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So far I've tried building my presentation from elements, each of which is differently paced.

A lot of tutorials are given just by rapidly writing fully worked solutions on the board, thereby leaving little room for actual discussion of concepts. I don't like that kind of tutorial.

The last tutorials I taught were based on the fact that the maths school would release fully worked solutions electronically at the end of the week and encouraged their tutors to have a more "lesson" based presentation.

I ended up with a presentation based on putting skeleton solutions comprising a list of sub questions that just referred to the appropriate concepts from the lectures. Then I asked the class to come up with the answer to each sub question so that they could see how a solution was built up and could contribute to its construction.

Then of course, even though you are keeping the ones who want to move faster happy by giving them the opportunity to participate and thus not get bored, you still have to regulate who's answering and how often and how to encourage the weaker students to have a go.

Any thoughts from others on how they handle this are welcome.

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This situation sounds very similar to many of the classes that I have taught. Like you, I dislike the kind of tutorial that is just based on worked solutions - but, what I found was that the worked solution, could be a part of a dynamic tutorial.

This is what I have done - initially, I go through a worked example on the board (interactive whiteboard tends to work well), all the time encouraging questions about each of the steps. (I colour code the processes as well).

Then, I have the students in small mixed ability groups (i.e. I choose the groups based on abilities) go through separate questions, following the worked example as a guide. Gradually, having the groups determine solutions without a worked example on the board - using the skills, they have acquired.

I found that the smaller groups were less confronting as the quieter students started to become more vocal within the groups (and occasionally in front of the class) with more tutorials. Also, I found that the students became more confident to tackle questions on their own for practice/homework.

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Perhaps the real solution to the conundrum is to take out one of the restrictions. Limit the material (it is just impossible to cover much material in any depth in 50 minutes), stretch time (perhaps by asking the students to prepare beforehand, or leave homework(ish) questions for later), or match abilities (ask the more quick students to help out others, perhaps by setting up work groups for each class).

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Get them to work on problems in small groups (4 to 6 students) and walk around watching/listening and giving advice. This works best if they are working at a white/black board. If you don't have enough board spaces, dark whiteboard markers work well on windows, and butcher's paper is ok at a pinch.

You need to make your expectations clear on what is supposed to happen. I also tell them that no-one is allowed to move on without helping the others to understand. (EDIT based on recieved comment: Perhaps a better thing is to tell them they need to check if everyone is ok with moving on before moving on, rather than requiring that people help others to understand.) I also tell them it's always ok to ask someone to how/why they did something, as long as they're polite about it. When you are watching/listening, look for problem-solving strategies you can help them with, misconceptions you can fix, and also good things you can point out to them to increase confidence. If everyone is stuck at the same place, tell them to leave that one for later and you'll do it at the board at the end of the class.

At the end of the class, do some presenting to the whole class, perhaps doing a problem they all struggled with. Another thing to do in the whole-class time is ask them to volunteer things they learned today. This sort of "cognitive closure" is good for them.

One practical advantage of doing group work is that once you've set up how it works, students can start working even before the other students arrive, in some cases even before you arrive, so you don't have to waste time waiting for people to turn up.

In terms of learning, you get to actually see what their understanding is and have a hope of fixing misconceptions. This can't happen very well in any whole-class setting. Imagine the situation where you "got through" the wole amount of content but half the class lost you at the first example. All the rest of the examples are wasted. This way what they work on is more easily targeted at what they need. Also, students at different ability-levels can engage at different levels, sometimes by self-selecting into different groups, sometimes by the stronger students acting as teachers for the others.

Finally, it's more fun for everyone!

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  • $\begingroup$ A quibble: might be a little too much to declare that everyone has an obligation to help others understand. Yes, doing so is "a good thing", but to declare it a (moral?) obligation will alienate some students, and also tend to delegitimize the teacher. Instead of "requiring" it, "recommend" it, for all the pursuant virtues. $\endgroup$ Jul 17 '14 at 23:16
  • $\begingroup$ @paulgarrett Fair point. "Recommend" is probably better. However they should still as a group come to a concensus that it's ok to move on, even if some students don't fully understand. I'll make an edit. $\endgroup$ Jul 17 '14 at 23:20
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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.


To summarise:

  • 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.
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