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I'm using the word constructivist as it is used in this paper, not in the sense used in logic. The abstract should be sufficient to understand at least roughly what the model is. The important part, more or less, is that students construct their own questions and pursue them, rather than being dictated and tested upon a standardized curriculum.

Anecdote. When I was in college, I took a course on videogame culture. Many of the lectures consisted of a discussion of relevant current events, usually involving most of the audience, but pushed along by the professor when things stood still. Out of class, we wrote papers about topics of our choice, and there was a community blog that we posted on about other relevant issues. We were required to read, post, and comment in certain quotas every week, but many students became very engaged and posted more. I learned a lot in this class.

I have been fantasizing for some time about conducting a mathematics course with a similar approach. In my fantasy, my students post questions and answers to our own private StackExchange, exploring their own ideas about pre-calculus algebra in a mathematically meaningful way. One student makes up a family of polynomials and explores its properties in a series of well-presented questions, inspiring others to write answers, as well as further questions expanding on his. One student derives a cubic formula. Gosh, while we're at it, maybe another student asks what would happen if commutativity was dropped, stumbling inadvertently into some group theory. During face-to-face meetings, we look over everyone's work as a group, highlighting the best progress and supporting those who are stuck. It's a pretty rad dream, alright.

The thing is, I think this could even almost work with upperclassmen math majors, given the right tweaking and incentives. But what I would really love is to find a way to bring a constructivist learning model to lower level coursework, since many of the students are so unengaged by the conventional style. The main problem is that I suspect many of these students have insufficient skill to participate, despite the course being college level. I wonder if there is a way to adapt the model for this purpose. Could a partial integration solve this issue? Would it be impossible to achieve the literacy required to pass a traditional final exam?

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  • $\begingroup$ Out of curiosity: Are you thinking that the class is entirely online? Or just that it has some web-based platform (e.g., Moodle, blackboard, something like SE) to supplement in-person class meetings? $\endgroup$ – Benjamin Dickman Aug 25 '15 at 15:59
  • $\begingroup$ @BenjaminDickman I suppose that's part of the question- how online the course would need to be to be most effective. My inclination would be to have the course mostly online, with in-person meetings supplementing online writing. $\endgroup$ – Alexander Gruber Aug 26 '15 at 20:43
  • $\begingroup$ What sort of class do you have in mind? I'm teaching a very open-ended course that's essentially math appreciation for non-science majors. I could see something like this working there, not so much for precalc, calc, etc. $\endgroup$ – pjs36 Sep 6 '15 at 15:43
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I would be pessimistic. An open-ended approach works well in the liberal arts and social sciences -- such as the course on video game culture -- where there are a fairly small core group of skills (reading critically, researching, analysis, writing and commentary) that can be applied broadly to a large number of subject areas, and practiced in a spiraling manner.

In contrast, mathematics and other STEM courses are "dense" with new skills being added on a daily basis, and in most cases being chained from prior work in a particular sequence of prerequisites. It takes a lot of work on the part of an expert instructor to scaffold this kind of material in an accessible way.

The example of "stumbling inadvertently into some group theory", while fascinating, would be extraneous to the skills intended if we ask if someone has mastered precalculus. Moreover, if there is a list of required, distinct concepts and skills expected at the end of any mathematics course (and a relatively large list, compared with non-STEM subject areas), then it seems highly unlikely for the open-ended approach to meet that for everyone at the end of the course.

This is the kind of approach that would be great in an independent study or elective context -- and perhaps we should require everyone to experience that at least once! But for the core sequence of precalculus, calculus, etc., there is an irreducible conceptual ladder.

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  • $\begingroup$ @AlexanderGruber I would be concerned about sufficiently teaching the necessary skills in a precalc class as Danial discusses. However, I really like to try and engage other learning strategies in math classes. I think you should try to pursue strategies that allow students to discuss and present their work. IMP has some great resources that you can adapt, among others, but if the students are going to reach their learning objectives you will need to lead the class more than you're describing. $\endgroup$ – BBS Sep 21 '15 at 13:46
  • $\begingroup$ I don't completely agree with math requiring a much larger set of skills than say arts. In the core of it lies almost the same cognitive abilities, such as compare and contrast, generalize, apply, synthesize, etc. One does learn new math daily, but not new math SKILLS, necessarily. $\endgroup$ – Behnam Esmayli Apr 25 '18 at 16:51
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I don't know about the online part, but Discovering the Art of Mathematics definitely has some ideas that are low-level (non-major, even) but yet stumble upon very interesting things indeed. I'm sure they'd be interested in discussing an appropriate online pedagogy. The AIBL would be a good resource.

To be sure, one doesn't have to be a strict 'constructivist' or 'constructionist' or any of that to use these, but there are aspects of them that really would resonate with your question - including e.g. precalc or calc.

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