Of what use is mathematics? Hasn't everything in mathematics already been discovered? Secondary school teachers are often faced with students raising these questions. They deserve to be answered and future scientists in secondary schools should be encouraged to raise even more questions. In order to help teachers to stimulate curiosity, we have created at the Université de Montréal a new course, "Mathematics and Technology", which is mandatory for preservice secondary school teachers. It presents modern examples of applications of mathematics. The examples are chosen so that the students can understand both the modelling process and the mathematics. They can appreciate how the mixture of ideas and imagination with the use of mathematical tools, can be the source of technological breakthroughs. They experience mathematics research in action and that mathematics is a living discipline within science and technology. The point is not simply to provide them with examples and applications that they can repeat to their future students, but rather to give them the tools to formulate and develop real-world examples appropriate to their students. One message of the course is that basic high school and undergraduate mathematics form a remarkable toolkit, provided they are well understood and mastered, allowing students to readily explore their wide applications and, often for the first time, to discover the additional power obtained when integrating different tools. Ideas are a scientist's most precious commodity and behind most technological successes there lies brilliant yet sometimes elementary observations.
The subjects treated in the course include: cryptography, error correcting codes, GPS, robots, compression of images by jpeg format or through the use of fractals, Google’s algorithm PageRank, random number generators, DNA computers, maths and music. As part of the evaluation, the students do a project in which they explore by themselves a technological application of mathematics and learn to present both the problematic and the related mathematics in an accessible form.
Teaching this course has forced us to revise our usual pedagogical methods: here no subject is a prerequisite for further courses, the definitions and theorems are not the ultimate goals of the course and the problems are not drills. These factors can cause some anxiety on the part of the students. On our side, we are not specialists in any of the technologies we discuss here, a situation similar to the one these future teachers will experience in their classroom if they decide to present modern applications of mathematics. So we had to revise our teaching. We tried to create as many links as possible to technology. We encourage students to participate in the course. This allows us to evaluate their background relative to the mathematical tools being used. As for exams, we opt to reassure them from the beginning by stating that the exams are open book, non-cumulative and limited to the basic material. Emphasis is put on simple mathematical modelling and problem solving. Our sets of exercises focus on these skills.
In this lecture I will discuss the general objectives of the course and its format and I will present some examples. I will also discuss how we adapted our teaching for this particular course.
