Academic here. In my experience, many experts in any given area have massive amounts of intuition about their own area. But keep in mind that we do have to end up teaching outside the area. So it's safer to teach technicalities rather than a (most probably incorrect) intuition.
Some other issues:
- in many subjects, it is dangerous to work with the wrong intuition - they not only do not give a clear picture, they actually give you the false picture.
- even though math gets a bad rep, I think the only reason we are able to work with high-dimensional matrices and vectors is because we let go of geometric imagery at some point. Most people, including myself, have a hard time visualizing even 4 dimensions.
Sometimes I start with a geometric idea in 2D and see if it generalizes.
Other times I think of a 2D caricature that encapsulates some important operation but doesn’t necessarily relate to the actual idea. It's more like a heuristic visual than anything. A simple example might be if I’m thinking about projecting onto a linear subspace. The subspace is some convex blob in the plane in my mind’s eye, even though that’s not a linear subspace, it’s easier for me to conjure and the important part, the projection, remains intact. If the linear part was also important then I’d probably make it a line but I can’t think of a time I’ve done that.
Some concepts only made sense to me when, as you said, I stopped trying to visualize them. The most notable one to me was quaternions, because those are 4D, but even complex numbers only made sense when I stopped looking for physical intuition and realized they’re just a description of certain operations for points in the plane.
Now, I can suffer through a lot of abstract nonsense, but the worst classes I took were heavy on the abstract nonsense and light on answers to “why do we need all this machinery at all?” If I couldn’t imagine an application I cared about, I had a hard time figuring out what were the important concepts and how to fit them together.
When I was a couple years into my actual math education (about midway through vector calculus) I had a crazy moment when I was really struggling to understand some problem that required more manipulation than the 3 dimensional problems I'd been used to up until that point. My roommate was a postdoc and he told me the simplest, dumbest thing, which was, roughly "the math is a separate thing than the picture. You have to actually think about the math"
That one sentence blew my mind. Until then I had relied almost entirely on the geometric representation of a concept or a problem, and never knew to reach for the the actual concept.
I think about 75% of people check out of math completely super early; of the 25% that remain, 75% of those never quite "get" that math is a separate thing from graphs and charts. The rest are usually successful scientists, engineers, and similarly technical folk. (I won't venture my opinion on the difference between mere number manipulators vs. actual prov-ers of mathematics, but I think it's probably a similar ratio)
Same here, but not with calculus and geometry. My problem was with statistics: I had always been able to solve high-school level problems by intuition alone, and never needed the theory. That is, until 3rd-year university, when the problems became too complex to grasp by intuition alone. I had a really hard time that year, because I basically had to re-learn 4 years of foundational theory.
So I'd say that teaching intuition first is a dangerous path: students may fail to fully understand the theory if they can get by on intuition. And intuition is not a solid foundation to build on.
3/ Show how maths prevent the corner case of the intuition
When I study maths alone (even after a course), it's more like:
1/ Understand the technicalities of maths (very tedious because many teacher "leaves the details out"), without bothering much about the subject.
2/ After that huge effort, I try to play with the math to get an intuition of what it does. An intuition == a way to explain what it does in layman's terms, with drawings, physical/real examples, etc.
3/ Revisit the maths, and especially their corner cases (what happens if this denominator goes to zero, what if I cannot invert that matrix ?) and confront that to my intuition. The maths are the safest way to make sure you get your intuition right.
Some other issues:
- in many subjects, it is dangerous to work with the wrong intuition - they not only do not give a clear picture, they actually give you the false picture.
- even though math gets a bad rep, I think the only reason we are able to work with high-dimensional matrices and vectors is because we let go of geometric imagery at some point. Most people, including myself, have a hard time visualizing even 4 dimensions.