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					                        TEACHING STATEMENT – NORMAN DO                                            May 2009

1    Teaching experience

Over the past ten years, I have had the opportunity to teach mathematics to students at a variety of
levels. My teaching experience can be very briefly summarized as follows.

    ◦ University: I have tutored several undergraduate courses in mathematics. Apart from delivering
      tutorials, this has also involved marking assignments and taking consultation sessions.
    ◦ Olympiad: Since 1998, I have been heavily involved in the activities of the Australian Mathemati-
      cal Olympiad Committee. This has included lecturing at residential training schools and mentor-
      ing several of Australia’s International Mathematical Olympiad team members. I have also been
      involved in marking, creating problems and setting papers for mathematics competitions.
    ◦ School: For several years, I have taught mathematics enrichment classes at a number of schools. I
      usually meet on a weekly basis with small groups of talented and motivated students.
    ◦ Other: I have tutored at several National Mathematics Summer Schools at the Australian National
      University. In 2008, I was involved as a lecturer, which required me to design and deliver a one
      week course on algebraic geometry for talented students about to enter university.
      I have also participated in a program where I was asked to design a mathematical research project
      suitable for highly able school students and mentor them through the project over six months.

In the future, I would be very keen to expand my teaching experience to include higher level pure
mathematics courses, such as abstract algebra, complex analysis and geometric topology. For further
details of my teaching experience, please refer to my curriculum vitae.

2    Teaching philosophy

As a student and practitioner of mathematics, I have attended countless tutorials, lectures and seminars
over the years. As an educator of mathematics, I have also had the opportunity to deliver a number of
these myself. With this in mind, I believe that one of the most important teaching skills is the ability
to recognize mistakes and learn from them — not only those made by others but also those made by
One common error committed by mathematics instructors is to concentrate solely on communicating
to students what they should think about at the expense of explaining how they should think about
it. Even those that do contemplate both of these issues will often neglect to impart to students why
they should be thinking about it. So I believe it is of the utmost importance to provide motivation
when teaching mathematics, whether it be for the definition of an object, the idea for a proof or the
introduction of a new concept. For example, it is logically rigorous to begin a group theory course
with the definition of a group as a set with an associative binary operation, an identity and inverses.
However, a far better approach would be to precede the definition with its motivation. Of course, this
would entail an exposition of the importance and ubiquity of symmetry in mathematics, brimming with
examples already well-known to the students. It is then a simple matter to motivate the definition of a
group and the fact that it captures, in some sense, the abstract idea of symmetry. It is not only easier to
remember, but also easier to understand concepts when they are contextualized in this way.
The issue is even more aptly demonstrated by a particular case involving one of my former students.
Her linear algebra lecturer had taught her to calculate the determinant of a matrix by row reduction.
After taking some pains to memorize the algorithm, the student had no trouble performing it on a given
matrix. Fortunately, however, she had both the curiosity and intelligence to come to me and ask why
on earth anyone would want to calculate this magic number known as the determinant. As her tutor, I
found it truly amazing that she would be taught such a mindless algorithm without some prior motiva-
tion for its existence. We rectified the situation by discussing the, unbeknownst to her, related problem
of determining the volume of a parallelepiped formed by three vectors. Rather simple deductions led to
equally simple rules that such a volume function must obey. The student was then able to use these rules
to calculate several examples which demonstrated that there was an algorithm to determine the volume.
Of course, the student quickly pounced on the fact that they were simply calculating the determinant
of a square matrix by row reduction. This geometric viewpoint very naturally yields the theorem that a

                         TEACHING STATEMENT – NORMAN DO                                             May 2009

matrix of column vectors has zero determinant if and only if the vectors are linearly dependent. Also,
at no extra cost, this intrinsic definition guarantees that the determinant is always the same, no matter
how the row reduction is performed. I think it is safe to say that the student will always remember what
a determinant is, but only after being taught how to think about it and why anyone would bother to.
It is becoming increasingly common for mathematics lectures to be complemented by written material.
Some lecturers will go so far as to give a detailed account of every aspect of the course, which may
not be a bad idea. However, one begins to wonder what exactly the role of a lecture is. Could we not
simply hand out a textbook to the students on day 1 and ask them to return to sit the exam on day
N ? I firmly believe that this is not the case. Therefore, a lecture must consist of more than recitation
performed by the lecturer and dictation by the students. Indeed, the crucial advantage of a lecture is
that it is not constrained to be linear or set in concrete, as a book is. There is the possibility to interact
with the students, make digressions, revisit previous material, clarify ideas further, gauge the level of
understanding of the audience and make amendments to the organization and structure of the course.
For this reason, it is important for the classroom to be an interactive arena, with information passing in
both directions.
An important aspect of learning mathematics is the dichotomy between understanding concepts and
solving problems. A common complaint of students is that despite understanding the material, they
have difficulty with assignments and tests. Conversely, a student may be able to solve problems —
often by pattern recognition — but without a great deal of understanding of the relevant concepts. A
true grasp of mathematics requires both, and this should be reflected in the way mathematics is taught.
Therefore, it is necessary to intersperse any discourse on theoretical ideas with concrete examples and
My views on teaching are intricately related to my belief that mathematics is more than pushing around
symbols according to some predefined set of rules. It must stem from intuition and have connections
and applications to other parts of mathematics or the world around us. I believe that it is extremely
important to convey this to students of all abilities at all levels. Of course, it is easy to wax lyrical on
educational theory as many before me have done. Here, I have touched on just a small number of the
pedagogical issues which I personally feel strongly about and which distinguish me from many other

3    Teaching practice

It is quite incredible that so many people continue to make fundamental mistakes in the mechanics of
teaching. Such simple considerations as being punctual, speaking clearly, writing legibly, making eye
contact and being organized remain stumbling blocks for many teachers. I would certainly like to think
that my teaching practice is completely free from such simple errors. In fact, I believe that my wealth of
teaching experience has made me very comfortable in front of a class.
I have taught in a variety of situations before, including one on one consultation sessions, small classes,
tutorials organized around group work and large lectures. I have also used technology in the classroom,
mainly in the form of graphical calculators and mathematical software. With respect to such different
methods and approaches to teaching mathematics, I have no strong opinions barring the fact they should
always be used appropriately. For example, the use of mathematical software may demonstrate an
illuminating example which would otherwise be difficult to imagine. However, its misuse may draw
students away from the actual content involved and appear as little more than a gimmick.
In terms of my interaction with students, I always strive to be entertaining as well as approachable. To
achieve this, it is important to understand their backgrounds and reasons for studying mathematics.
However, regardless of who they are, my goal is always to infect them with my enthusiasm. In the past,
I have found that I learn a great deal from my students and from the need to explain new concepts to
them in the best possible way. In fact, I have developed the mindset that, after learning a new piece of
mathematics, it is beneficial to think about how it may best be passed on to others. Not only has this
aided my teaching but also my own understanding of mathematics.