A Calculus class for 1st year students may have two subclasses (with two different lecturers) :
The main class (which covers the theories and concept),
and the 'response' class (which provides and explain how to solve problems in exercises)
For the latter, what is the ideal way to teach?
Some logical ideas that have been think of are :
Give them problems (homework) and call some students forward to work on the board in the next meeting
Give open book quizzes (open book so that the students are also studying at the same time)
Give problems and explanation as many as possible? (sometimes does this makes students lazy or not..?)
Some facts : several students that rarely focus on the lecture (skip classes, or even working on other things inside the class) have higher grades. I presume that online resources (such as Khan Academy, Youtube, ... works better for them?
Reserve time for questions. In my experience this will not actually take much of the time. Don't be afraid to not answer certain questions. For example, "what do you think is on the next test" is usually not wise to entertain. Unless you are writing the test and have a specific sense of how to use that question to leverage studying. For example, I will sometimes ask in return, "well, tell me something you're not done studying".
Prepare a few examples to give lecture style. Perhaps just 1/3 of the time for this, but it assures there is real content delivered. This could be at the beginning, middle or end, in fact all three of these can be rearranged as you see fit for your audience and setting.
Put all the students names on 3x5 cards. Write about 5-10 problems on the board and randomly select names to work those problems. Give them 10 or so minutes and emphasize that everybody should be trying to work through the problems to follow along. When the time is up, go through the problems critiquing both answers and presentation. Mark the cards with dates and short description for your record. Next time, go to new students, or for fun repeat to keep them on their toes. Eventually everybody comes up front to work problems in this fairly low-pressure setting. This works best if you have some course points to assign. If you have no influence over their grade then sadly I have not much hope. I mean, try talking to people anywhere about math for 2 hours when you have no control over their grade. You'll find yourself alone in a room long before the end. (statistically, people are rarely mathematicians, so the sentence before is said in that manner of thinking)
The benefit of 3. is it gives them incentive to work on the class regularly and it gives you a chance to warn against common mistakes and/or to show better ways to solve the given problems. Finally, it is probably useful for the students to see that everybody (for the most part) struggles with the material. Too often students refuse to ask questions because they think they are alone in their confusion. In fact, the confusion is the rule. Ideally, this process helps some of the students to start asking good questions. We probably need to teach them what is a "good question", but I'll leave that for another post.
To the extent the class does not cover problem solving, you should cover a a few in lecture style as examples. I don't totally believe (maybe don't want to believe) that the regular class is all derivations and theory and no example problems. But if so, definitely showing some examples would be helpful.
Emphasis even in case of 1, should be on drill still though. Pop quizzes, work together assignments, kids to the board, etc.
When explaining example problems, you will need to be efficient. Don't have time to cover all, so cover things that get messed up most often or are most likely to be covered on tests.
I would keep the tone in your section more collaborative and informal and friendly than the normal class. More "buddy telling you how to get it done" and less "herr doktor professor". Students love feeling like they are getting the inside scoop and cutting through the pretensions.
Use the board rather than slides. If you want super extra credit A+ points, than compose your remarks ahead of time. But still do the exposition on the board, rather than slides. Slides are a crutch. And they are so much different in connection than writing as you go. When you show slides, it's like showing the book. When you write on the board, you are a fellow warrior fighting in the trenches. Maybe a little bit better of one. But still "in the fight".
Talk to other TAs and get their experience. Coming here was a nice step. But do it IRL. And get to the nitty gritty (time management, discipline, loudness, handwriting, etc.) Not just high level stuff.
You avoided the question about country and student ability. Realize that some issues of pedagogy are the same from CalTech to RN JuCo. But in other cases, it makes a difference what the student limitations are in brains, desire, time, prep, etc. Think about these variables. Pedagogy is not a linear y(x) function but nonlinear, multifunction and full of stochastic noise (and student ability, etc. are confounding variables to some sort of same for everyone solutuon)