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Example 2: (Things that aren't graphs)

Another problem geometric thinking faces is that modern math often seems to regard pictures as not being proofs, even if they are more convincing, so there is a bias regarding how to choose to spend class time. Let's say you want to differentiate $x^3$. You can draw a cube, and a slightly larger cube, and then look at the difference of the cubes and subdivide it into a bunch of small regions, three larger slabs taking up most of the volume. Algebraically, this subdivision corresponds to multiplying out $(x+h)^3$; collecting the terms uses the commutativity, which corresponds to rotating the various identical pieces. It is no different to write this proof out algebraically, the difference is that the algebraic one is a "proof" but the geometric one is.. not? Even if it's more convincing. So it's like it's the picture is only there for culture.

Maybe I have the lecture time to teach thatboth, I will. But I would like to go farther than that. When I differentiate the cube root function, the same cube appears and I go through it again if I feel like it just to convince myself of the truth. Actually, every time I ever use the product rule I always picture the same rectangle with a slightly larger rectangle. My point of view is that one important "definition" of multiplication is in terms of areas, and that a linear function is not necessarily a graph. When you think of a linear function, you should also picture things like rectangles, sectors, similar triangles like the kind that come up when "proving" basic differentiation formulas. Differentiating the integral may seem like a magical trick, but it's really just a continuation of the point of view that multiplication can look like an area/volume and differentiation means taking a small change in the input.

Now, I'd like that point of view to be absorbed, but it's not exactly in the textbook, or completely consistent with what students' other teachers taught them. It's hard to go against the idea that "you should think graphically" -- if I ever think about the sine or tangent function now, it might be the area of a triangle, it might be the length of some vertical line segment, but it's basically never using the graph, which contains basically no additional information. If I have more than one shot at it, I'll try to explain both, but is it really of service to go around saying all the time why graphs aren't the end-all-be-all?

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People have mentioned examples which are hard to share due to some kind of prerequisites. Here's one: I learned PDE from a professor who, in his mind, was always thinking about distribution theory, but officially could not talk about it until after he covered the material relevant to the exams. In distribution theory, whenever you see an integral over a domain $\int_\Omega u(x) dx$ you actually picture the characteristic function $\int \chi_\Omega(x) u(x) dx$ or $\int H(f(x)) u(x) dx$ if $f$ is a defining function for $\Omega$ and $H$ is a heaviside function. From this point of view, you imagine that all functions are smooth and compactly supported (or you can imagine their approximations), so that if you integrate by parts on $\int \chi_\Omega \nabla u(x) dx = - \int \nabla \chi_\Omega u(x) dx = \int \delta(f(x)) \nabla f(x) u(x) dx$. The boundary terms come when the derivative hits the characteristic function. Same thing for Stokes' theorem, Gauss's divergence theorem. It's pretty handy to compute this way.

For a little while this was all I understood until I later found out what was going on. The limit of difference quotients of $\chi_\Omega$ is clearly supported on the boundary of $\Omega$ and it's clear, especially if you picture an approximation, that $\nabla \chi_\Omega$ points in the direction of increase of $\chi_\Omega$ -- i.e. the inward normal. More simply: there are two points of view -- if you were to take difference quotients of $u$, you use a Lagrangian point of view in which the point at position $x$ moves in the direction $i$, and you observe a change in $u$ between those points; instead, you can take an Eulerian point of view, (where the adjoint difference quotients go on the characteristic function) and you can instead look at movement of the region with $u$ fixed.

Until I understood this point of view in a simpler way, it would not really be sensible to explain it to others. But now I know that giving a watered down version of the same proof when "proving" the fundamental theorem of calculus / Gauss's divergence for a calculus class in fact does not lose any key ideas (except for technicalities like how you need the mean value theorem to ensure the difference quotients are bounded). Of course, I would also talk about characteristic functions to any math student, since it is a nice point of view.

By the way, in the calculus of variations, when your $u(x) = L(x, \phi(x) )$ is a Lagrangian and $\phi(x)$ is a solution to the Euler-Lagrange equations, and you take difference quotients using the flow of a vector field whose flow preserves the Lagrangian (a "symmetry"), you end up with Noether's theorem through only this one variation (there are only boundary terms in what I called "the Lagrangian point of view" because you vary through a family of solutions except at the boundary). So it's also a nice way to prove conservation laws in one swoop.

My point: for a little while, distribution theory seemed like a magical theory with prerequisites that made it unexplainable in everyday talk, but once I really understood the ideas I could usually discard the vocabulary (actually, the whole theory can often be replaced by cutoffs, partitions of unity, Taylor expansion, and changes of variable -- although I still think it's great to learn). I suspect that this phenomenon is not uncommon for elementary applications of "fancy" mathematical theories. I believe that often once one has a more basic understanding, one can throw away the new words but still fully reveal the ideas (but maybe that's completely due to my own background). People here have talked about Feynman -- he was good at doing this in the context of physics. If you watch his (outstanding) lectures on Project Tuva you will see more or less the proof of Noether's theorem about which I just wrote.

A second point:

Another thing I think happens to me is that I feel some pressure not to convey just how often I rely on geometric modes of thought, especially when they go against the usual way of explaining things, or the background of a typical student, and are not completely necessary.

Example 1: When you row-reduce a matrix, you make a bunch of changes (most importantly some "transvections") in the basis of the image space until a few of your basis vectors (say $v_1 = T e_1, v_2 = T e_2$) span the image of the matrix $T$. When you picture the domain of $T$ foliated by level sets (which are parallel to the null space of $T$), you know that the remaining basis vectors $e_3, e_4, ...$ can be translated by some element in the span of $e_1, e_2$ (i.e. whichever one lies on the same level set) in order to obtain a basis for the null space. Now, this is how we visualize the situation, but is it how we compute and explain? Or do we just do the algebra, which at this point is quite easy? If the algebra is easy and the geometry takes a while to explain and is not "necessary" for the computation, why explain it? This is a dilemma because once algebra is sufficiently well-developed it's possible that the necessity of (completely equivalent) geometric thinking may become more and more rare; and algebra seems to be more "robust" in that you can explore things you can't see very well. But then, when students learn the implicit function theorem, somehow I feel like having relied on that kind of foliation much more often would help understand its geometric content. Still, even if it's in your head and very important, are you going to draw a foliation every time you do row operations? We know the geometry, know the algebra, but it would take a while to repeatedly explain how to rely on the geometry while executing computations.

Example 2: Another problem geometric thinking faces is that modern math often seems to regard pictures as not being proofs, even if they are more convincing, so there is a bias regarding how to choose to spend class time. Let's say you want to differentiate $x^3$. You can draw a cube, and a slightly larger cube, and then look at the difference of the cubes and subdivide it into a bunch of small regions, three larger slabs taking up most of the volume. Algebraically, this subdivision corresponds to multiplying out $(x+h)^3$; collecting the terms uses the commutativity, which corresponds to rotating the various identical pieces. It is no different to write this proof out algebraically, the difference is that the algebraic one is a "proof" but the geometric one is.. not? Even if it's more convincing. So it's like it's only there for culture.

Maybe I have the lecture time to teach that. But I would like to go farther than that. When I differentiate the cube root function, the same cube appears and I go through it again if I feel like it just to convince myself of the truth. Actually, every time I ever use the product rule I always picture the same rectangle with a slightly larger rectangle. My point of view is that one important "definition" of multiplication is in terms of areas, and that a linear function is not necessarily a graph. When you think of a linear function, you should also picture things like rectangles, sectors, similar triangles like the kind that come up when "proving" basic differentiation formulas. Differentiating the integral may seem like a magical trick, but it's really just a continuation of the point of view that multiplication can look like an area/volume and differentiation means taking a small change in the input.

Now, I'd like that point of view to be absorbed, but it's not exactly in the textbook, or completely consistent with what students' other teachers taught them. It's hard to go against the idea that "you should think graphically" -- if I ever think about the sine or tangent function now, it might be the area of a triangle, it might be the length of some vertical line segment, but it's basically never using the graph.

Also, while I can express the pictures in my head one at a time, the fact that I repeatedly, repeatedly see this pictures is something that I feel is harder to express. After all, can't you just do algebra and get through this stuff more quickly? The algebra is "easier" too; it takes up less space.