I have been a TA for 6 courses, and I have observed the following general pattern of the weekly 55-minute tutorial sessions:
The instructor assigns problem sheets a few days in advance. The problem sheets contain:
- routine exercises
- problems that can be solved by an application of an important theorem taught that week
- problems that need to be broken down into multiple steps by fitting together two or more previously known results
- multi-step problems that cover ideas beyond what was covered in class
The instructor emphasises that everyone should do their best to attempt every problem in the tutorial sheet before attending the tutorial. During the tutorial, I solve some selected problems on the board in detail.
First, I pick one routine exercise of each type and display all the steps in my solution, and point out how the template works for similar problems.
Then, I pick the problems that demonstrate the application of a theorem, and again solve one problem of each type in detail and point out the signs that a particular theorem is to be applied, and also how to adjust the solution in similar situations.
This takes around 20 minutes. Sometimes the students offer alternative solutions or ask questions to clarify some steps, so I spend additional time as necessary.
- For the problems that require multiple steps, I go over each one slowly, explaining each part of the breakdown carefully. In this part, I specifically ask the students for their ideas. Their suggestions are usually on the right track, so in such cases I invite them to elaborate further, and ask them to help me by suggesting what I should write next on the board. Even when a suggestion is incorrect or unfruitful, it offers a nice teaching moment to try and explore why that method won't work. The final step in such problems is usually some computation, either numerical or symbolic, that I leave for them to verify on their own. I have found the students to be confident of their computational skills, so they are comfortable with this: I do give them the final answer or expression for them to verify that they have got it right.
By this time, there is usually very little time left to cover the difficult multi-step problems, so I open the floor for discussion. Students usually ask questions related to previous problems, or about the theory discussed in class that week, and that closes the tutorial session.
The final exam never tests the students based on the difficult multi-step problems. So, those who were unable to attempt them are not at a disadvantage. At the same time, the bright students enjoy the challenge posed by those problems. I always stay back after the tutorial to answer any further questions, and I usually receive a couple of questions about those problems, which I'm happy to discuss.
This method of running the tutorials seems to bring good results. In my experience, the weakest students need clear worked-out solutions to the simpler problems, which they can refer to later on. So they remain attentive for the first 20–30 minutes, and start zoning out a little after 40 minutes.
Asking the bright students to solve the routine problems instead does not seem to help, because they can't express their work in a clear manner right off the bat, which is quite understandable. I also find that these students are able to find sufficient internal motivation to get through the first 20–30 minutes of easy problem-solving without losing much focus. So, I have a fully attentive class for at least 30 minutes, which is great. Then, when I engage the bright students in solving the more difficult problems, they are quite eager to offer their insights when attacking these problems, which is great because these are meant to develop their intuition and their understanding of the course material.
Somewhat contrary to your description of your class, I find that the students resist engaging in discussion groups, and that any attempt to force them to do so results in a dissatisfying tutorial experience for everyone involved.
- I give detailed solutions to each "type" of routine exercise and application of a theorem. This usually takes 20–30 minutes.
- I invite discussion on the more conceptual problems, taking suggestions from the students on how to tackle them. This takes another 15–25 minutes.
- If any time is remaining, I spend that answering further questions on the tutorial problems or the theory covered in class.
- I also choose to stay back after the tutorial is officially done to answer deeper questions, and any other questions that could not be taken up during the tutorial due to lack of time.
- I specifically avoid making the students work in groups, and also avoid making the bright students explain the solutions to the routine exercises to the rest of the students.