Yale, Stanford and Toronto
As an undergraduate at the University
of Toronto I tutored first year maths (Calculus and
Algebra) in both my third and fourth years. As a PhD
student at Yale University I taught several classes
of Calculus. I taught for two years at Stanford University
(1984-1986) and for three years at the University of
Toronto (1986-1989), where besides calculus and linear
algebra, I taught group theory, differential equations,
vector calculus and algebra. In 1990 I arrived at UNSW
in Sydney Australia, where I am currently Associate
I have been teaching at UNSW since 1990, generally
have excellent teaching evaluations, and have played
an important role in structuring courses in Pure mathematics
To lower years I have taught Calculus, Linear Algebra
(Higher and Ordinary), Discrete Mathematics, Vector
Calculus, Higher Geometry, Differential Geometry, Logic
and Computability, Higher Real Analysis, Higher Complex
Analysis and probably some others I can't remember.
Fourth year (honours) courses I have designed and taught
here over the years include Algebraic Number Theory,
Advanced Combinatorics, Information theory and Codes,
Lie groups, Algebraic Topology, Representation theory,
and Themes of Classical Mathematics and Geometry.
In 2007 I taught (and was in charge of) the engineering
course MATH2069 Maths 2A which includes Complex Analysis
and Vector Calculus. I taught the Complex Analysis strand,
using Saff and Snider as a textbook.
I also taught MATH5785 Geometry, which was a special
treat for me, as I used my recent book `Divine Proportions:
Rational Trigonometry to Universal Geometry' as a text.
We did Euclidean geometry (the correct way, using quadrance
and spread so that all theorems hold over a general
field), projective geometry, spherical/elliptic geometry
and hyperbolic geometry. Even these latter subjects
can be described purely algebraically, and doing so
clarifies them considerably. This is a very important
point that anyone interested in geometry should attempt
Next session I will be teaching MATH1241 Higher Linear
In my opinion here are the keys to successful mathematics
teaching at the university level, in order of importance.
These might be useful to young lecturers who are starting
out on their teaching careers.
The fundamental requirement for succesful teaching
(actually at any level) is to have
something to teach. In mathematics,
this means content that is accessible,
useful and interesting.
Material should be aimed at the appropriate level
for students, it should be seen to be useful by them,
and it should stimulate them at the same time. All
three aspects are necessary, and getting the balance
right is not easy. Well chosen examples are particularly
important for mathematics teaching.
The second most important factor for succesful mathematics
teaching is careful preparation.
You must know the material, have your examples worked
out beforehand, and have thought about how best to
structure the content, both at the global and local
The third most important factor is effective
delivery. Without distorting your natural
self, you should strive to be enthusiastic, friendly,
and to talk and write clearly. There are many specific
skills to be learnt in this direction.
Mathematics is an active subject. Students learn not
only by reading and listening, but also by writing
and more crucially by doing. It is
thus valuable to structure mathematics learning so
as to provide opportunities for students to take lecture
notes, work actively in solving problems, and to write
up and discuss solutions.
The right tools make any job easier. A well-designed
book is an important aid in succesful long
term learning, particularly if written by an expert
who has thought deeply about how to best structure
the subject. Be skeptical about the merit of quickly
put together notes, especially common on the web.