Teaching A-level for the first time is a big step for many otherwise experienced teachers, especially in such subjects as mathematics, where rustiness sets in quickly.

Last year, Claydon High School and neighbouring Thurleston High School in Ipswich, set up a sixth-form. This year, we hoped to appoint two mathematics teachers, both able to take A-level classes. We wanted an experienced teacher and a confident newly-qualified teacher. Despite advertising nationally, and contacting a dozen teacher-training institutions, we interviewed only two NQTs and some experienced teachers keen to begin A-level. At interview, it was clear that all would need support to teach beyond GCSE. We appointed only one, a good 11-16 teacher with A-level potential.

A mathematics teacher needs a depth of subject knowledge and understanding, an ability and willingness to make students think mathematically, and enthusiasm and involvement.

More subject knowledge is needed to teach A-level mathematics than to pass it (yet many 11-16 maths teachers have only A-level qualifications). Teachers need to understand how parts of the syllabus relate to each other and to more advanced topics. For example, “completing the square” can be related to inequalities, solving equations, complex numbers, transformations of graphs and, later, to quadratic forms and their associated matrices.

Teachers should also know different ways of approaching each topic, for instance, logarithm arises as the integral of the reciprocal and as the inverse of exponential. Good teachers strive to extend their repertoire of approaches and understanding of connections. They try to convey the power and excitement of mathematics to their students - its unique mode of thought.

Teachers can improve students’ understanding by making them think mathematically.

Consider the trigonometric equation: cosxx = c. Its general solution is: x =qa + 360n, where n is an integer and a = cos-1c.

You could tell students just to learn this result, but understanding comes from sketching the cosine graph and thinking about its symmetry and periodicity (Figure 1).

Another aspect of mathematical thinking is the encouragement of good questions. I want mathematics students to show curiosity and to consider it natural to ask questions such as:

* Does cosine have an inverse? (leading to questions on domains).

* What is its derivative? (leading to a discussion of radians).

* Is it linear? (cos(a + b) - cosa + cosb, but this leads into trigonometric identities).

I also want students to distinguish between conjecture and proof - plotting an approximate gradient function for cosine suggests the derivative is sine, but is not a proof. Intellectual honesty is essential in mathematics and we do students no favours by ignoring rigour.

How do mathematicians show enthusiasm and involvement? Music teachers play an instrument, conduct and listen to music and direct productions. English teachers read widely and watch plays. Art teachers carry a portfolio of work to show us. What is the equivalent for a mathematics teacher? Several possibilities come to mind:

* Read popular mathematics books such as Life’s Other Secret (Ian Stewart) or The Man Who Loved Only Numbers (Paul Hoffman);

* Read the mathematics subject pages in The TES’s Friday magazine or journals such as The Mathematical Gazette and Mathematical Spectrum;

* Run a mathematics club and support the various Mathematics Challenge competitions run by the UKMathematics Trust;

* Join the Mathematical Association or the Association of Teachers of Mathematics, and attend conferences and local meetings;

* Follow up questions arising in class, discussing them with colleagues and students or writing them up in journals;

* Maintain an interest in an aspect of mathematics such as number theory, geometry or the history of the subject.

Unfortunately, evidence of any of these forms of involvement is rare.

Where will we find enough teachers with enough of the desirable qualities? We have to attract better students and train them better.

Recruitment problems are forcing schools to tolerate mathematics teachers with little knowledge or involvement in the subject. Finding an applicant with a good mathematics degree (or even a grade A at A-level) is unusual. Even then, the nature of many undergraduate courses does little to foster enthusiasm or involvement. University teaching also needs improvement.

Training courses are too short, especially for mature trainees who have not studied mathematics recently (although they may compensate with increased determination). Trainee teachers need more time on their subject - even those with recent degrees in mathematics may fail to appreciate its internal structure.

The mathematics national curriculum is no help - a trainee could get the impression that ratio is just another “number” topic to be covered at level 6, rather than a key idea underlying many aspects of school mathematics (including percentage, proportionality, enlargement, similarity, trigonometry, gradients and probability). The Initial Teacher Training national curriculum has helped focus attention on underlying structure. Training institutions have time to cover only a few key concepts of school mathematics. Hopefully, they make newly-qualified teachers aware of their obligation to keep exploring the network of ideas. But with no time to cover important A-level concepts in initial training, teachers must discover the underlying ideas for themselves. This is almost entirely done “on the job” as in-service training for prospective A-level teachers is rare.

How then can we raise standards in mathematics? Teachers without mathematics degrees should be enabled (even compelled) to learn more about the subject, in return for a reduced timetable. They need something like the Open University mathematics degree, modified to deepen understanding of elementary ideas important in teaching.

Similarly modified masters’ degrees should be available for graduate mathematicians. These courses should include reading and writing mathematical articles and giving mathematical presentations. Successful completion would attract pay increments.

Mathematics teachers would get involved in mathematics and improve their understanding. The nation would get better-informed and more enthusiastic teaching, with increased recruitment of able mathematicians. And we would get our A-level teacher.

Steve Abbott is deputy head of Claydon High School, Ipswich, and editor of ‘The Mathematical Gazette’