Sheila Evans with Malcolm Swan document the progress of a project designing problem-solving lessons
This article describes how the design of a problem-solving lesson that incorporates a range of sample pupil work evolved over the course of a project. The content of the sample pupil work is shaped by genuine pupil work, but is written by a designer. The article will highlight the advantages of using sample pupil work and how issues that arose through their use have been addressed. This may be of particular interest to teachers who use, or plan to use, real or 'manufactured pupil work in their mathematics lessons.
Over the last two years, a team of mathematics educators at the Centre for Research in Mathematics Education at the University of Nottingham have designed mathematics formative assessment lessons to support US schools implementing a new curriculum. The team is led by Malcolm Swan and comprises a number of designers, including Sheila Evans. The activities, and the teacher guide, are carefully designed to maximise opportunities for pupils to make visible to themselves, and the teacher, their current understanding and reasoning. In so doing teachers are able to be particularly adaptive and responsive to pupil learning needs over the course of a lesson, as well as facilitate their planning of future lessons (Black and Wiliam, 1998; Swan, 2006). These secondary school mathematics lessons are freely available on the web: go to http://map.mathshell.org.uk/materials/index.php
About a third of these lessons involve nonroutine, problem-solving tasks. They are designed to assess and develop pupils' capacity to apply mathematics flexibly to rich, unstructured problems, both from pure mathematics and from the real world. During these lessons pupils have to make sense of a problem, to decide on the mathematics to apply to the problem, to do the mathematics, and finally have to interpret and communicate the solution to their peers and their teacher.
Problem solving lessons present teachers with an important pedagogical dilemma. As the whole purpose of the lesson is for pupils to develop the ability to select, apply, and compare appropriate mathematical methods, the teacher cannot be sure which methods pupils will choose. We have found that pupils are unlikely to choose methods that they have only just acquired, and usually prefer concrete numerical or graphical approaches to algebraic approaches. This presents the teacher with a dilemma, how does one reveal the power of an algebraic approach without 'forcing" pupils to use it, in which case the lesson is no longer a true problem-solving lesson, but a mere exercise in algebra? One possible solution is to follow pupils' own attempts to solve a problem with a critiquing activity in which we offer pupils pre-prepared 'sample pupil work' that show alternative attempts to solve the problem, all of which are imperfect, and which invite pupils to try and improve, and complete. As different approaches are contrasted and compared, ideas are combined and refined into collaborative solutions.
The structure of these lessons has changed over the duration of the project. To begin with, we gave pupils a problem at the start of the lesson, but found that it was difficult for teachers to understand and follow pupils' reasoning and then to follow this up immediately. Now pupils are given a problem to tackle on their own, before the lesson. The teacher then reviews their work, but instead of grading it, which promotes competition between pupils (Black and Wiliam, 1998) and distracts attention away from the mathematics, the teacher formulates questions for pupils to answer in order to improve their solutions. The lesson starts with pupils using these questions to review their own work. Then in small groups, pupils evaluate fellow pupils' solutions, with the aim of producing a joint approach that is better than their individual efforts. After discussing, as a class, the range of strategies used by different groups, pupils then assess the sample pupil work. …