The five per cent solution

Incremental improvement can mean better maths learning, says Hugh Burkhardt.

Writing sums on whiteboardPolicy-makers find it all too easy to identify problems in the education system. Having done so, they try to solve them – going through a process of discussion and consultation before choosing ‘the’ solution. Things don’t turn out as they expect but by then they will have moved on, either in or out of government.

Examinations are no exception to this. Indeed, exam reforms are often the most hotly debated of all, and rightly so. The types of task assessed in examinations have enormous influence on what is taught in our ‘high-stakes’ system. But little support is provided to equip teachers to meet the new demands as they come along.

I’d like to revisit a very different approach to change that actually led to the intended outcomes, and was popular with mathematics teachers and their students. The approach was based on gradual step-by-step improvement, and was developed in the 1980s by my team at the Shell Centre for Mathematical Education at the University of Nottingham, working with AQA’s ‘grandmother’, the Joint Matriculation Board. It was known in mathematics education as ‘The Box Model’ (Shell Centre, 1984, 1985, 1987-89).

The model’s core design feature was that one new type of task was introduced each year, representing one question in the examination, five per cent of the two-year mathematics syllabus, and about three weeks’ teaching (Burkhardt, 2009). Schools were given two years’ notice and, to enable teachers to prepare their students, were offered a box of materials including five exemplars of the new type of task; marking guidance and examples of student work; materials for three weeks’ teaching; and materials to support do-it-yourself professional development.

This approach proved popular with teachers, who enjoyed the three weeks of challenging but well-supported new teaching. And exam results showed that students learned the new stuff. The limited scale of the materials meant they could be imaginatively designed on the basis of research, and carefully developed through trialling and revision. One box, The Language of Functions and Graphs (Shell Centre, 1985), later won the first ‘Eddie’ – the prize for design excellence awarded by the International Society for Design and Development in Education.

Five per cent of change per year may not seem rapid. But over a number of years, it would represent a faster rate of classroom improvement than usual. The timescale for change at classroom level is at least five years when new pedagogical and mathematical skills are involved.

So, how could this approach be useful today? Consider the current revision of the GCSE. Designing and accrediting examinations that realise the admirable assessment objectives AO2 and AO3 is proving difficult for Ofqual and for the awarding bodies, not to mention teachers. The ‘criterion referencing’ of the 1989 National Curriculum in Mathematics has ensured that for the past 25 years GCSE Mathematics has consisted entirely of short part-tasks, each taking successful students about 90 seconds. AO2 is about reasoning and critiquing reasoning, and should demand substantial chains of connected reasoning similar to essay writing in English. AO3 is about solving non-routine problems, where working out how to tackle the problem is as much a part of the challenge as carrying it through. Again, this demands substantial chains of autonomous reasoning. There is no recent experience of designing, accrediting or teaching for such tasks.

Ofqual has continued to try to maintain inter-board comparability by fragmenting the three assessment objectives (AO1, AO2 and AO3) into 22 criteria, with detailed rules for how many marks should go on each aspect for each tier. This makes it very difficult (Swan & Burkhardt, 2012) to design rich tasks that really give students “the opportunity to show what they know, understand and can do” (Cockcroft, 1982) in mathematics.

A process of gradual improvement like the one described above would convey many benefits. For teachers, it offers a pace of change that matches the timescale on which their professional expertise develops. For the awarding bodies, it would allow time to do trials – the only reliable way to discover the level of difficulty of complex tasks. And the politicians could point, year by year, to gains in student learning in the important skills that AO2 and AO3 represent, which have been missing from school mathematics for many years.

Hugh Burkhardt, formerly director of the Shell Centre, has directed a wide range of projects in both the US and the UK, often working with test providers to improve the validity of their assessments. In 2013 he was awarded the ISDDE Prize for educational design, for lifetime achievement.

  1. Burkhardt, H. (2009). On Strategic Design. Educational Designer, 1(3).
  2. Cockcroft, W. (1982). Mathematics counts: Report of the Committee of Inquiry into the Teaching of Mathematics in schools. London, UK: HMSO.
  3. Daro, P., & Burkhardt, H. (2012). A population of assessment tasks. Journal of Mathematics Education at Teachers College, 3(1).
  4. Shell Centre. (1984). Problems with Patterns and Numbers. Manchester, UK: Joint Matriculation Board and Shell Centre for Mathematical Education.
  5. Shell Centre. (1985). The Language of Functions and Graphs. Manchester, UK: Joint Matriculation Board and Shell Centre for Mathematical Education.
  6. Shell Centre. (1987-89). Numeracy Through Problem Solving. Harlow, UK: Longman.
  7. Swan, M., & Burkhardt, H. (2012). A Designer Speaks: Designing Assessment of Performance in Mathematics. Educational Designer2(5).


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