Contrasting 2D and 3D in my field (Discrete & Computational Geometry1) is essential. For example, every 2D polygon can be triangulated (with vertex-to-vertex diagonals), but not every 3D polyhedron can be similarly tetrahedralized.

I experience significant push-back when I move to 3D (with undergraduates in the US). Typical student reaction: "Whoa!" (Most have not had Calc III.)

Q. How do you ease the 2D$\rightarrow$3D transition in your teaching?

I use polydrons, but still it's a scary transition for many students. And I have to prepare them for the leap to dimensions $> 3$.

1 Handbook of Discrete and Computational Geometry. Chapman and Hall/CRC, 2017.

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    $\begingroup$ I love that you use polydrons. Do you also work on the idea that there are an infinite number of regular poygons but only a few regular polyhedra? $\endgroup$ – Sue VanHattum Feb 9 at 3:28
  • $\begingroup$ @SueVanHattum: Yes, that's a proof we do first intuitively, and then more carefully. $\endgroup$ – Joseph O'Rourke Feb 9 at 11:59

After teaching Calculus 3 for many semesters, your question certainly resonates. I think it is empowering to coach students in how to sketch surfaces in 3D.

Their tendency from curves in 2D is to first draw the $(x,y)$ axes, and then superimpose the curve. That's how I sketch. But for surfaces in 3D, it's best to first draw the surface, and afterwards, superimpose the $(x,y,z)$-axes.

Start with a hyperboloid of one-sheet, then progress to planes, spheres, etc.

At first blush, this is a trivial matter. But learning how to sketch surfaces builds confidence.

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(long comment) I think you just have to give them time.

  1. Solidify your algorithms and the rest of it within the 2-D world. Don't mention 3-D at all then. Lay a solid foundation on the tools themselves.

  2. Then move to 3-D and start by doing some refresher on solid geometry itself (for some first time as solid geometry gets very short shrift in schools versus all the SAS triangle stuff). Don't even do the computer crap, but just solid geometry. And I would do a first lesson that was just "names of animals in the zoo" (pyramids, prisms, platonic solids, etc.) Give them a homework or in class assignment to label the creatures. Then you can do a little stuff with vertices and like in another lesson (but still just solid geometry).

  3. Then do whatever computer stuff you are doing with the cells or whatever it is in 3 dimensions. And give this some time. And don't you dare mention 4-D. Keep focused on the immediate hurdle.

  4. Then go to 4-D. Show a little pictures of shapes and the like for the cool factor. Don't dwell on it. But for many it will be first time seeing the polytopes. Wiki has a good descriptive article. Maybe don't have to do as much remedial here as it's not so important they know all the names of the creatures here. But do a little orientation before getting into the calculational work.

  5. And then get into whatever math you do with them. And how to keep track of the stuff analytically (I guess with coordinates with 4 numbers). Stick to 4 at this time until it's down solid. No 5 or higher.

  6. Then generalize the methods already taught in 4 space to 5, 6 or whatever. Should be simple if they got the 4 down.

P.s. I find you very engaging and suspect your students love you. But I also kind of get the impression you are a little "hard" and don't do enough progression or expect too much in terms of preparation. Just a comment, not an insult. If it resonates great, if not, ignore. Of course a mature mathematician or very strong new student can handle multiple dimensions at first exposure. But I think this is not your demographic. So be much more step by step progressive and take the time needed. [I don't think there is some simple "aha" trick to get the higher dimension awareness. Need to take the time.]

P.s.s. I do think that hand drafting (especially isometric view, but even plan and sides) gives students a lot of practice in thinking 3-D. Unfortunately technical drawing has been rather deprecated in the modern curriculum. There are great YouTube videos on this stuff though. I don't see how you can use this insight, but just mentioning it because I do think it helps with thinking spatially. But also requires large amounts of time.

P.s.s.s. Just scanned the book. That looks really hard for a startout (even the intro) for kids who have only had frosh calculus and may not be top of the pack. Also, a handbook seems too long and too unsynthesized (multiple authors are a problem even for specialist encyclopedia tomes...would prefer to read Greenwood and Earnshaw to some cobbled together set of articles by 65 chemists) to be the right way to instruct new students. They need an easier, integrated approach. Then they can go buy that handbook if they care enough, or really if they work in the field.

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  • $\begingroup$ Thanks for your remarks. One point: I only linked to the Handbook of D&CG to indicate the area. I use a much more elementary textbook. $\endgroup$ – Joseph O'Rourke Feb 9 at 13:03
  • $\begingroup$ For the 15 hour question: Look at the test. Try to get some statistics/research on common errors. (If not, use your intuition...i.e. make a best guess.) Design something (lecture and practice) to hit common booboos. Don't reteach theory/concepts in some organic manner. You don't have time for that and it will be a turnoff. But you will have their attention if you say (and mean it) that you will horse them up for the test. Try to make the practice approximate the tests as much as possible. (If old exams are public, mine those directly. If not, mimic them.) $\endgroup$ – guest Feb 9 at 17:08

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