Improving Learning in Mathematics

This is probably the most exciting and innovative resource for mathematics teaching that I have seen for a very long time. And what is most surprising is that this resource was funded by the government through Success for All, which is a partnership between the DfES Standards unit and the Learning and Skills Council. Not only that, but you get all this for free!

To download a copy of the book from the resource go here. The book explains the rationale of the resource, outlines some of its uses and offers practical advice about adopting the approaches suggested.

What is interesting when you visit these websites is that the resource, which has been largely developed by Malcolm Swann and Susan Wall, is described in the context of “skills for life” – the Maths4life website describes itself as a “numeracy and non-specialist maths project”. I suggest that the DfES does not realise what a wonderfully rich resource they have produced – a resource which provides a wealth of challenging activities for all post-14 learners in any institution and at whatever level they are working.

But the resource does not just provide the activities. Through the videos and the professional development guide it provides teachers with excellent support to help them adopt the suggested approaches in their own classrooms. This material would be a most valuable component of any initial training programme for teachers.

So what are the qualities of the teaching and learning suggested? “Active” rather than “passive” learning and “challenging” rather than “transmission” teaching is exemplified, advocated and supported. The activities included in the resource are designed to encourage best practice and have undergone several rounds of observed trials in a variety of learning contexts: colleges, schools, prisons and work-based learning environments. But, as the authors say, the materials themselves do not guarantee effective teaching. The following principles for good teaching need also to be followed:

Build on the knowledge learners bring to sessions
Expose and discuss common misconceptions
Develop effective questioning
Use cooperative small-group work
Emphasise methods rather than answers
Use rich collaborative tasks
Create connections between mathematical topics
Use technology in appropriate ways

The resource provides a lot of general practical advice about how to put some of these principles into practice:

Arrange the room to facilitate discussion
Get students to respond to questions using mini-whiteboards
Use posters to stimulate thinking
Use card-matching activities to focus on interpretation

The activities themselves are provided in two ring binders and as pdfs on the DVD-ROM. The materials do not attempt to “cover” any particular course, but instead provide a range of activities, together with detailed lesson notes, targeted at learners working at Level 1 (GCSE E to G) through to level 3 (AS and A2). There is also advice – accompanied by templates on a CD – to help you develop your own activities using the same principles. If you think that mini-whiteboards, posters and card-matching activities are not suitable for your A-level class then just look at what the students say in the numerous video clips provided with the resource.

All the activities focus on developing mathematical knowledge and skills, while at the same time fostering mathematical thinking. They are of five types.

Type 1: Classifying mathematical objects

The activities include “odd one out” tasks and tasks where objects are classified using two-way tables (Carroll diagrams). These help learners to recognise properties and develop mathematical language and definitions.

Type 2: Interpreting multiple representations

Through card-matching activities learners draw links between different representations and develop new mental images for concepts.

Type 3: Evaluating mathematical statements

In deciding whether given statements are always, sometimes or never true learners develop rigorous mathematical arguments and justifications, using appropriate examples and counter-examples to defend their reasoning.

Type 4: Creating problems

Learners devise their own problems for other learners to solve.

Type 5: Analysing reasoning and solutions

Activities include: finding as many different ways as possible of solving a problem; examining a complete solution and identifying and correcting any errors; reconstructing an “extended chain of reasoning” by putting together in the correct order each step represented separately on a set of cards.

Some of the suggested activities use purpose-built applets. Some of these were developed from applets available on the Freudenthal Institute’s website: www.fi.uu.nl/wisweb/en/. I particularly liked the Building houses applet, which challenges learners to explore connections between 3-D models and their plans and elevations. Here is my solution to one of the problems on the DVD-ROM, which certainly challenged me! You have to build a model to match the given plan and elevations using the correct number of cubes. The beauty of the software is that you can move your 3-D picture about to check that your plan and elevations are correct.

Finally, if the activities themselves do not convince teachers or managers that the approaches advocated in Improving Learning in Mathematics will do just that then they only need to spend time looking at the video clips of lessons and listening to the positive comments from the teachers and, most importantly, the learners who have been involved in the development of this material.

Barbara Ball

Improving Learning in Mathematics
DfES Standards unit
September 2005

You can order the materials direct from DfES Publications on 0845 6022260 quoting reference T&LMATHS.