I’ve recently read a book of L.Kudryavtsev, a widely known mathematician. He was the head of maths department of Moscow Institute of Physics and Technologies, and an author of course books on mathematical analysis. The book title is roughly translated as “The Principles of Teaching Modern Mathematics”. This post is intended as a collection of advices and thoughts I’ve picked from there.
Note that I do not give my personal opinion on the usefulness of these: my intention is to provide the summary of the book itself; however, there is very little if at all that I could disagree with.
On teaching in general
 Encourage, not punish.
 Help if a student needs it.
 Prevent bad practices.
 Always recognize students’ accomplishments.
 Combine trust and regular control of students’ progress.
 Do not blame students for flaws in your teaching.
 Be your worst critic.
 A teacher needs time to teach well. Side activities can prevent you from teaching well.
 Almost anyone is able to learn your course.
 Students’ lack of motivation is most often a result of poor teaching in past.
 A student should be in awe of the beauty of mathematical concepts. Inspiring him can be hard: older scientists and teachers have a better understanding, but are not as excited; younger teachers can be more inspiring.
 The productivity is measured based on how many knowledge a student retains, not how much knowledge was he exposed to
On classes
 If your students take notes during the class, it might be of a profit for you to review these notes from time to time. Check, which parts of the lectures were written down. If several records did not hold whatever you wanted your students to memorize, you might want to rethink the structure of your classes. It is also a great opportunity to help those of your students who do not know how to write down lecture notes in a most efficient way.
As a side note, in Russia there is a bad practice of punishing people for not writing notes during classes. It is quite useless and just makes students waste time copying someone else’s notes.

The best professionals are not always good teachers. They can contribute by mentoring students, teaching specialized courses (not broad topics such as Analysis or Linear Algebra), doing lectures on concise topics.

It can be harmful to expose students to different points of view on the similar subject at a time. Only when they got familiar with one point of view should they be exposed to others – and in this case it is a good thing.
On seminars
 Seminars are very important. It is a collective form of work (no more than 25 students). It is a time to build a trustful, relaxed relationship with your students, as you will get to know each other more, than during lectures. An atmosphere of mutual trust and respect is a factor I can not stress enough.
As a side note, I had a little experience of teaching programming to future engineers in France, and the typical relationship between me and a student surely looked somewhat more distant. I can not attribute that to language barrier nor am I able to extrapolate it to the whole world outside Russia. But a reader should be aware that the book author comes from Russia where traditionally the relationships between future mathematicians and their teachers is often very personal.
 Methods based on knowledge of human perception, psychology, are very promising to further increase the productivity of students without exhausting them. However, they are still underdeveloped and not systematized. Talented teachers manage to pull off some tricks, but that’s it.
I am not aware of what is the situation today (2017), the book was published in 1980
 During seminars, you can inspire students to be better selves, to improve the good traits of their personalities etc.
 Talk with students one on one. Use individual assignments as a reason to start discussions, after all, this is the only sure way to know that the student has understood what he has done.
On exams
 Everyone should know, which knowledge will be evaluated and what books and other resources are available. Do not waste their time on sjjjearching appropriate books.
 Giving an adequate grade is not enough. The student should fully understand why did he get one. It should be also obvious for him that working on lectures, solving problems, understanding the concepts makes passing the exam successfully much easier.
 Be aware of how the information was presented during the classes.
 The way you handle exams influence the students view on their further studies.
 The atypical problems are not an adequate tool of evaluation unless they are very easy. Not everyone is capable of doing great under pressure.
 You should check if the knowledge is superficial by asking such questions that demand connecting the facts.
 Avoid vague and badly formulated questions.
 Writing down additional questions and discussing them with your colleagues might be a good idea.
 An exam is a way to evaluate knowledge, not quick thinking or ingenuity. These should be developed and challenged during the seminars.
 Do not punish students if they present you a correct picture not based on your class.
 Do not make it last hours.
 Looking at the past grades is often useless and harmful. In most contexts you should grade the students independently on its previous classes.
 When done correctly, an exam boosts student’s motivation and confidence.
 Try to find out what does the student know, not what does he not know.
 A good idea is to combine a written part (which is less subjective) with a discussion (which allows you to quickly understand how competent is the student, given you have some experience).
On mathematical education
How do we see the students after they graduate? Not speaking about pure mathematicians here, we want them to be able to:
 Construct mathematical models
 Pose mathematical problems
 Chose the appropriate mathematical method and algorithm to find the solution
 Use computational tools at their disposal.
 Use quality mathematical methods of research.
 Go back from mathematical models to practice.
 Semesters 15 should be dedicated to fundamental courses and programming.
 Semesters 68 should be dedicated to applied mathematics and specialized courses on concise topics.

The final part of the education consists of more specialized courses which should be applied in thesis.
 Discrete mathematics can not replace continuous mathematics “because the world is discrete”: both of them are but useful approximations.

Mathematics cultivates the thinking, makes it clearer and more concise. It enhances the ability of logical analysis; it teaches to select precisely the things necessary for a research and throw out the unnecessary ones.
 The essence of mathematical concepts is universal and does not depend on the applications. Explaining their very abstract meaning is very important. Teaching engineers adapted courses where the meaning is explained only through practical application to one specific domain is crippled.
 Mathematics for engineers should be though as a narrower selection with less details, compared to pure mathematicians. But the spirit should stay the same.