I totally agree. There are certain explanations out there which have orders-of-magnitude differences in ease of understanding (seeing i as a rotation, seeing radians as the "mover's perspective", seeing integrals as "better multiplication"). My personal mission is finding these aha! moments which unravel years of confusing symbol manipulation. Calculus is definitely more easily understood with infinitesimals vs. limits (ask any physics major or engineer).
Personally, I'm looking forward to a world where the very best explanations / analogies can bubble to the top. It's ridiculous that 200+ years after Calculus was invented, we still teach it poorly, and nearly everyone struggles.
Rigor & Intuition have a delicate balance in math. I see it as language: children can speak fluently, even if they don't know the "rigorous" rules of grammar & spelling [which are left to linguists]. I suspect the reason most adults have trouble learning languages is because they try to start from rigor (vocab lists and grammar structure) vs. absorbing an intuitive notion of what's going on (and later refining with rigor, "me want food" => "I want food").
>Personally, I'm looking forward to a world where the very best explanations / analogies can bubble to the top.
Please do keep in mind that there isn't one best explanation per subject, just a "local maximum" explanation that is best to a class of people sharing a similar way of thinking. Personally, the normal explanation of "representing (approximating) a complex cyclic function as a linear combination of complex trigonometric functions" made the most sense, while your explanation reads like a convoluted mess of analogies. Therefore, try to have a few completely different explanations per topic, rather than just the one which makes most intuitive sense to you.
Definitely, appreciate the feedback here. It's easiest to share the explanations that come to me, but I love finding a few different ways to look at things, and as they emerge I like to include them.
As a simple example, here's the formula for adding the numbers 1...n:
The explanation I was originally given (pair the first and last items, and count the pairs) seems gnarly because you have even/odd issues, off-by-one errors, etc. There are others which click better.
I agree here. The solution for me on not quite getting concepts in calculus and engineering was to delve into greater abstraction. When you get linear algebra, you understand the inverse function theorem in analysis. When you understand Hilbert spaces, the Fourier series makes a lot more sense.
A lot of the simple analogies and explanations I can use to talk about determinants or Fourier transforms now I can only conjure because of the complex study taking years to soak through.
"Calculus is definitely more easily understood with infinitesimals vs. limits (ask any physics major or engineer)."
I don't think this is true, and suffers from the same fallacy as the opposite statement that limits are easier to understand than infinitesimals.
For example, I've always found limits very easy to understand and infinitesimals confusing, even before I was asked to work with them rigorously using epsilons and deltas. I'm also a math major. In the physics classes I took, I sucked at mechanics (like, almost-failed sucked) but love relativity and QM.
Talk to me in terms of Lie algebras and invariant subgroups and I'll be ten steps ahead of you. I realize that's not usual, I'm just pointing out that there's no absolute "easiest way" to understand something.
Warning: personal theories of learning and pedagogy ahead, stated with more certainty than warranted.
I think a key to understanding is presenting the same idea using multiple models/representations. The learner already has some picture of the idea you're trying to explain in their head. The picture might be confused and poorly formed, but there's some version of it nonetheless.
I'd say, for that learner the "easiest way" to understand something is to find an accurate picture of the idea you want to explain and relate it to an idea the learner already understands clearly.
I have a picture I want to put into your head. Your version of that picture is fuzzy and confused. I need to relate the clear picture I want to give you to clear pictures of other ideas you already have in your head.
Ceci n'est pas une pipe, the map is not the terrain, the signifier is not the signified, etc.
Great article!, I´ll say it is very easy to follow till the spike part. From there is it possible to follow what you are explaining but not as easily as before. Maybe is because you are introducing notations that are not clear yet. I don´t know. But keep at it, really great work!.
I totally agree. There are certain explanations out there which have orders-of-magnitude differences in ease of understanding (seeing i as a rotation, seeing radians as the "mover's perspective", seeing integrals as "better multiplication"). My personal mission is finding these aha! moments which unravel years of confusing symbol manipulation. Calculus is definitely more easily understood with infinitesimals vs. limits (ask any physics major or engineer).
Personally, I'm looking forward to a world where the very best explanations / analogies can bubble to the top. It's ridiculous that 200+ years after Calculus was invented, we still teach it poorly, and nearly everyone struggles.
Rigor & Intuition have a delicate balance in math. I see it as language: children can speak fluently, even if they don't know the "rigorous" rules of grammar & spelling [which are left to linguists]. I suspect the reason most adults have trouble learning languages is because they try to start from rigor (vocab lists and grammar structure) vs. absorbing an intuitive notion of what's going on (and later refining with rigor, "me want food" => "I want food").