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I am trying to implement some version of a Project or Problem-Based Learning methodology (which I will just refer to as "PBL"[1]) in Linear Algebra and I am not sure how classes are conducted in it.

tl;dr/Question How are classes conducted in PBL? Should I just give them the problems/projects and let them figure out the words and concepts mostly by themselves, answering questions, discussing their understandings and so on? (And perhaps giving class-wide explanations of common questions, such as explained here)

I have looked up several articles on PBL, but this part is never discussed in detail. Moreover, in math courses, it seems the problems need to be guided more to make sure the necessary skills are developed (e.g. here and here).

I am apprehensive on simply throwing students linear-algebraic words and concepts, as the language itself is usually one of the main problems with them. However, I understand that this could just be me fearing deviating from my own experience.

My initial background is in pure Math. Throughout my whole formation (2010-2018 for undergrad and grad) all my classes consisted of lectures + lists of exercises (generally without a grade associated to them) + 3ish exams. There were a couple of projects here and there for the more applied courses, but they were only complementary to the lectures and exams.

I am teaching engineering courses, and the current lectures-exams approach is frustrating for several reasons which I will not get into, so I am planning on changing the teaching methodology.

I have also started studying Computer Science (at another university than the one I work in, of course) on my free time during COVID, and have found that classes have many more projects which me and the other students find much more enjoyable. To my surprise as a professor, seeing things from the student perspective, I noticed that cheating is also much less rampant in projects than I would expect. These projects are used along with exams for evaluation.

So I am trying to adapt this kind of methodology to my courses. Currently, I have written down 8 "small projects" or "hard problems" which are to be completed throughout a 18-week semester (2-ish by project) and cover all the topics in the description of the Linear Algebra Course. I think I have managed to avoid routine and/or artificial problems. I plan on using them along with standard exams for evaluation.

For example, one projects guides a group of students on how to encode two phrases with the Hill cypher; Another group receives one of the phrases in both encoded and decoded form, and another phrase is simply encoded. Then the second part of the project guides the students to use a change of basis matrix to decode the second phrase.

In this example, would the PBL approach be simply giving them the problem related to change of basis matrices, without explaining what they are beforehand, and let them look them up by themselves in the textbook?

Thanks in advance.

[1] Yes, I am cheating by conflating the two notions :)

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    $\begingroup$ I would caution against using an exotic and experimental teaching methodology based on hearsay, without having very concrete and complete treatment of it in hand (and ideally hard evidence that it's beneficial, but that's unlikely to be available). It's possible that this question is too large in scope for a complete SE answer. Perhaps asking for references to a complete description and evidence would be better. $\endgroup$
    – Daniel R. Collins
    Commented Jul 31 at 15:42
  • $\begingroup$ As a former math PhD student and medical student enrolled at a PBL-centric medical university program, I can tell you that I really dislike PBL and anything related to it. $\endgroup$
    – Andreas Rejbrand
    Commented Aug 1 at 17:37
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    $\begingroup$ @AndreasRejbrand Would you care to give a more detailed account on what was/is the methodology of the courses you've taken which have used PBL and why you really disliked it? $\endgroup$
    – Luiz Cordeiro
    Commented Aug 1 at 23:10
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    $\begingroup$ I don't have time for a full response right now but the University of Michigan's proof-based linear algebra course (Math 217) has been taught in a completely IBL (inquiry-based learning) format for the past 8 years and has been quite successful. I'll share more information in a day or two $\endgroup$
    – mweiss
    Commented Aug 2 at 22:38

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Ålborg (or Aalborg) university has been doing lots of project-based mathematics education. I have read this article from 1995; maybe it leads to further reading via the usual literature search methods:

Vithal, R., Christiansen, I. & Skovsmose, O. Project work in university mathematics education. Educ Stud Math 29, 199–223 (1995). https://doi.org/10.1007/BF01274213

I do not currently teach pure or applied mathematics, but rather didactics of mathematics, so I have not tried a corresponding approach myself in a relevant context.

However, a word of caution: if you are unsure about your approach, and if some of the students do not like it (for good or bad reasons), you are likely to meet resistance from them. Depending on your institutional culture and your personality, this might or might not be a big deal.

I would humbly suggest moving in smaller steps. Maybe try out a single project at first and see how that goes.

You might also check if your university, or some nearby or otherwise affiliated institution, has people working on mathematics education or university-level education or similar themes. A collaboration could be valuable for everyone involved. Some mathematicians have a very negative view on didactics, but you seem to be interested, which could offer possibilities for them to guide you towards relevant literature and ideas, and for you to provide a setting for doing shared research on didactics.

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  • $\begingroup$ Thanks for the answer and sincere suggestions. Actually, I have a couple of acquaintances who are coming back from vacation next week and have PhDs in related themes (student-centered teaching in undergrad mathematics and such), and we've scheduled meetings to see what we can get out of that experience. and how to implement it properly. $\endgroup$
    – Luiz Cordeiro
    Commented Aug 1 at 23:11
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I used many of the projects at the Inquiry-Oriented Linear Algebra site. I typically gave one day a week to these, and did a more conventional lecture and examples class session the other days. (Although I tried to include a lot of call-and-response, asking students to provide the next step.) These projects enriched my classes substantially. [I was impressed at how well they aligned with our work from our David Lay textbook.]

You have to ask for a teacher account, and it may take a few days. Once you have that, there is lots of support for each activity (they call them 'tasks'), including slides showing typical student work (I suggest you get students working on whiteboards if you can, there is research on how much this helps in getting participation), and videos showing classes working.

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You might be able to draw on this book: Linear Algebra and Geometry:

Jacket

It does not directly match your needs, but its approach seems quite amenable to PBL. (Caveat: I have not taught a version of your course.)

Here is the jacket copy:

Linear Algebra and Geometry is organized around carefully sequenced problems that help students build both the tools and the habits that provide a solid basis for further study in mathematics. Requiring only high school algebra, it uses elementary geometry to build the beautiful edifice of results and methods that make linear algebra such an important field.

The materials in Linear Algebra and Geometry have been used, field tested, and refined for over two decades. It is aimed at preservice and practicing high school mathematics teachers and advanced high school students looking for an addition to or replacement for calculus. Secondary teachers will find the emphasis on developing effective habits of mind especially helpful. The book is written in a friendly, approachable voice and contains nearly a thousand problems.

Here is a typical question, Ch.4, Matrix Algebra, p.152:

![TSP

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  • $\begingroup$ The language style alone in your quoted passages makes me immediately think of some BS involved ;-) But you are right: if one hasn't tried something in honest, one is not qualified to judge it :-) $\endgroup$
    – fedja
    Commented Aug 3 at 0:45
  • $\begingroup$ I see nothing unusual in the language style. I like the book because of (a) its emphasis on geometry, and (b) it's many "field-tested" questions (I snapshotted one from my copy.) But to repeat the caveat: I have not taught this course with this book. I'm only suggesting it as a source. $\endgroup$
    – Joseph O'Rourke
    Commented Aug 3 at 23:48
  • $\begingroup$ I see nothing unusual in the language style It is not unusual at all nowadays, quite the contrary: it is a typical language style of a TV commercial advertising some patented junk no one really needs. It might be just that the authors didn't know how else to express themselves, but the impression it gives is very negative, IMHO. The problem sample is fine but nothing out of the ordinary: just a standard discrete math. question. $\endgroup$
    – fedja
    Commented Aug 4 at 1:43

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