Using Doenet Interactives to Facilitate Inquiry
This post describes work of the author funded by National Science Foundation awards #2011807 and #2449139.
An essential element of my Team-Based Inquiry Learning (TBIL) classroom is that, as much as possible, students whould be discovering knowledge for themselves, rather than just being told.
However, I only have 150 minutes with my students each week! And while I’d love to give them the expeirence of my own inquiry-based learning classrooms (a list of problems to solve and theorems to prove, a pat on the head, and a “go get ‘em, show us in class next week what you were able to figure out”), that level of unscaffolded exploration is frankly not appropriate for the majority of my students, who are mostly non-mathematics STEM majors.
To this end, a TBIL classroom emphasizes the need for “Scaffolded Exploration”, where students are given enough tools, preparation, and context to authentically engage with an activity using inquiry, without leaving them rudderless and unable to even begin to puzzle things out.
One mechanism to do this is to provide students interactive applets to help them explore mathematical concepts. While many of us who went on to teach in the mathematical sciences are quite comfortable with drawing several examples to explore a new concept, or even visualizing things in our own heads, many students have not yet developed such skills, or at least are not prepared or confident enough to run to a whiteboard or their paper to do so in a small group of their peers.
I found this particularly tricky when I first introduced geometric properties of linear maps in my sophomore (engineering-majority) linear algebra course. Asking students to consider how changing the length of a parallelogram’s base affects the change in its area isn’t a terribly deep question, but it’s one that definitely benefits from visualization.
So in Activity 5.1.8 (2024 Early Edition) we provided such a static image:
Well, it’s certainly better than nothing. But for all the abstraction of multiplying the base by $c$, that $c$ value sure looks fixed at around maybe $1.4$… Would it be nice if students could see the effect for many different values of $c$, or even observe the effect for the $c$ values of their choosing?
Enter Doenet, a language and platform for authoring interactive mathematics visualizations. There’s much to be said about Doenet (and I’m sure to do so in my future posts!), but for today I’m just going to focus on the end product (which, of course, is 100% open-source markup powered by 100% open-source software). Let’s see how Doenet can helped us create an interactve figure to replace the above static image in Activity 5.1.10 (2025 Edition):
This has worked so much better! I can send this link to my students’ computers, and together they will move the slider, observing the original area when $c=1$, the doubled area when $c=2$, and the halved area when $c=0.5$! By removing the cognitive overload of visualizing various values of $c$ in their heads or on paper, students can dive directly into the mathematics, particularly my true goal to make them realize that $\operatorname{det}([c\vec{v}\hspace{0.5em} \vec{w}])=c\operatorname{det}([\vec{v}\hspace{0.5em} \vec{w}])$ without me telling them this myself!
To see how we explore the determinant’s other geometrical properties using Doenet interactives, check out Section 5.1 of our 2025 Edition! In a future post, I’ll explore how to create a great Doenet interactive for yourself at Doenet.org. (Or if you’re inpatient, you can of course see the Doenet source for this interactive yourself here on GitHub.)
As you probably expect, TBIL activities and interactives aren’t trivial to always come up with alone, which is why I’m excited to work with a great team of collaborators at GitHub.com/TeamBasedInquiryLearning to create high-quality, free and open-source TBIL activity books for Precalculus, Calculus, and Linear Algebra. If you want to get involved, ping me in the MathTech.org Discord, or even better, join our TBIL Zulip chat!
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