Universities bemoan the poor quality of maths candidates, but lecturers can do more to help understanding by showing students the importance of making connections between ideas, says Peter Kahn.

Survey after survey speaks of the difficulties university students face with mathematics. A recent report lambasted the poor maths grounding of first-year engineering students and called for remedial tuition. But it could easily have focused on students studying physics, economics or mathematics itself.

Too many young people shy away from studying mathematics, which leaves them unprepared for employment in a knowledge economy underpinned by mathematical patterns of thinking. Solutions, such as the introduction of AS/A levels, were meant to encourage more students to study mathematics. But nearly one-third of pupils failed the new AS level in mathematics this summer. In universities, meanwhile, more time has been devoted to discussing the content of maths courses and the applications of mathematics.

But the way ideas are taught and learnt is equally important. Too often, lecturers simply present students with finished mathematical products. Students are shown proofs or mathematical models in their polished form, but they have no sense of how these ideas relate to other ideas or why they are given in the form they are.

Lecturers need to concentrate more on the process by which students come to understand mathematical ideas. For instance, stating the definition of an idea and providing a couple of examples is one way of trying to get the idea across. In contrast, getting the students to generate a range of their own examples of the idea is more likely to ensure they engage with the meaning of the definition. Getting students to draw out their own definition from a range of examples of an idea goes a further stage.

Evidence suggests that only when students engage with mathematical ideas at this level will they be able to make the connections that lead to understanding.

If lecturers are too concerned with presenting the products of mathematical thinking, students will likewise be too concerned with finishing the mathematical tasks. If, however, a student completes a task without trying to connect the ideas involved to other relevant ideas, their learning is usually only of short-term use. Given the way in which advanced ideas build on more basic ideas in mathematics, this approach is disastrous for most learners.

Unfortunately, we cannot simply tell them that they need to make connections. When higher education took in only an elite, students might have been expected to see the value of making connections between ideas. All you needed to do was introduce the necessary ideas and ask students to solve problems involving the ideas. In solving the problems, they would naturally see the value of making the connections for themselves. But we now have a mass system of higher education. We need to promote strategies that help students to engage with mathematical tasks on a genuine mathematical level.

Take the task of solving a problem. Students need to do far more than memorise standard solutions. Studies of expert problem-solvers indicate that students will not only need to try out a given method of solution, they will also need to clarify their understanding of the problem, to plan how they might solve the problem and to review the effectiveness of their attempts. Taking this analysis of effective problem-solving further, we can see that these stages involve even more fundamental mathematical thought processes. Understanding the problem, for instance, might involve connecting a visual image to the formal statement of the problem or analysing a complex idea so that the student can make sure they understand all of the contributory ideas.

This kind of approach to solving a problem exposes connections between ideas. Students need to realise that it is more important to make connections between ideas than to complete tasks. And lecturers need to realise that they can no longer get students to make connections simply by presenting polished mathematical ideas and by requiring the students to solve problems. Only as we find practical ways of applying this understanding will an end to the mathematics problem in our universities be in sight.

Peter Kahn is teaching development officer at the University of Manchester. His book, * Studying Mathematics and its Applications * , was published by Palgrave in October. * Effective Learning and Teaching in Mathematics and its Applications * , co-edited with Joseph Kyle, will be published by Kogan Page in January.

** Strategies for effective maths learning **

- Focus on the underlying thought processes mathematicians use to complete tasks
- Generate examples of concepts
- Connect visual images to related ideas and their symbolic representations
- Pay attention to the precise meaning of symbols
- Spot which basic ideas contribute to more advanced ideas
- Make sense of the logical ideas involved
- Systematically look for connections between ideas more widely.

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