Corrinne Manogue at OSU is the source of this one:

You teach upper-level physics. Say, you want to teach your students about eigenvectors. You could

(A) Introduce the word "eigenvector" before or at the beginning of lecture, explaining what the term means and where it comes from. And then lecture on how to solve for eigenvectors, and then have students practice.

OR

(B) Put students in groups: give each group a different (carefully chosen, of course) matrix and ask them to see if they can find any vectors that don't change direction when you multiply it by the matrix. Let them explore, remember how to perform matrix multiplication, encourage them to draw (not just do algebra), watch them develop an intuition for what each matrix is doing, and try guess and check, encourage them to use geometrical insight to rule out or hone in on solutions, let them struggle with whether there can be more than one solution. Then let them share their solutions with their peers. Point out important similarities and differences across problems and solutions strategies. Point out important things you'll need to bring up later. Then, then introduce the word "eigenvectors". Draw on the insights they have (and haven't) made and present the formal method for finding eigenvalues.

The argument for doing A could be this: "Students don't have any intuitions about eigenvectors and linear algebra. It's a weird word that's distracting. If I introduce the word before lecture, it will help them focus on the mathematical structure and methods I want to teach, rather than on the weird vocabulary."

The argument for doing B is this: "By drawing on what students do know and can do, you can quickly build up a set of intuitions that orient students to the concept of eigenvectors. Since, they are not likely to formally develop all the methods on their own, I can capitalize on what they end up doing to anchor the formal instruction to their own ideas and methods."

Anyway, what do you guys think?

I don't think those are the only two options, of course. If I had to choose, I'd do B because students understand the basics of rotation.

ReplyDeleteWhen I taught this concept from the perspective of the inertia tensor and the principle axes of rotation of an object this past semester, we spent a day in class asking how much information you have to give to figure out the orientation of a chair at some crazy angle. I took what they produced (not exactly the Euler angles but they worked) and made a screencast showing the similarities and differences between their approach and the book's approach. I went on to talk (in the screencast) about how you can build up weird rotations as combinations and even showed the basics behind eigenvectors. Back in class we re-tackled the chair (every student had a chair they got to play with) and we made great progress.

Like Andy, I think this is a bit of a false dichotomy. I've been doing JITT for a couple years, so that's what I would do, i.e. none of the above.

ReplyDeleteOne problem I have with B is that it's not obvious that the appropriate context exists and that the students will really understand why such vectors are interesting. In my mind, that's the real trick with applying inquiry methods as part of the introduction of new concepts.

There is definately an (A+B)/2 or C option as well. It involves some front-loading of concepts and vocabulary so that everybody has some common ground. In the example of eigenvectors, you could show them how to find eigenvectors for 2x2 matrices, then give them some 3x3 matrices and ask them to do a similar exploration to option B.

ReplyDeleteRemember they may never have even multiplied matrices together; making option B even more challenging.

All these things lie on a continuum and the most appropriate activities to use at any given time are a delightful mix of previous student learning, prep time available, experience teaching the course or topic, classroom culture, resources available, courses which will follow, compatibility of instructional strategies with the rest of your department and so on. For my next lecture or my course coming in the Fall, I have to optimize based on all the above factors. Over time and after many iterations/optimizations, I might find that my courses are filled with option B. I would honestly love that (and look forward to a time when much of Paradigms has become portable).

For now my next iteration looks most like my option C. Give them some basic tools and then use class time to explore the phase space of the advanced tools. I have the "let them explore the phase space of the advanced tools" resources somewhat in place and am now trying to improve how I give them those basic tools.

I'm fine with it being a false dichotomy. I'm happy that there are option Cs, or D, or whatever. I'm all on board with the point that having an delightful mix of appropriate activities requires time, experience, resources, support etc. I don't have time, experience, or resources to do so myself.

ReplyDeleteI'm responding to the claim that "students don't have appropriate intuitions at upper level, so exploration before vocabulary is only be appropriate at the introductory level."

Joss, what course are you teaching eigenvectors in where students have never multiplied matrices before?

I think one problem with this and similar conversations is the word "only" in your quote above, Brian. I know that on twitter I responded hastily to the notion that vocab up front should "never" be done. We all, in practice, never work on those fringes but our words get us in trouble.

ReplyDeleteFor the record, I think a lot of what Corinne has done with Paradigms is fantastic. I met her a couple months ago for the first time and we had a great conversation about how to teach various mid- to upper-level topics.

What I love about all of these conversations is the exposure I get to so many different ways of doing things. Finding the right balance is for me to do, rather than there being a right answer out there.

On this topic, I like to set the table a little for students for some of the crazy, weird stuff they likely haven't seen before. Sometimes that's vocab, sometimes that's a cool simulation that I can narrate in addition to giving them the link to play with. I've been sharing in Jerrid's blog about the benefits of a backchannel so that the conversation can (sort of) continue outside of the classroom. I'm really excited to explore that more in the future.

My thinking is that the path to Option B is much more straight-forward in the case of introductory courses. Of course you have a lot of problems with the opposite issues there. Students come with some ideas related to these concepts, which are unfortunately mostly rooted in algorithmic problem solving instead of conceptual understanding. It seems that those students would often make the exploring then naming even more challenging than if they came in with very little intuition or previous knowledge related to the idea.

ReplyDeleteOur program is a very "don't put your balls in somebody else's vice" program so linear algebra is suggested but not required. I know that I had students last time that I taught Quantum Mechanics I that didn't know how to multiply matrices, but don't remember if they had never known or had simply forgotten.

I don't think it's necessarily more straightforward to develop good inquiry tools in introductory classes, it's just that's where most of the work has been done. There's plenty of theory involved in Newtonian mechanics; as much or more as in linear algebra and eigenvectors.

ReplyDeleteWhen I teach students about eigenvectors, it's usually in the context of differential equations, and I use a tool like the Matrix Machine at the Interactive Differential Equations website (http://www.aw-bc.com/ide/idefiles/media/JavaTools/mtrxmach.html) to develop the conceptual understanding and context necessary for inquiry.

Chris, I love that simulation (hadn't seen it before), thanks!

ReplyDeleteCould you do this: Ask your students to play with the simulation before class, asking them to find situations where the input and output vectors are the same and to try to generalize when that's possible. Then in class monkey around with the theory about it with them. How good of a job would they do?

Andy, I didn't take you to be taking a hard line of only / never. In general, I won't hold you to your twitter words, if you don't hold me to mine ;) But, all of that doesn't make the claim itself not worth exploring. Because I do agree with you and Joss about this: It feels like option B stuff (as unfairly and dichotomously as I present it) is easier or more straight-forward with intro courses. The question is, "Is it really?" Or have we just not thought enough about the kinds of intuitions and ideas that are useful for learning upper-level stuff. I mean obviously, there are some differences. But carving out those similarities and differences is not easy.

ReplyDeleteFull Disclosure: Anything I say about upper-level is crap, as I've never had to teach it. My experience with learning upper-level physics was this: Everyone who was good on the math spent about 2-4 hours per week doing homework, everyone else spent 16-20 hours per week. I'm not sure I learned any physics until after grad school was over.

@Chris: In term of developing things for intro vs. advanced courses, I factor in things like how often I teach the course (lots more for intro ones), which existing work am I using or building on, and how solid my own understanding is. From this perspective it is easier for ME to develop for intro courses, but you are completely correct that it doesn't generalize to intro courses by their nature being more straightforward to develop for.

ReplyDeleteI've been thinking quite a bit about your thought, Brian, about "is it really?" lately. I've really been struck by the differences between intro and upper level teaching. It seems, in the quantum case, that there are expts we can't let our students do, and, in lots of cases, math we can't wait for our students to learn. At the intro level that's true in some cases, of course, but people have put a ton of effort into ways of getting around that. I'm thinking of the pseudo-recent study of actual demos versus simulations in classes, for example.

ReplyDeleteSometimes I feel like I'm simply filling up my students tool bag to let them go tackle cool things. I don't feel that way as often when I teach intro courses because there are a lot more fundamental cool ideas at that level. Hmm, now I'm not sure if that last sentence is right. Oh well, let's pretend I said it on twitter ;). Seriously, though, these are great conversations to have and it's so much fun to find a community passionate about this stuff.

Hi Brian et al,

ReplyDeleteJust to be a little difficult, I am going to say that I think the essence of Brian's question is NOT a false dichotomy. For any given topic and any given student, the vocabulary is either introduced before the concept, or the vocabulary is introduced after the concept. There really is no middle group (I suppose you could shout "eigenvector!" in the middle of a student's matrix multiplication, but that is just silly).

In theory, I think that concept-then-vocabulary is probably better. In practice, I think that I tend to do vocabulary-then-concept. This is an indication of why I need to read these blogs.

Andy, Joss---prove me wrong (as usual).

Bret

I'm not sure that the intro/advanced distinction is so clear-cut. One thing some of my intro students really struggle with is the dot product: they just don't get it, see what it's good for, etc. I think this is because the inner product of two vectors is really pretty sophisticated mathematically, yet it's part of the "intro" curriculum. So, I guess one question is, are there good inquiry tools for introducing students to the dot product? What do they look like?

ReplyDeleteHere's how I think about it, Bret. There's some wiggle room in how you present a vocab idea. On one extreme you can point to something and name it, on another extreme you can carefully describe the context and details of the concept. For me, I like to work with students in the class to do the latter but with a common set of ideas to frame our conversation.

ReplyDeleteHere's an example: Calculus of variations. I'd like to be able to discuss in class the notion of minimizing a functional but I'd love to use pre-class time to make it clear what a functional is. I make a screencast saying "Imagine walking from here to there and at every step noting the height of the roof above you and its slope. After noting all that, let's, I don't know, add them and divide by 7. Who would care, right? Well, let me tell you that we can model roller coasters this way, just bear with me! We'll call our crazy calculation f(y, y'; x)."

Then, in class, we can tackle such things as, "how can value and slope be independent", or "how would we find the best one" or "why would we do this in the first place."

It seems to me that by defining the functional I save a ton of time in class. But in class is where we really tackle the calculus of variations.

I'll admit that this year I didn't quite do all that. Given the time constraits, I did both the def'n and then some euler equation stuff in the screencasts. However, that's mostly because I care way more about the particular functional (KE-PE) and that's where we spent a ton of class time.

Does that help?

@Bret - Like you, in practice I tend to do vocabulary-then-concept.

ReplyDeleteAndy's example touches on the point I was going to make. Often a concept involves multiple new vocabulary terms and sometimes refinement of previously defined terms. Defining some of the vocabulary can really help the subsequent discussion (Andy's functional) and other vocabulary is simply a name tag that you need to put onto it at the end so that when it comes up again in the future everybody can nod in agreement (Andy's calculus of variations).

The fact that vocabulary for a concept is often a collection of terms is part of why I see it as a continuum. We can choose when we start to use each of these terms, so they don't all have to come before the concept or all come after the concept.