Question 1: Evidently a high school graduate should be able to do basic arithmetic so as to be able to fully function, every day, in society. It is desirable for high school graduates to have a basic notion of algebra, especially the use of variables, so that they can read articles in the newspapers and magazines about "breakthroughs" in mathematics.

Most important is that students are trained to be accurate, and to recognize when they are correct (as opposed to hoping!). It is important that students display clarity, and are able to succintly explain what they have done in a calculation and why. Our feeling is that not much knowledge should be required, but a graduate should know what they are talking about when it comes to simple applications of mathematics.

In three words, high school mathematics should strive for clarity, correctness and accuracy.

Question 2: It is important that students are trained to express themselves clearly. We find that the biggest obstacle to most students doing well in calculus is their inability to express themselves. If students can't even explain an idea to themselves, what chance to do they have of explaining the idea to an examiner, or using it in practice?

Students need to work on thinking through problems. Working on problems that are not given in standard form, perhaps even with incomplete data, is more like what people will need in life. If we can train people to competently deal with such questions we will have given them a good lesson in how to organize their thoughts. Old questions like "what is the most area you can enclose with a given perimeter?" are still effective, because they get students to think, and the feedback from that is usually encouraging to teachers.

Students learn by doing. The key to good teaching is to persuade the student that they are capable of figuring out how to do the problem and then getting them to do it. Students must study for themselves. Thus the precise content of material is much less important than the teacher's ability to inspire the student to think about it, and to recognize that this is worth doing.

We'd like to see more emphasis on "counting arguments" in high school curricula, such as come from combinatorics, and on simple logical thinking, such as using induction arguments.

Some years ago students were "inspired" by the threat of bad grades. This is no longer the case, so we need to make the material seem worthwhile. Too many "reformers" believe that this is done by watering down content. We believe this is best done by having teachers that are better mathematicians, and better trained to be teachers.

Question 3: When very few students coming into universities, like ours, need to go through several remedial mathematics courses of high school algebra. When businesses stop complaining that they do not have enough qualified applicants (how often does one see press reports of business people complaining that applicants cannot write well and are not basically numerate?). When newspapers and magazines begin writing regular articles about breakthroughs in theoretical math and physics, without fear that everything will be over their readers' heads. When it is no longer socially acceptable for educated people to say "I never could get the hang of doing math"; instead educated people would be as embarrassed to say this as to say "I never could get the hang of reading".

Question 4: Mathematics teachers should have a solid background in basic mathematics. They should be confident of their skills in arithmetic and algebra. They should be at least competent at calculus and linear algebra at the university level.

The key thing is that teachers should be enthusiastic. They should impart to their students that they have fun doing mathematics, and that it is nothing to be feared--that once the students get the hang of it they will enjoy it too. Too often teachers act as if the material is very tough and that just getting a passing grade is a big achievement; clearly no student can come out of such a class believing that they can truly master mathematics. Perhaps the worst educational reform of the last few decades has been to give qualifications to far too many people who are uncomfortable with the basics of mathematics. We need teachers that will help students overcome their fears and learn to enjoy math; not the vast horde of teachers now being produced that seem unenthusiastic, and have such difficulty with courses like calculus.

We need teachers who allow students to learn by doing mathematics, and succeeding. Teachers should be sufficiently self-confident to not rely too heavily on their textbooks, but to work with the students that they have as individuals. It is the bane of teaching that pedagogical theories come and go, inspired by bureaucrats who naively believe you can only progress through change, whereas there is scant evidence that this improves matters in the schools. On the contrary, many teachers are unable to fully appreciate what the new pedagogy requires of them, and so do it badly. Let's give teachers more room to find their own way of teaching best, perhaps working towards a common examination.

Our pedagogy is simple: Students learn mathematics by doing mathematics. Teachers best teach by directing their energies to content and the students, not to clever new pedagogies.

Question 5: (Granville): I was good at it and was able to do many simple calculations at an early age. At high school I was as interested in history as mathematics, and later worked a bit in script-writing, TV production and desiging computers; however experience showed that mathematics is a much easier career option, where ability is more likely to be well rewarded.

(Hollingsworth): At university I began by studying electrical engineering. A crisis came when I was in a course that I was totally uninterested in, and I started to wonder why I hadn't enrolled in mechanical engineering instead. After some thought, I realized it was because I preferred the preponderance of mathematics in electrical engineering; that being the case I then switched to mathematics and never looked back.