When I was taking linear algebra and calculus in university, I found that there was a lot of focus on deriving formulas from underlying principles, with the notion that this constituted a "fundamental" understanding of the mathematic concept. I got quite good at being able to derive formulas for anything, and did well enough to scrape by on my exams. However the concepts didn't really stick with me. Going through Khalid's site I quickly discovered I had a terrible intuitive understanding of mathematical concepts, almost embarrassingly so. Somehow, derivation from first principles doesn't quite capture intuitive insights for me, especially once I start worked at higher levels of abstraction removed from easily understood foundations (i.e. multi-dimensional vectors).
The two things that I found most helpful in relearning math is (1) building up a foundation for mathematical concepts through betterexplained's intuitive method and (2) turning it into code as soon as possible. For the latter, I have a side project that is a sort of platform to test all my various ideas, from city performance modeling, to procedural form generation, where I am constantly trying to rework or tweak with new math formulas. It's amazing how much more efficient and useful this is as a learning method.
Kalid from BetterExplained here, it's awesome to hear the approach is working for you. I had very similar reservations, I could derive many results (like a robot) but had little intuition behind e, i, pi, radians, etc. (let alone how they all came together in something like Euler's Formula).
My litmus test became: If I can't intuitively describe i^i (an imaginary number to an imaginary power) I don't understand it. I don't care if I can derive the equation 15 different ways. If I couldn't spit out some properties of i^i (positive or negative? Real or imaginary? Big or small?) after a glance then I knew I didn't know it. (Why can I spit out properties of 2^3 or 3^(-4) in a few seconds, but not i^i?)
Code is an excellent way to practice these ideas; the bugs in your logic correspond to the bugs in your thinking, and you see (very explicitly) where to correct them.
Kalid: you should really check out geometric algebra (a.k.a. Clifford algebra). It will give you a much deeper understanding of what i is and what the exponential function is, and how they generalize to higher dimensions and more complicated models, and it will help stitch together the weird inconsistent little fragments of understanding provided by imaginary numbers, quaternions, matrix algebra, Lie theory, differential forms, etc. into a more unified/cohesive model.
This is the model all high school and college students will be taught in 100 years, or perhaps even in 50 years, and it will prevent an enormous amount of confusion and misunderstanding. It’s already becoming the practical tool of choice in many geometric computing problems, and among niche groups of physicists.
(Or books might be better sources for going in depth. Search for New Foundations for Classical Mechanics, Geometric Algebra for Computer Science, Geometric Algebra for Physicists)
Unfortunately, university lecturers in many subjects, including mathematics, are often doing that job only because they have to as part of their main research position, not because they actually want to or have any aptitude for it. There are some great lecturers who are happy exceptions, but they are few and far between in my experience.
The sooner traditional universities and their ineffective, anachronistic teaching practices die out as the main form of tertiary academic education, the better. They could be replaced tomorrow by some combination of recorded lectures given by presenters who are actually good and in-person tuition given by people who can actually teach and care about their students' education, and I suspect absolutely nothing of value would be lost. The students would be both better educated and probably much better off financially, while the academics with a talent for research but not for teaching could make better use of their time and skills.
Problem: Who wants to spend a lot of time learning to teach when you don't really see this as more than a requirement to get that job? Which I completely understand, I don't blame the professors. Having tried a little bit of teaching myself, it's just very, very time-consuming to prepare yourself for doing it really well.
At university in the 1990s I preferred learning from (good) textbooks over most lectures, and today there's the Internet - I've actually gone back to learning with >50 courses on Coursera and edX over the last few years. So I think the problem has shrunk significantly since you can choose to learn from great teachers more easily than ever. But I'm a "pull" person when it comes to learning, meaning I don't even want someone to tell me what I'm supposed to know, I prefer to go out and look and select and get it myself.
It sounds like we're of a similar generation, but my experience was that unfortunately finding good textbooks for mathematics was also very hard. The average textbook was a dry, boring regurgitation of the standard definitions and results in a field and some associated proofs, with no added insight or motivation or context or example applications whatsoever. Many of them were written by the same academics whose lecturers were similarly bland and uninspiring.
Most of those academics could have done with learning about the ADEPT method we're discussing here and similar ideas. Unfortunately, they just had no interest in doing so. As you say, teaching well takes time, but it also takes a willingness to try to teach well.
I remember a particular meeting at the end of an academic year where the teaching representatives of the faculty at my university were seeking feedback from the undergraduate students. When one of the students boldly (but entirely fairly) asked why the presentation skills of most of the lecturers were so bad and why they weren't required to undertake training to improve when the university's teaching so fundamentally depended on them, the reply was essentially "We know and we agree, but they wouldn't accept it." In almost any other profession, the response to substandard performance of a key job function and refusing to undertake measures to improve would be getting fired.
These days, with students here in the UK paying thousands of pounds in fees every year on top of what we used to have, I can't imagine that official response would go down any better than it did for us, but as you say, these days there are more promising alternatives. This is why I think universities need to stop being the main tertiary academic education, at least in anything like their current form.
My problem is that I hate learning from video lectures. Do you know any MooC sites that don't require videos i.e. that present the material in written form (either chiefly or to duplicate lectures)?
Your first paragraph perfectly captures how I felt when I attempted to work through Linear Algebra Done Right, by Sheldon Axler. It's an awesome book, and the derivations and exercises show the beauty of maths.
But I had no mental model on which to hang the various results, and struggled to keep it all in my head, particularly after a break of a few weeks. IIRC, I stopped around the point when eigenvectors are introduced.
Oh, and please share examples of your coding+learning approach :)
I'd love to seen an ADEPT method explanation of Laplace Transforms.
I got through most of math fairly easily by having a mental model of what was going on and could always check that I was on the right track as it made sense in my mental model. However, when I got to Laplace transforms I never figured out how to visual what that meant. Everything collapsed into transform into the magical space where you can do some things easier and then you can transform out into a new place. I could never be sure of how I got from a to b without a tedious examination of every step to ensure I applied the rules correctly.
I'd love to have a mental model for Laplace Transforms.
As a generalization, how does one explain things where no good mental model exists?
Kalid from BetterExplained here, my quick intuition:
The Fourier Transform breaks a signal into its "cycle recipe" (what circular paths are present?). The LaPlace Transform breaks a signal into its "spiral recipe".
Circles are made from a type of exponential (given by e^ix), and spirals are the more general version, where the radius changes (if s=a+bi, then e^is = e^a * e^bi, aka a circular path where the radius changes exponentially).
The LaPlace transform actually deals with decaying spirals (negative s) -- why is this useful?
Well, perfect circles that never decay (the Fourier Transform) are nice for analyzing audio samples, as in music. (Repeated drumbeat throughout the song.)
Decaying spirals model things in the real world, where friction, etc. dampen the signal over time. The Laplace transform can cleanly represent this scenario, whereas you need an infinite number of cancelling terms in the Fourier Transform to represent the "decays over time" setup.
Engineering applications prefer Laplace, compression mechanisms may prefer Fourier. (Separately, Laplace/Fourier make differential equations easier to solve by writing functions in terms of exponentials, which are easy to derive/integrate. The Laplace transform is more general and powerful in this regard, since it can handle any rate of decay, including 0. The Fourier Transform is embedded within the Laplace.)
Just some quick thoughts from an amateur on this :).
> As a generalization, how does one explain things where no good mental model exists?
Papert et al. are convinced that computers ought to help with this problem. The idea is that visualizations, even if they are interactive, aren't very useful. Instead, the learner should iteratively build and play with a simple version of a model (a microworld) until the learner gets into the full-fledged model. Preferably by programming the model itself.
It's sad that this idea never caught much traction beyond educating children.
Great article! This is the most intuitive description of imaginary numbers I have ever read. I'm going to try the same technique with Software Defined Networking (SDN), which has been on my "must figure it out" list for a while. :)
It seems the main point being made by the article is that even if you are going to teach principles, it can be more effective to start by giving some context and motivation before getting into the detailed technical explanations. Deriving results from first principles may well result in better understanding than simply learning results and proofs by rote, but it's still useful to have some indication of why those principles might be useful and relevant as well.
Excellent! I always think that people, like myself, that work with somewhat abstract and/or complex concepts need to improve our ability to explain them. Really happy to see a concrete method to do this.
I see one thing missing from the ADEPT method, which is "what's the story?" in the literal sense. Storytelling is one of the most powerful learning techniques and even a self-learner can create a story about the concept to be learned. The story doesn't need to be picturesque; even a dry story is much more useful than just the facts and description.
Some of the examples they provide are stories ("Academic progress on imaginary numbers took off only after the diagrams were made!"), but the notion of teaching with a story isn't made explicit in the steps of the method.
Fourier transformations clicked for me from your previous articles, this outline really shows the skill of "sharpening" rather than "building" knowledge. Helpful for young kids to adults alike.
I had the same experience. However later when I encountered new properties of fourier transforms, such as the convolution theorem, fast fourier transform and the fourier transform of boolean functions, I felt my acquired intuition no longer fitted. Did you also experience this? If so, how did you get past it to that "next level"?
Cool. I am currently learning something new and something that have stuck with me is the "if you can explain it, you know it". So while I am learning I am writing a set of instructions about what I am learning and some of what I have written down seems to resemble parts of this article.
I agree, if you want to learn something, you have to spend your time and focus on it. There is really no other way. I can accept it can be more digestible for many if it is packaged nicely in method X. Helping people learn is a very noble goal, but it should come with a fair warning there is no shortcut in understanding.
The two things that I found most helpful in relearning math is (1) building up a foundation for mathematical concepts through betterexplained's intuitive method and (2) turning it into code as soon as possible. For the latter, I have a side project that is a sort of platform to test all my various ideas, from city performance modeling, to procedural form generation, where I am constantly trying to rework or tweak with new math formulas. It's amazing how much more efficient and useful this is as a learning method.