Teaching
@ Caltech
A (very
biased) sample of student comments [png]
 Applied Linear Algebra
Spring 2016, Fall 2016
[ACM 104]
 Introduction to Probability
Models
Fall 2016
[ACM/EE 116]
 Introduction to Statistical
Inference
Winter 2016
[ACM/CS 157]
 Introduction to Matlab
and Mathematica
Spring 2016
[ACM 11]
Past
Teaching
University
of Liverpool
 Uncertainty, Reliability
and Risk
Winter 2014.
University
of Southern California
A (very biased) sample of student comments
[png]
 Mathematical Statistics
Spring 2012, Spring 2013.
[Math 408]
 Mathematics of Physics
and Engineering
Spring 2012.
[Math 245]
 Fundamental Principles
of Calculus
Fall 2011, Fall 2012.
[Math 118x]
Hong Kong University
of Science & Technology
 Mechanics of Materials
Spring 2007, Spring 2008.
 Statics and Dynamics
Fall 2006, Fall 2007.
Moscow State University
 Classical Differential
Geometry
Spring 2005.
Notes in Russian [pdf]
Summer Schools
Teacher
of mathematics [jpg]
Summer School
Moscow Center
for Continuous Mathematical Education
Summer 2000, Summer 2002. 
Teaching Philosophy
Teacher:
Children, write down the proposition:
“The fish was sitting in a tree.”
Pupil: But is it true that fish sit in trees?
Teacher: Well . . . it was a crazy fish.
 Arkady
and Boris Strugatsky, "Monday Begins on Saturday"

 Emir Kusturica and Goran Bregovic,
"Arizona Dream" 
What are my
general goals as a Teacher?
I believe that the most effective way of education is selfstudy.
This is especially true in mathematics: the only way to learn mathematics
is to do mathematics. It takes time and effort to deeply understand
the main oncepts of a mathematical theory by constructing examples
and ounterexamples, solving exercises, and finding analogies with
other already learned subjects.
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However, the importance of the Teacher's
role in the process of Student's selfeducation truly cannot be overstated.
Obviously, selfstudy is extremely effortconsuming and, therefore,
requires strong motivation. So, my first objective is to inspire my
students, to show them why the subject I teach is interesting and
useful, and so motivate them.
Imagine a classroom of brighteyed
Freshmen. Having just entered university, they are highlymotivated
and eager to learn. How many of them will still be motivated and inspired
after the end of the course? Frankly, not all of them. There are several
reasons for that, and clearly the quality and style of lectures plays
a crucial role. So, my next very important aspiration is to design
my lectures in such a way that it allows me to keep the interest of
Students and the integrity of the knowledge simultaneously.
Finally, whenever someone learns a
new subject plenty of questions appear. Undoubtedly, I consider it
my responsibility to help Students to tackle their difficulties arising
during the course. To put it in a nutshell, my main goals as a Teacher
are to motivate and inspire my Students, to provide them with welldesigned
and highquality lectures and to be open for questions, discussions,
and collaboration.
How do I motivate and inspire
my Students?
Study without desire just ruins the memory. So, I always try
to explain to my Students why I teach what I do, why the topic is important
and interesting, and what kind of problems we are able to solve using
the mathematical theory under study. I hope that eventually my inspiration
will be transmitted to the majority of my Students and that they will
become intrinsically motivated.
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However, I fully realize that there
is no way to motivate all Students in the classroom. There will always
be students without intrinsic motivation. I will try to help them
by extrinsic motivators in the form of rewards. People like to repeat
behavior that is rewarded and Students are no exception. A joint coffee
break or simple verbal encouragement can be a kind reward for good
performance. And the most important point is that extrinsic motivation
can, after a short period of time, produce intrinsic motivation.
What do I mean by good lectures?
First of all, a good course of lectures must reflect the current
state of the subject. I form a timelimited course in such a way that
it contains all the main concepts, important theorems, and illustrative
examples. Although my ambition is to cover as much important content
as possible, the quality of knowledge is definitely more important.
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Next, the content is not what a course
is all about. I believe that the quality of knowledge highly depends
on the manner and style of lecturing. In my teaching, I combine two
popular lecture modes: using slides and using a piece of chalk. I
use slides to convey the most important points of the lecture and
for demonstration of visual information (e.g., graphs, figures, and
tables). I use blackboard for all computations and explanation technical
details. This allows students to follow the lecture in realtime and
gives them more time to digest new information.
The lecture style is also important.
When teaching a course for undergraduates I take into account that
their abstract mathematical thinking is usually not welldeveloped.
So, I prefer to use more examples and pictures in order to describe
the ideas behind the theorems and proofs. Unlike undergraduates, graduate
students have more background. In this case, I try to focus more on
theorem applications, on different techniques hat can be used for
the proofs, and on open research problems. Also I always remember
that it is difficult to pay attention for more than 20 minutes during
a lecture. So, timetotime I slow down and try to create space for
a new breakthrough. From my point of view, for this purpose there
is no better means than humor. Especially since there are plenty of
mathematical jokes.
Finally, a wellmade examination culminates
the course. I prefer openbook exams rather than closedbook, since
I like to test students' ability to solve problems using new knowledge
from my course rather than test their memory.
How do I know that my Students
are learning?
I like questions: I like
to ask, and I like to be asked. So, I always let my Students ask questions
during the lecture. It gives me immediate feedback on how students are
thinking, what they understand, and what is difficult for them. In addition,
asking questions helps students feel more omfortable during the lecture.
I consider the absence of questions as a first sign of a lack of understanding.
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Sometimes I ask students questions
during my lecture. For example, instead of a direct statement of a
theorem I can consider an example that catches the main features of
a theorem, and then ask Students to guess the right formulation. This
stimulates active thinking and develops mathematical intuition. Moreover,
it is much easier to remember an answer you have to think about rather
than rote knowledge. Also I like to spend time with students in an
informal atmosphere, whether it is a coffeehouse or a hiking trip.
Occasionally, in such unofficial activities some students may discuss
their concerns more freely.
How can I improve my teaching?
Feedback from both students
and colleagues plays an important role in teaching improvement. In the
middle of the term I ask Students to complete a short questionnaire
on how to make my lectures and assignments better, which aspects of
my teaching are strong and which need improvement. From timetotime
I invite my mentors to my lectures and ask them for a feedback on how
to improve my teaching.
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Tests, quizzes, and exams are equally
important for both learning and feedback. The results of examinations
help me to understand which parts of the course are too difficult,
and which concepts are still too fuzzy. This understanding allows
me to improve my teaching for next term.
When we teach mathematics we
teach just mathematics, don't we?
I always keep in mind
that by no means will all of my Students be professional mathematicians.
This leads to a natural question: what benefits from my course can
my Students get for their future life?
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Mathematics adjusts our minds. Learning
mathematics develops logical, creative, and abstract thinking. There
is no lie in mathematics and this teaches us to be honest. Mathematical
complexity and beauty nurses our spirit. I believe that even if many
formulas, theorems, and proofs are forgotten with time, my Students
will still have very useful skills for the rest of their lives.
