark to aim for).

#### Feedback

What is learnt is fed back to all other maths teaching staff in the school or the local teaching alliance. You might consider re-teaching the revised lesson plan, perhaps with video clips of the key aspects, before making this more widely available. Not only is this process important for professional development, it will also help in the sustainability of innovations in schools and colleges.

### Moving forwards

During this period of change in mathematics education, and education in general, real opportunities exist to teach and learn in a way that is different from the transmission model that has for so long dominated teaching and learning in the UK.

With opportunity, however, comes risk. We will have to take some risks to further change the landscapes so that teaching to the test does not dominate, but innovation does. Change is necessary if we want both our teachers our learners to enjoy mathematics and reach their potential as mathematical thinkers, with the confidence and competence to apply their newfound skills and knowledge.

### About the Author

David Burghes is a Professor of Education at Plymouth Institute of Education, part of Plymouth University, and directs the Centre for Innovation in Mathematics Education, that aims to support and inspire teachers of mathematics through dissemination of good practice from around the world. He is currently working for CfBT as the teaching and learning specialist for the Core Maths Support Programme. This article expresses David’s personal views and should not be seen as DfE policy nor necessarily that of the Core Maths Support Programme.