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First, some context: I'm a PhD student, and this term, I'm running a bunch of seminar groups for first years (I believe these are what would be called "discussion sections" in the US - essentially, each group is 20-ish students out of a much larger course, and I'm doing stuff to support the lectures. I'm marking, but not setting, the assignments). My previous teaching experience has all been with what I believe would be "upper division" courses in the US, and thus far I've tried to approach this in essentially the same way, but there are some pretty significant differences that lead me to think that I'm not using the time well. Specifically:

  • In previous teaching, I've been able to rely on students (at least after a little prior prompting) turning up with questions that we can discuss, being able to tell me which parts of the course they're struggling with, etc. and dealing with these issues has taken up a large part of the available time (essentially, because dealing with the things that they're definitely struggling with seems more reliable than me trying to guess what they're struggling with). This doesn't seem to be the case with these students.
  • Previous students have been able and willing to get involved in discussions, either in small groups or all together, and have even started such discussions. The default expectation from these students seems to be that they can sit there and do very little and I'll somehow pour mathematics into their brains.
  • Previous courses have all been heavily proof-based, and I've spent a good part of the available time helping with details of proofs that students have not been comfortable with, providing alternate proofs, and trying to drill down to the underlying concepts. This course, thus far, consists entirely of matrix algebra, and the expectation seems to be something far more computational.

As far as what I've done so far, besides the usual start-of-course administrative stuff, is briefly summarise the content of the course thus far, provide a couple of worked examples, and get them to work on some practice questions (theoretically in groups, but I haven't managed to overcome their unwillingness to talk to each other very successfully yet), while I walk around and provide support. I also experimented with asking a question involving slightly more thought ("so we've seen that it is not always the case that AB = BA, but we've also seen that this does hold when either A or B is the identity or zero matrix. Can we say any more about when AB = BA does or doesn't hold?"), which they found significantly more difficult than I had anticipated, and needed a lot of help to even start.

In general, this doesn't seem like that effective a use of the available time: in particular, there's very little that I've done that couldn't have been done by handing them a textbook with plenty of practice questions. I also haven't gained much information about concepts that they struggle with - most of the errors that I've found, both in their assignments and in the class, are purely failures of mental arithmetic, rather than any conceptual issues.

Essentially, I'm looking for ideas of how I can adjust my teaching to make it more productive. Ideas for encouraging more active participation would also be much appreciated.

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    This might be a better fit at matheducators.stackexchange.com – Johanna Jan 23 at 11:56
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    @Johanna, actually, the question also applies to many other fields, not just math. – Buffy Jan 23 at 12:21
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    What are "first-years?" First-year undergraduates? Is this meant to be a gender-neutral alternative to "freshmen?" Are these students mostly engineering majors? Math majors? – user1482 Jan 24 at 14:11
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    @BenCrowell Yes, first year undergraduates. That's just what they're called in parts of the world that are not the US. "Majors" aren't a thing here: they're all maths students. – user3482749 Jan 24 at 15:17
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If you were the professor for the course, I'd give a different answer, but consider the following.

I'll assume that you have the same group for each meeting, rather than a random selection of students that changes randomly over time. This lets you set expectations and a general flow.

One problem you may be encountering is that the students haven't yet learned how to learn. They haven't learned how to take notes and study properly. It is a lesson that needs to be taught, actually. Many students at this level have had an easy time of their earlier education. For some, at least, this is now changing and they need better procedures to capture and retain learning.

If you aren't getting questions, then ask a lot of questions yourself. They can be simple or complex. One of the simplest is to ask the class for volunteers to answer "What was the most important point in the professor's previous lecture?". Ask if anyone has anything to add to that. Evaluate and supplement any answers.

Use metaphor and examples to explain/illustrate things rather than being pedantic. Try to give them insight. While precision is good, it sometimes masks what is really at the heart of things. Why do some mathematical systems obey the commutative law and others not?

It takes some experience, but you can turn some things into games. Even competitive games. I once created a Jeopardy like game for teaching key ideas. It was a computer program that automated the questions. "Square Matrices for 10, Professor Buffy". Teams were formed and the students had fun trying to outdo other teams. This might solve your "working together" issue to some extent.

But just repeating what the professor said isn't likely to be effective. If they don't know what the professor did say then they need some instruction on effective note taking. Also on lecture summarizing, etc. Show them what tricks you used to learn hard stuff.

In fact, if you want an "ice breaker" question, ask what it was that the professor didn't make clear enough in the previous lecture.

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If I'm understanding the context correctly, this is a group of students who are first-year undergraduates, probably mostly engineering majors, taking a linear algebra course that comes in their lower-division math sequence after a year of calculus.

Re this specific type of course, this may depend on the text and the professor, but usually they start with a lot of very concrete calculations. Students spend a vast amount of time doing stuff like multiplying two matrices whose components are given as integers. Only later in the book, and the course, are they introduced to things like the axioms of a vector space. This approach, for better or for worse, seems to be pretty standard, and I guess is meant to make the class easier for the students by delaying abstraction until they have enough experience to make sense of the abstraction. If this is an accurate picture of the course you're teaching, then there is a limit on how much you're going to be able to deviate from that structure.

In general, what you're trying to do is referred to using terms like "active learning" or "interactive engagement." You can google on these and related terms. A good survey paper is Freeman et al., "Active learning increases student performance in science, engineering, and mathematics," http://www.pnas.org/content/early/2014/05/08/1319030111 .

Students at this stage are generally not an the intellectual level where they can come up with their own interesting, abstract ideas or profit from unguided discussion. Successful active learning techniques usually involve very structured activities that set students up for success. Almost always, the activities an instructor comes up with end up being much too hard for real-world students, who are beginners who are approaching things as beginners. For this reason, it can be very helpful to find other people's published activities that have already been classroom-tested.

In your example where you asked about what matrices might commute with other matrices, I would make this into an exercise where they start out with some concrete examples, which they compute in randomly assigned groups. Then, once they have some examples, you give them a really easy, concrete task. E.g., if they've already established that the 2x2 identity matrix commutes with some other matrix B, ask them to find one other matrix that commutes with B.

Another style for creating active learning exercises is Mazur's peer instruction method, or similar methods such as think-pair-share.

A lot of these methods involve group work. Often one possible big advantage of group work, compared to a dialog between you and the class, is that it creates a situation where students are expressing their own ideas in their own words, to each other, about concepts. In my field, physics, there is a great deal of research showing that this is a crucial ingredient for achieving better conceptual understanding in a freshman course: words have to come out of the students' mouths, and the words have to be about concepts (not the details of an algorithmic computation). Make sure that you know what the students are saying in their discussions, by walking around the room and by asking them to write specific things down.

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One tiny suggestion that works wonders: Instead of asking "Any questions?" ask "What questions do you have for me?" This is an invitation for questions. I tried this out this semester and have been amazed at how many more questions I get because they feel that it is okay to ask questions.

Another suggestion is help them get to know each other. I put people in random pairs in exercise sessions either with a bunch of lengths of string that I hold in my hand. Everyone grabs one side and works with the person on the other end of the string. Or we use Memory cards, or pairs of playing cards to choose partners. By the end of the semester they know about a dozen others and feel much more at home in class and ready to learn.

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