A usual/cliche kind of teaching that is followed all over India is the following.
The contents will be from a textbook.
The teacher will come prepared for every class, and
Does a presentation of the topics.
The students listen, try to understand what the teacher explains,
Some students may ask questions/doubts, and
Everyone takes notes.
The teacher gives some homework (usually exercises mentioned in the textbook).
When the class is over, the students just stop thinking about that subject, and the next class starts, the same things as mentioned above repeat.
They go home and read/learn the notes (nothing else, not even the textbooks), and do the homework.
The next day, the teacher asks if there is any doubt about what is learned in the previous class.
And the story repeats.
In contrast to the above-mentioned style of teaching, teachers can try the following practices. One primary challenge in doing so could be the tight syllabus; teachers are supposed to complete a huge syllabus within a limited time. However, some of the following must be easily handled. The first few points are helpful in keeping the class active.
Before going to the points, I should make this one very important remark. The main aim of an undergrad course is not to teach so much of math, but to get them ready to do it in a postgraduate course. It doesn't matter how many courses a student does before getting graduated; usually, a student doing a regular B.Sc. degree may not even learn half of the subject that a student of B.Sc. Hons. does. Still, both can compete evenly. When one learns a subject, it is not the definitions or results that he/she learns, but the techniques used in the course, and that is the reason for having 4-5 months long semesters. One can easily learn all the results and their proofs given in a normal course within a period of 1 or 2 weeks, but it takes pretty long to get familiar with the techniques used in the course. Teachers often forget this while running for completing the syllabus!
Encourage students to ask questions. Usually, students are either shy or afraid to raise questions. The main reason for this tendency is that if the question turns out to be too silly or stupid, the teacher and hence the other students may laugh at it, or if the question is too difficult that the teacher was unable to give a solution, the teacher may get angry. (A teacher should never make such a scenario in the class; still, people usually do that). One way to encourage them to ask questions is by asking them questions. You may not ask a particular student, but just raise the questions so that they will think, and learn what kind of questions are to be asked. Learning to ask questions is very important for a student. Usually, an undergrad course in India is of 3 years duration. During the course, let them ask whatever stupid/silly questions, never discourage them. By the time they start post-graduation, they will be ready to pose very relevant and remarkable questions in the class.
The teacher may request one or two students to present on the blackboard, the exercise that was given as homework. This can be done on a daily basis. Instead of choosing students randomly, the teacher may assign them prior to the class. Again, ask the audience to ask questions after the presentation. Usually, students wouldn't do that, because they never want their colleague in trouble (Because their turn will also come!).
Promote group discussions, at least once a week. A group must consist of only 4-5 students; don't let them choose the group members!
Apart from giving the routine exercises, give them some challenging questions, it need not be directly connected to the course, again, on a weekly/biweekly basis. Choose one or two best answers, and post them on the department notice board. That is much better than giving a toffee as a token of appreciation!
Promote them in using online resources like the MathStackExchange, softwares like Mathematica or Matlab.
Train them in writing Math. There's a way of writing, which we do not practice during our school life. If the teacher follows better notations, correct way of writing, etc. the students will slowly follow that. Still, the teacher can look at their notes, exam answer sheets, and note-down/explain-in-class the mistakes that they make in writing. For example, using the 'implies' symbol (=>) in between lines, and not writing complete sentences are in fact time-saving, though such practices should be avoided. Another example is in relation to the use of quantifiers, students usually have confusion when and where to put them.
Write-up assignments may be given, and ask them to get it typeset in LaTeX. (When we were students, we used to decorate the front page of assignments/projects in MS Word, such a stupid thing.)
Encourage higher education. There're many ways a teacher can do that. Here is a list.
Take feedback from the students, every month if possible.
Please visit the following link, and give your suggestions, which I shall add to the above points.
Some of the main challenges that I think we all face are the following; please give ideas on these too (use the above link).
How to effectively promote group discussions so that they practice the art of learning from their peers outside the class hours?
Apart from teaching math, how do we prepare them (or more correctly, help them in preparing) mentally, spiritually and politically? Is it only the duty of the high school teachers to put life values in students? Are we different from those teachers? Should we assume that we're not teaching kids, but 'mature' fellow humans? Of course, we consider our students as fellow human beings and respect them. But, how far are we responsible for this kind of parenting?
This is also related to the previous point. If we notice that few students are not listening/following the class, or not showing any interest at all, should we somehow force them to attend the class or punish them with writing impositions etc.? Our teachers used to punish by throwing us out of the class, and in fact, we always enjoyed the punishment!
How to identify and help students who are facing financial or other personal problems? We can't expect that all students are financially fit, for example, we may usually expect that everyone has access to computer/internet (or even electricity!) etc., but that may not be the case.