Wow... you left me wondering why signal processing teachers in college don't teach Kalman Filters with this simplicity. I know mathematical concepts are best taught mathematically, but that does led to information loss for those who don't have the background requisites.
I used to teach Discrete Cosine Transform and Wavelet Transform through images alone and would always find this teaching method of "intuition before rigor" work better than the other way around.
> why signal processing teachers in college don't teach Kalman Filters with this simplicity
> those who don't have the background requisites
Any college student studying signal processing should have the background prerequisites.
That said, it is easy to forget fundamentals. I have a couple theories for why professors don't use the intuition-before-rigor approach.
1) The professors themselves do not have great intuition but rather deep expertise in manipulating numbers and equations. Unlikely theory, but possible.
2) Professors do not generally get rewarded for their teaching prowess. And breaking mathematical concepts down to an intuitive level requires quite a lot of work and time better spent writing grant proposals or bugging your doctoral students. Cynical theory but probably true in many cases.
3) Once you understand the math, it is so much easier than the intuitive approach that the lazy human brain will not allow you to go back to "the hard way". I like to think this is the primary driver of skipping an intuitive teaching approach.
I would say #3 applies much more broadly than mathematical education. It is the difference between expertise and pedagogy. That is to say being an expert in a thing is a completely different skill than teaching others to become competent at the thing. Say you want to improve your golf game with better drives - should you learn from the guy at the range who hits the ball the farthest? Probably not. You should figure out the guy who has improved his drive the most. Eg, the guy who started out driving 100 yards and now consistently hits 300 is better to learn from than the guy who is hitting 350. (Credit Tim Ferriss for driving this concept into my head).
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.
>3) Once you understand the math, it is so much easier than the intuitive approach that the lazy human brain will not allow you to go back to "the hard way". I like to think this is the primary driver of skipping an intuitive teaching approach.
I found that developing a intuitive understanding was required for me to remember something. I could then remember the Math. The Math is nice to give a mathematical understanding but I found the intuitive knowledge much more important.
I majored in mathematics in college and was frequently irritated by the "intuition before rigor" approach, so I have a personal point of view on this.
First, intuition is an unreliable indicator. While solving a problem, or searching for something new, you are guided by intuition, but it isn't infallible. Sometimes you get the feeling that something will work, get very excited, and then you try to make it rigorous and it falls apart. There is a whole category of cranks who think they have created a perpetual motion machine or a theory of quantum gravity and don't know enough math to figure out that they haven't, so they live perpetually in that initial feeling of excitement. So, intuition is exciting, but you develop the instinct of not letting intuition get too far ahead of rigor without checking in. And vice-versa, if you go too far with calculations and no longer have a feeling for where it's going, you start to feel like you should stop. Lectures have to respect this yoking of the two together. If you just give me a bunch of calculations, my brain will revolt because the calculations aren't motivated. Likewise, if you give me a bunch of intuition without rigor, my brain will revolt because the intuition isn't grounded.
Second, intuition develops through practice. If you want me to have an intuitive ability to use a mathematical idea, don't hide it from me! Like, if you want to teach a kid to play baseball, there's a lot of talking involved, but maybe hand them a ball first and let them hold it in their hands while you talk, feel the size and texture of it. Even better, just let them try to throw it, and work up to playing catch. The motivating, big-picture idea of baseball includes things like the thrill of winning, camaraderie, suspense, and post-game snow cones, but their eyes are going to glaze over if you start like that. You get a little kid hooked by putting a ball in their hands and letting them throw it. Some mathematical ideas are fun to build from the ground up, and you can discover the snow cones and the cheer of the crowd later.
The third factor is irrational, but deeply felt for some people. Even though we know and experience every day that rigor and intuition exist equally and symbiotically, each rather pointless without the other, it is expected to give pride of place to intuition, as a way of catering to people who didn't enjoy math in school or are insecure about their abilities. "The really important thing is the idea, the intuition... calculation closes the mind and dulls imagination... the real geniuses have creative breakthroughs, and we are never short of less imaginative people to clean up the unimportant details afterwards." That's how, as someone studying STEM, you learn to talk to people outside of STEM, and it's chronically irritating and demoralizing that you have to denigrate an important aspect of what you do. So you tend to be hypersensitive to that chauvinism creeping into a "safe space" like a class in a technical subject. Of course each STEM subject has its own approach to mathematical rigor (mathematicians and physicists notoriously don't see eye-to-eye) but we all appreciate the value of it.
Also I think your third point is important. If you only get the math, you can use it, and the intuition will come. If you only have the intuition, you're in a much harder position to get going and develop the math.
Totally agreed. One of the favorite professors in college would introduce a theorem, and instead of jumping into proving it, would first show us useful applications of the theorem, and then prove it afterwards.
I can _only_ learn top-down. Facts without context are much less interesting and much more difficult to internalize. The incentive to understand, inspired by said context, is a prerequisite to mustering the motivation needed to to truly learn with sustained focus without losing interest. My ADHD brain will stop cooperating otherwise, and I’ll find myself daydreaming against my will!
This is exactly what I did as a grad student when I taught classes for my prof. She preferred bottom up and I preferred top down. I wanted to give people a reason for learning what they were getting ready to learn rather than "let's start from first principles" that almost all my professors wanted to do.
I prefer the exact same approach for learning to play board games too. Your 20 minutes of rules explanation will go completely over my head unless you start with "the goal/win condition of the game is to get more points/reach the end faster/collect all the macguffins".
Aw god, Yes, Yes, Yes! I understand applications not abstractions so show me it's value and I'll understand it immediately, or give me a massive motivation to understand it if I don't. If they'd taken that approach at my schools and university I'd have been so much better off.
I used to teach Discrete Cosine Transform and Wavelet Transform through images alone and would always find this teaching method of "intuition before rigor" work better than the other way around.