This paper discusses some of the design approaches used in 40 years of work at the Shell Centre for Mathematical Education – and some examples from other groups – for which there is some evidence of efficacy. These range from detailed tactics for designing student activities to more “political” strategies for fostering change at system level. The paper is structured as a short overview listing and summarizing the strategies, each of which links to a page containing more details, examples and – where possible – onward links to the actual materials.
The Shell Centre for Mathematical Education was founded in 1967, largely as a result of the efforts of the Professors of Pure and Applied Mathematics in the University of Nottingham. Some mathematicians at that time believed that the keys to improvement were a deeper understanding of the subject by teachers and a focus on these fundamentals in teaching. So initially the Centre mainly provided professional development workshops for teachers in the region. When Hugh Burkhardt arrived as Director of the Shell Centre, far from convinced that we really knew how best to help teachers teach better mathematics more effectively, he decided to adopt a broader, research and development brief. The vision was, and is, to focus on research, design and development that will have direct impact on what happens in school classrooms. The small team of permanent staff meant that large-scale influence can only be achieved through reproducible materials that support more effective teaching. This suggested an important role in our work for imaginative lesson design, based on research-in-depth, that would avoid the imitative procedural nature of the ‘demonstrate and practice’ teaching and learning, observed in most mathematics classrooms. The Centre has been able, over a long period, to attract exceptionally talented designers of educational materials - notably our late friend and colleague Malcolm Swan, with his genius for the design of learning activities.
From time to time we have analyzed the design strategies, tactics and techniques that have proved effective and written some of them up in journal articles (see, e.g, Burkhardt 1981, Bell 1993, Swan 2006, Burkhardt and Swan 2014, 2017). This article arose from a talk in Hong Kong launching the 2019 conference of the International Community for the Teaching of Modelling and Applications (ICTMA- 19). The talk focused on the sequence of Shell Centre projects and the gradual abstraction of design principles from them. It then seemed useful to organize these into a catalogue, presented at ISDDE 2019. This article adds brief explanations and, most important, links to illustrative examples from various projects from the Shell Centre and elsewhere.
These elements of design have been developed mainly with teachers and students aged 7 to 18, working in school and college classrooms settings, along with associated assessment and professional development. Nonetheless, we believe some of them are more broadly relevant. We hope the collection may prove interesting, even useful, to readers of Educational Designer, particularly those who design tools for others to use. Comments and suggestions for other entries are invited.
Some strategies are ‘hard sells’, often for political reasons (Burkhardt 2019). For example, politicians often see examinations as ‘measuring for accountability’, with little concern for what is measured and its effect on classroom practice. So it pays to work with policy makers as well as practitioners, whenever you can (Burkhardt 2019). Focus on ‘their’ problems, offering insight and modest step-wise win-win solutions even when they, as so often, want to ‘fix the problem’.
It is useful to distinguish (Burkhardt, 2009):
Strategic design focuses on the design implications of the interactions of products with the system they aim to serve.
Many otherwise excellent innovations fail because of the neglect of strategic design questions including: What kind of users (often teachers) is this designed for and developed with? What changes does it require in their current practice? Is adequate support provided, within the materials or otherwise? Above all: Why should they change? Are the reasons among their day-by-day priorities?
Tactical design focuses on structures within the materials that help teachers achieve learning activities that effectively forward student thinking.
This raises parallel issues and questions: What demands of professional expertise and time does using this tool make on the target user initially, then after using it several times? Does the tool provide effective support for this transition? Is there enough detailed support and guidance for users in the target group? Is there so much that many won't read it?
Detailed design focuses on helping the typical user to achieve high-quality realisation of the learning goals – for example, a typical teacher to handle a powerful learning activity sequence that they can enhance from their own talents and experience. It is here that creativity can inject magic into learning – in design the details matter. Here there are two overriding questions:
- Does the design set out an activity sequence that works well with students in forwarding the intended learning goals?
- Does the design communicate with target users so that they can realise with their students a variant on the activity sequence that is compatible with the design goals?
An iterative program of trials, with rich and detailed feedback guiding revisions, should provide the answers to these questions.
While detailed design is not the focus of this article, it is perhaps the most important factor in design of all materials for learning since the quality of the tasks that students work on sets an upper limit on what they can learn. Tasks for learning and assessment should be varied and rich enough to require thinking – not just remembering – and constructing chains of reasoning based on recognising connections within mathematics and to a range of contexts in the wider world. The many and varied tasks in the links below have been chosen to exemplify high quality.
Note: Online materials
This printable document only contains the overview – more complete descriptions of each strategy and tactic, with examples and links to online resources, can be found online - see https://educationaldesigner.org/ed/volume4/issue13/article53/links/ for the index.
- Look for ‘leverage’ points that answer: “Why should they change?” "Is there enough help?"The Shell Centre's Testing Strategic Skills program (1980-88) provides our best realization of these strategies, exemplified by Problems with Patterns and Numbers (‘The Blue Box’) and The Language of Functions and Graphs (‘The Red Box’)
The next 4 entries provide explicit examples of leverage strategies.
Take a constructivist approach to professional development – Professional development, like all learning, should be activity-based, with teachers learning primarily by reflecting on high-quality student learning activities in their own classrooms. See ‘The sandwich model’ under ‘Design tactics’ below.
For teaching materials
For professional development materials
For assessment materials
A challenge in designing non-routine problems, particularly for high-stakes situations, is to know how far from each student's social and mathematical experience the problem is – the transfer distance. The next two examples are methods of handling this through giving students a known prior experience. (Guidance to teachers on how intensively to prepare the students may, or may not be followed if the examination is high-stakes.)
Enormous thanks to the many people with whom these strategies and tactics have been developed - and, of course, to fine designers who have used them so powerfully.
Bell, A. (1993). Some experiments in diagnostic teaching. Educational Studies in Mathematics, 24, 115-137.
Burkhardt, H. (1981). The Real World and Mathematics, Glasgow: Blackie, reissued Nottingham: Shell Centre Publications 2000.
Burkhardt, H. (2009) On Strategic Design. Educational Designer, 1(3). Retrieved from: http://www.educationaldesigner.org/ed/volume1/issue3/article9.
Burkhardt, H., & Swan, M. (2017). Design and development for large-scale improvement. Emma Castelnuovo Award lecture in G. Kaiser (Ed.) Proceedings of the 13th International Congress on Mathematical Education, pp 177-200. Cham: Springer International Publishing
Burkhardt, H. (2019) Improving Policy and Practice. Educational Designer, 3(12). Retrieved from: http://www.educationaldesigner.org/ed/volume3/issue12/article46/
Swan, M. (2006). Collaborative Learning in Mathematics: A Challenge to our Beliefs and Practices. London: National Institute for Advanced and Continuing Education (NIACE) for the National Research and Development Centre for Adult Literacy and Numeracy (NRDC).
Swan, M., Burkhardt, H. (2014) Lesson Design for Formative Assessment. Educational Designer, 2(7). Retrieved from: http://www.educationaldesigner.org/ed/volume2/issue7/article24/
Additional references can be found in the individual ‘More…’ sections. We have only provded formal references for academic articles and books – there are many other links to materials on the web.