Mathematics in teacher education -
what do students need for science studies?

It is said that a serious problem for mankind is the inability to understand the exponential function. We remember the story of seeds on the chess board - but still, we have difficulties understanding the consequences of the fact that an annual growth of only 2% will lead to a doubling in 35 years. A mathematics teacher thus has an important role to give the student and understanding and familiarity with the exponential function, as well as other components in a mathematical understanding of the world. In teacher education, mathematics is also important as a beautiful tool of science.

My list, below of what I would like students to learn in math is based primarily on contacts with students in teacher education and with first-year undergraduates in the introductory physics course in the programme "Problem Solving in Natural Sciences". Many of the skills could have been learned already in high school, but my experience tells me that, in general, they are not.

A Mathematical Way of Thinking

Even more important than the details given in the next section is probably that the students develop a mathematical and scientific way of viewing the world. They should be able to recognize significant ideas, as well as the power of generalisations and of mathematical proofs. I would like students to develop a sense of wonder and delight, faced with the power of simple equations to describe many different and complex phenomena. I would like them to appreciate the creative aspect of mathematics, science and research. They should also learn to appreciate "beauty" in a mathematical proof or theorem or in a phenomenon - even if this aspect is not so easily examined!

A physicists shopping list from the mathematics store

In addition to the general considerations above, likely to be desirable also for a mathematician, there are specific techniques, ideas and concepts that are essential for a proper understanding of science, but some of them also for a more general societal aspects. (I often forget things on my shopping lists and do not claim this one to be complete.) Even if science is not taught to pupils in schools in this mathematical way, I think a teacher needs to be able to see the mathematical way as a complement to more "hand-waving" approaches.

Derivatives
to describe e.g. position, velocity, acceleration
The definition of derivatives should be understood and students should be able to make approximate evaluations from a graph or a tabulation
Integrals
as analytical expressions and objects in tabulations, as well as with the relation to derivatives. It is also important that students can see an integral as a summation of small partial contributions - as a preparation for numerical evaluation and as an essential ingredient in writing down an integral, when needed.
Differential equations
to model equations of motion, and to understand oscillations and exponential growth, e.g.
Series expansions, (MacLaurin, Taylor)
for error estimates and error propagation,
sometimes for numerical evaluation (involving e.g. sin or cos for small angles etc)
and to recognise limiting cases
The exponential function
for population growth and radioactive decay, electronics and the cooling of a cup of coffee.
Understanding geometry, including:
Euclidean geometry, for proofs and mathematical thinking
Spherical coordinates, helpful, e.g., in studies of planetary motion and atoms
Trigonometric functions: to understand e.g. circular motion, optics, forces on an inclined plane and other vectors, radiation balance
Analysis with many variables
to be able to deal with complex physical situations - but also socio-scientific issues, including environmental problems
Complex numbers
for wave motions and electromagnetism
grad, div, curl
useful in mechanics, essential to understand electromagnetism, wave equations and quantum phenomena.
For the higher physics courses, many additional mathematical techniques are, of course, very useful. Complex analysis (including residue calculus), Fourier expansion and other expansions, FFT, matrices and eigenvalue problems, partial differential equations and techniques for solving them, are a few of these tools.

Mathematical work-out for better physics

Often the students show clear signs of lack of practice - even after introductory mathematics courses. I think they need the stamina to carry out long calculations, and sufficient practice to be able to get a correct result in a calculation involving many steps without mistakes. They need to trust themselves to be able to carry out a derivation of a formula they need. Exercises with various functions including graphical representations and their expansions for small values will help them recognise relations and special cases (e.g. that Einstein's equations reduce to Newton's for low velocities). Practicing order-of-magnitude estimates will sharpen critical skills in societal debates, in addition to being useful in planning experiments etc. (Physicists often refer to this type of problems as "Fermi questions" - after Enrico Fermi, who asked a physics PhD candidate about the number of piano tuners in Chicago.) Many amusing examples can be found in "A mathematician reads the newspaper" by John Allen Paulos

On my wishing list, I would finally like to add modelling skills, using analytical as well as numerical methods. (Examples can be found in the "High-School Computational Science" project). A teacher should be able to write a small computer program to model simple systems, and not have to rely solely on ready-made black-boxes!

The image of mathematics and science

If the students could combine all these skills with a familiarity and proficiency in writing and talking about mathematics, where they find a balance between words and equations, maybe we could even, albeit slowly, change society's view of mathematics and science. Mathematics can possibly claim innocence - after 1945, physics cannot. We share a responsibility to ensure that science is viewed as an indispensable part of our lives and not as the root of all evil. Even if science alone cannot solve the problems of the world, the analytical skills trained in mathematics and science are certainly needed in any solution. "Knowledge without compassion is dangerous - compassion without knowledge is meaningless" (Victor Weisskopf). Let us include both aspects also in the teacher education in our subjects!
http://fy.chalmers.se/~f3aamp/lektor/matkrav.html
Ann-Marie Pendrill, 1998-09-08
AMP är biträdande professor i Atomfysik och "expert" för mat-nat-fak för Lärarutbidningsnämnden, LUN
This collection was prepared as a contribution to a LUMA (Lärarutbildning i matematik) conference 23-25 september 1998