Mingfeng Zhao (mingfeng@math.ubc.ca)
Teaching Statement
We are educators, we are born to make a difference.
—Rita Pierson
My teaching goal is to help students master the fundamentals of both lower level and advanced
mathematics courses. The training of thinking processes needed in solving problems, no matter what
fields students would go into, is of great importance.
I would like to lecture in a way that allows all my students to interact. I believe that students can learn
better when they are involved to solve a problem from intuitive thinking to the general and rigorous
proof .Asking questions and seeking answers from students is a good way to encourage thinking and
is also a good assessment of interaction in class. For example, when explaining finding the volume
of a solid in a rectangular region using double integral formula, I ask my students how to compute
the volume of a rectangular parallelepiped. I graph a rectangular parallelepiped with a rectangular
RR
base R and height h together with the notation R hdxdy. I then graph another surface f (x, y) in
the same region. By asking questions for each step I perform, students soon realize that they need to
integrate f (x, y) over R. To further enhance the thinking process, I draw the third graph with the
same surface but on a different region. I walk around the classroom and ask the students to think
of the question independently. I then ask them to discuss in small groups and share their ideas with
their fellow students. The lecture moves on with questions and answers. At the end of the lecture, I
summarize the material and take questions from my students.
Thinking and learning should not be constrained in class, but should also be emphasized after class.
The interaction with my students after class is valuable in many aspects. It gives students an opportunity to get to know me as a person, not as a teaching robot in front of the blackboard. It also
allows me to assess their understanding of the class material. If a lot of students come with the same
question, I would know I was not doing a good job explaining the concept in class.
While teaching the ordinary differential equations, I found that many students liked to use formulas
they remembered to solve problems and ignore the concepts in the questions. But we know that
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Mingfeng Zhao (mingfeng@math.ubc.ca)
learning mathematics is to learn the thinking process not just memorize formulas. One student came
to my office hours to ask questions about the integrating factor method for solving the general first
order differential equation. He said that he could solve the exact equations, but it’s very difficulty
to memorize the formula of the integrating factors for the equation. So I told him that he did not
need to remember formulas, he could do it based on the definition of the integrating factors. For
example, let’s consider M (x, y) + N (x, y)y 0 = 0. First let’s multiply any nonzero function r(x) (in
this course, we only focused on the integrating factors which are functions of only x), we got an
equivalent equation: r(x)M (x, y) + r(x)N (x, y)y 0 = 0. Now we are looking for a special function
r(x) which is an integrating factor. Just by the definition of the integrating factor, we know that
r(x)M (x, y)+r(x)N (x, y)y 0 = 0 is an exact equation, so we can get
∂
[r(x)M (x, y)]
∂y
=
∂
[r(x)N (x, y)],
∂x
that is, r(x)My (x, y) = r0 (x)N (x, y) + r(x)Nx (x, y), which implies that r(x) satisfies an separable
differential equation r0 (x) =
My (x,y)−Nx (x,y)
N (x,y)
· r(x). Hence we can solve r(x) easily. After getting the
integrator r(x), we can find the solutions of the exact equation r(x)M (x, y) + r(x, y)N (x, y)y 0 = 0.
When the student saw what I did, he said that it’s much easier when we did every simple step, and
we did not need to remember anything, just need to understand basic concepts, even though it took
a longer time to solve problems. Then he tried an exercise independently, he could solve the problem
easily based on what I told him before.
Teaching mathematics and doing mathematics research bring me great pleasure. I believe that solving
mathematics problems can also bring great satisfaction to my students once they can truly understand
the process of getting results. To bring such accomplishment to my students, I always show my passion
for mathematics and my willingness to help my students find the same pleasure.
In the future, I would love to teach all levels of undergraduate and graduate mathematics courses. I
am excited to lead students research in all fields in mathematics. I am happy to get students involved
in my research area-partial differential equations and I can also work with them in other research areas
in mathematics.
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