Background

The starting point for Bowland Maths was a real concern, expressed by Tony Cann, a businessman and philanthropist, about the state of maths education in England. In late 2005, Tony approached Quentin Thompson, an education consultant with considerable experience in education policy and management, to explore how he might best invest some of his funds to improve the situation.

Quentin’s analysis identified a major issue during the early years of secondary education (Key Stage 3) when many pupils lost interest in maths. It is the point at which mathematics becomes a little more abstract and pupils find it harder to see its relevance to their lives. KS-3 is also a critical period during which a young person’s attitudes to maths are often set for life.

This loss of pupil interest was set against a societal context in which the role of mathematics is becoming ever more important. Modern life requires some form of mathematical thinking almost every day. Thinking with mathematics is often essential for helping to understand a situation and for making well-based decisions. Employers are also increasingly complaining that new recruits cannot do "simple jobs" that involve quantitative reasoning.

There had thus been a growing concern that school mathematics was not adequately equipping young people for an age where the ability to think in a mathematical way is becoming critically important. Planning the use of money is one example, but there are many others: for example, perceptions of risk, fed by dramatic stories in the media – often quantitatively innumerate themselves, are very different from reality; such misunderstandings affect peoples' lives – who are encouraged to think that life should be risk-free, while also having a grossly exaggerated fear both of specific risks and of risks in general

More generally, it is worrying that the sentiment "Of course, I'm no good at maths", was - and still is - a socially acceptable, even fashionable, confession among many otherwise-well-educated people; the equivalent “Of course, I'm no good at reading” is rather less socially acceptable!

If these concerns were to be met, the disjunction between "school maths" and the outside world had to be bridged. There was a critical need to help teachers and their pupils respond to this challenge, successfully and enjoyably. In early 2006, Bowland Maths defined its objectives as being to introduce and promote context-rich problem solving in teaching mathematics at KS-3 that pupils would find fun.

A three pronged approach

The initial analysis suggested that the Bowland Maths initiative should adopt a three-pronged development strategy to tackle the issues:

• Teaching resources that encouraged pupils to engage with substantial and context rich problems, often extending over several lessons, with extensive teacher support and guidance
• Professional development that helped teachers with the new pedagogical challenges involved
• Assessment tasks which (a) provided teachers with feedback to help them monitor pupils' progress and so adapt their future teaching, while also (b) demonstrated that assessment is possible and feasible for context-rich problem solving

The first step was to develop a series of ‘Case Studies’ based around rich contexts which would make mathematical problems more interesting and/or relevant for pupils, with an emphasis on open-ended problems that required a range of process and problem solving skills to solve. The term case study was inspired by its use in institutions such as Harvard Business School, where it refers to a problem, usually taken from real life, that takes several days for students to address. By means of an open competitive process, advertised widely, more than 200 proposals were received from all over the world, from which an eventual 18 case studies were initially developed. Each of these case studies was designed to take pupils between 3 and 5 maths lessons and included extensive teacher guidance materials along with illustrations of pupils’ work taken from school trials.

The emphasis on process skills, combined with a pioneering push to make maths more engaging for pupils, was consistent with the views of maths education experts at the time, and reinforced the key processes that were being emphasised in the mathematics programme of study of the 2008 National Curriculum.

The second step for Bowland Maths was to design Professional Development materials for teachers. It was clear from the early days that, to use the case studies effectively, but also to teach key processes better more generally, teachers needed new pedagogical skills. Bowland Maths thus developed a series of interactive video resources for teacher development with a view to making professional development opportunities ‘accessible’ for all teachers over a long period.

In summer 2008, the Bowland Trust put 18 Case Studies and five Professional Development modules onto a web site. Shortly afterwards, all these materials were also put onto a DVD, five copies of which were made available free to all secondary schools in England, through their local authorities.

The third step was to develop a series of shorter ‘Assessment Tasks’, the development of which started in 2008. These were published on the web in 2010, with each one designed to take pupils between 20 and 60 minutes. The teacher materials for each task included illustrated samples of actual pupil work, as well as a ‘progression grid’ to help teachers see how pupil progress in process skills could be assessed.

The assessment tasks have since been ‘discovered’ by teachers as valuable teaching materials in their own right. They are often used as a ‘starter kit’ for problem solving, as they combine simple questions with teacher resources, such as illustrative pupil responses and progression grids, which can help teachers to plan their lessons.

In addition to their direct use in classrooms, the second aim in developing and testing the short ‘assessment tasks’ was to demonstrate to maths education experts the kinds of questions that could be used to assess pupil mastery of key processes. Bowland’s underlying motivation was to encourage developments in exams that would include ideas of problem solving; this was (and still is) a major stumbling block to curricular changes in the classroom. There are now some gradual changes in exams in the direction of problem solving, which may become a driver for more teaching of it.

In 2012, the Bowland Lesson Study Project was launched as a fourth strand of Bowland Maths. This was primarily to boost the Professional Development component because the approach to professional development used in Lesson Study addresses some of the key issues that Bowland Maths encountered in its earlier efforts: the need for a clear approach to PD for process skills; better demonstration and diffusion mechanisms for innovative ways of teaching; a more systematic mechanism to improve teaching materials based on practice.

Mathematical diagnosis

To remove the disjunction between "school maths" and the outside world required an understanding of why it had become so prevalent. A major contributing factor was that, in the past, school maths had been fairly heavily focussed on ‘basics’ - the mathematical skills and concepts that form the 'toolkit' for mathematics. But to link mathematical thinking to life in an effective way requires the school experience to link the following interacting components:

1. Knowledge and skills (the ‘basics’): the provision of a mathematical toolkit, including procedural skills, with knowledge of the concepts that underpin them
2. Strategies and tactics: for using these tools to tackle new problems, knowing what each tool is good for
3. Feedback and checking: pupil discussions with each other and their teacher about their approach and the realism of what they are finding
4. Attitude: an attitude inculcated in pupils that recognises the value of analytical approaches to problems, including those of mathematical reasoning

When faced with a new problem, an individual needs confidence (from 4 above) to think about and then work out how to tackle it (2), then to start carrying through a plan (using 1), while regularly reviewing progress or lack of it, and changing the plan accordingly (3).

Such an approach is valuable for thinking about any problem in life – even writing an essay – and good performance in mathematics has the same features. Yet school maths has focused on the first point above, with little attention to the others.

This needed to change. Bowland Maths was designed to support that change, addressing all these aspects in a coherent and effective way and helping teachers and pupils to respond to the challenge, successfully as well as enjoyably.

Curriculum implications

This diagnosis suggested that there were three neglected elements needing to be added to what is taught in schools:

1. Non-routine problems: Much school mathematics was currently about learning, remembering and then reproducing what has been taught. But problems are rarely routine, so recall is not enough; to be effective requires thinking with mathematics. In the classroom, this suggests that pupils should tackle a variety of problems that they have not been shown how to solve, together with some explicit teaching of approaches that can be used to help solve such problems.
2. Substantial problems: Most important problems involve chains of reasoning that take much longer than the minute or two characteristic of most maths tasks in schools - again, there is a parallel with the reasoning used for an essay. In the classroom, this suggests that pupils should tackle substantial, more complex problems, perhaps requiring a whole lesson or more to complete.
3. Multiple connections: Effective performance depends on building multiple connections between different parts of maths and with practical situations that can be modelled. For example, pupils need to be able to represent situations in words, in numbers, with graphs and using algebraic symbols - and to be able to move freely between these representations, looking for the clearest way to find and express meaning. In the classroom, the time-sequence of teaching often uses a logical sequence of topics, with, at best, each one linked only to the one immediately before it. The multiple links that are required for robust understanding do not arise in such simple ways; they have to be built in. Solving non-routine problems depends on such links, and is the key to developing them.

Bowland Maths was designed to provide materials that would enable teachers to address these three areas and so reinforce the mathematical knowledge and skills of pupils.

A worry always was that problems like these might prove to be too difficult for pupils: after all, many pupils have trouble doing even straightforward exercises. Surprisingly, the answer is no; there is plenty of evidence - which the use of the case studies have reinforced (in trials and elsewhere) - that pupils can cope with richer, more complex problems, provided that they have the appropriate range of learning experiences in the classroom.

Pupil motivation

Bowland Maths addresses another important challenge. Whenever attitudes to school maths are surveyed, a depressing picture emerges. Maths generally has a high profile and while successful pupils are usually proud of their achievements, the less successful tend to regard maths as an obstacle that they have to overcome in order to get on with what they want to do in life. The activities that many pupils undertake in school maths do not motivate them.

Motivation is a vital factor in learning; if it can be improved for mathematics in school, there are gains in performance. The types of problem that pupils are required to tackle are a key factor in their motivation. Pupils can be motivated by a great variety of problems - not only in the real world, but in fantasy worlds too, which can all be used to develop thinking and problem solving processes. Once they are motivated, their mathematical; abilities tends to improve too. There is substantial research that shows that such an approach can be successful and enjoyable for pupils - and for their teachers: thinking with mathematics about substantial problems - real or fantasy - can indeed be taught, learned and assessed.

These are some of the reasons behind the Bowland Trust development of the concepts in Bowland Maths - larger scale, open problems, interesting to pupils and using examples in forms that enable teachers to develop pupils' skills across the aspects of performance listed above. Key Stage 3 was selected as the initial target as that is the time which seems to be critical in developing attitudes to maths which can last for life.