# Multivariable Calculus Lecture Ideas

I am teaching a course in multivariable calculus this semester. We are covering the basics about $\mathbb{R}^n$, including dot products and cross products, curves, and quadric surfaces. After that we learn differentiation for functions $f:\mathbb{R}\rightarrow \mathbb{R}^n$ and partial differentiation for functions $f:\mathbb{R}^n\rightarrow\mathbb{R}$. We finish off with Lagrange multipliers.

My program so far has been to lecture on a chalkboard while they take notes. I am not really keeping their attention. I was wondering if anyone had any ideas to involve them more in class. We are in a computer lab, and have access to Maple. Your answer need not be computer based, but I have computers available.

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How big is the course? Why are your students taking the course? What are your students interested in? Based on their homework, what do they seem to be weak or strong in? What makes you say that you're not keeping their attention: is it that they are too chatty or that they are silent and unresponsive? If you're teaching face-to-face, your best bet is to start troubleshooting and you have an awful lot of ways to to find out what could be improved. – stankewicz Sep 19 '12 at 21:00
Speaking only for myself, if I was taking a 10-week course and this was all that was covered, then you would not be keeping my attention simply because the material doesn't seem like enough to fill a 10-week course. This gets back to stankewicz's question about whether your students are weak or strong at this material. – Ben Crowell Sep 20 '12 at 1:58
@stankewicz @Ben Crowell The course is twenty students. Most students are some kind of science major. They are unresponsive. It is an early morning class as well. – Joe Johnson Sep 21 '12 at 19:40

You could try to implement Eric Mazur's Peer Instruction. There is a great video about it here, and here are two articles discussing how it works. The latter has links to statistical evidence that students learn better with this method (for me, the jury is still out). I guess the basic idea is that classes start by asking students a multiple choice question, and they vote on it using clickers. It should be something controversial, about a conceptual point rather than a numerical computation. After the first round of voting, students talk to each other and try to convince one another of which is the correct answer. The class then votes again. The idea is to spend class time on absorbing the material rather than hearing it for the first time. It's expected that students will have done all the reading on their own before class. There are links on Mazur's website which are designed to help professors implement this. A computer lab should have everything you need.

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BTW: If you don't have clickers, you can let them poll using their phones at www.polleverywhere.com. If you have under 40 students this is free. – Jon Bannon Sep 19 '12 at 19:52
If implementing Mazur's technique, I recommend not following his advice to go whole hog, eliminate all lecturing immediately, and do nothing but conceptests for every lecture period. It takes time to get a feel for using the technique successfully, debugging the questions you want to use, etc. Also, the pedagogical literature in physics shows that any "interactive engagement" technique tends to work equally well when implemented well -- there is nothing magic about Mazur's technique in particular. My experience is that students are grateful if I let them do a variety of things. – Ben Crowell Sep 20 '12 at 2:56

Here is another answer, based on the way your question was phrased. It seems you're looking for ways to effectively use the computers in the room. David Perkinson from Reed College is great at this, and generally lectures from slides with images created in mathematica or maple. His slides can be found here and I imagine he would send you tex code or image files if you wanted. One very clever idea he uses is to leave blanks in the versions of the slides he gives to the students so they have to write on them. That way they have to listen actively, but also get to see things in a way which a textbook or blackboard might not be able to show.

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When introducing the cross product, one thing that always catches my students' attention is a demonstration of precession. We have a bike wheel with a heavy solid rubber tire. (You may be able to borrow one from your physics department.) I get it spinning with its axis in the horizontal plane, and then suspend it by a rope from one side. They are gratifyingly surprised when they see it precess in the horizontal plane rather than flopping. This connects to torque expressed as a vector cross product.

Another example I like to do is to have them imagine a spinning cylindrical space station and lead them through a series of thought experiments leading them to figure out the rules by which the centrifugal and Coriolis forces operate. E.g., a person standing on the deck releases a ball. In the frame of the stars, the ball travels straight. What appears to happen in the rotating frame? Once they see that the centrifugal force operates like a radially directed gravitational field, I do examples that show the existence of a velocity-dependent force. E.g., what happens if the person standing on the deck throws the ball opposite to the direction of rotation, such that he exactly cancels its motion in the frame of the stars? What if a ball is released on the axis? What if the ball is released on the axis with some small radial velocity? Based on this sort of thing, it's fairly easy to get them to conclude that the Coriolis force is an odd function of the ball's velocity, and also an odd function of the angular velocity of the space station. There is only one good way to get a vector-valued function with these properties, so it must be $\mathbf{F}=c\mathbf{\omega}\times\mathbf{v}$. They enjoy imagining the fun and strange things that happen on the space station, and it's a cool example where the uniqueness of certain mathematical operations makes it possible to determine the answers to problems without grotty calculations.

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I used this more for integration and change of variables, but still you might be able to use the idea.

I took my TA section outside to a (ground level!) skylight which was a mostly transparent spherical cap with some gridlines on it. It was sunny, and you could peer down to see the shadow of the lines on the floor below. I used this as an example of how projection might change the area, and how acting on each area element needed to be considered in doing the integration. You might be able to come up with a similar visual aid for the chain rule.

Gerhard "Ask Me About System Design" Paseman, 2012.09.19

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