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