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Comment by dandermotj | Hacker News Reader

The more advanced mathematics I learn, the more I realise that in many cases I do not have a fundamental understanding of many topics, which leads me back to the beginning. More often than not, Better Explained is at the end of that path. If only this was how mathematics was taught!

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I never got calculus.

I mean, I got the ideas behind it and could talk with my professors about it, but I couldn't solve much of these equations.

It was a strange time at university...

I even helped friends to get these exams done, but I myself always failed. Luckily we only had one math lecture about calculus, the 4 others were about different stuff.

And it was always the fundamentals which got me. The whole rearranging an equation stuff all the school life is about totally eluded me. And this is what the calculus exams were about. At least the professors saw that I got the other math exams right, so they knew I wasn't an imbecile, haha

Calculus didn't really 'click' for me until I had to apply it in my college physics classes.

That's when it all came together.

Physics felt like the word problems you'd see in elementary school math. Ie, taking the concept from a theoretical 3+4=? to "Jim has three apples and Mary gives him four more, how many does he have?" And bringing that to calculus.

Yes. Many mathematical concepts didn't click for me in the math lectures.

The "aha"-moments I got in "computer graphics", "audio engineering", and "artificial intelligence"

I've a similar story! Working through the Nature of Code[1] is exactly the point in my life where differential equations and vector algebra suddenly made sense to me. Once I became the 'creator' I could see how this math described the real world.

[1]: http://natureofcode.com/

Do you remember what aspects of Calculus didn't click for you at first, that is when it was presented to you initially as pure mathematics? For example, was it the "idea" of a limit, or just not knowing at first why anyone (even mathematicians) would want to know the rate of change of a function?

More of the latter. And how to really think of derivatives beyond just the slope of a function. Or seeing an integral as a continuous sum of infinitesimals instead of just the area under a curve (which is only one way of visualising). And what does that 'dx' really mean inside the integrand. And where integration-by-parts comes from.

And Taylor expansions too. My high school calc teacher just put the equation on the board for Taylor series one day. No explanation where it came from, or how to think about it. When I asked if he could 'prove' it to give some context and make it seem less like black magic, the other kids in the class said "I'm sure the formula's right if it's in a textbook". But I didn't care about it being right, I cared about seeing where it came from.(and it's not that hard to derive actually). Taylor expansion is like the screwdriver of physics, it's used everywhere.

Other than calculus, matrices also didn't make much sense to me until physics. Eg, why is matrix multiplication so complicated, and what do these things actually represent? Especially opaque was the concept of eigenvectors and eigenvalues, until we used them prolifically in quantum mechanics.

Thanks for the response, I asked it out of genuine interest in how we discover new ideas, sometimes ignore or are unable to see their importance, adopt them and then share them. Even forget those ideas and perhaps have to re-discover them later. I personally empathize with the stuff you listed. In particular, I had trouble accepting at first why matrix multiplication is defined the way that it is.

As someone who feels "closer" at this point to applied math than pure math - largely, because it is literally more tangible - I admit a fascination with pure math, as if it were the human pursuit of universal beauty, in its ineffable, permanent and probably unattainable form. And that same pursuit of reason and order has brought renown & fame to some, but mania, loss to many. It's both the pride of humanity and terrifying, in equal measure. Like staring at a Van Gogh, and finally perhaps realizing how marvelous, impactful, insightful the statement the artist may have intended to make, yet from such humble circumstances, and with so few tools.

I am never comfortable with my understanding of a topic until I can teach it (or at least pass it on). There's just always an annoying, niggling feeling that if I were to be faced with a student who kept asking "why", I'd be a bumbling mess. I was the same as you for linear algebra. Everyone was always so happy to learn the rules, answer the questions like its a script and pass the exam. This frustrated me immensely. Sat down last year and started from the beginning, and I'm glad I did. Sometimes the key is to wipe the slate of existing knowledge clean and start again!

For me, the "why" came from the word (or story) problems -- the exact problems that most people really hated. There is one problem from calculus that really stuck with me, and made me really understand limits:

"You are on an island, which is 10 miles from the shoreline. You can row a boat 4 miles an hour, and you can run 7 miles an hour. You have a friend in need that lives on the shore, 6 miles down from the part of the (straight) shoreline that is closest to the island. Where on the shore do you need to land the boat to minimize your travel time?"

So obviously, aiming the boat directly to your destination minimizes your distance, but you are only going to be traveling at 4 mph. And if you aim straight to the shore you minimize your time in the slow boat, but your distance is longer. The visual solution is to plot the total time based on various landing points on the shore, then find which one is minimal. The exact solution is to make a formula that represents both legs of the trip with the landing point being your variable, take a derivative of the formula and solve for 0 (if I remember correctly).

I definitely had similar issues with learning math when I was in school. You mention that you sat down last year and started from the beginning... would you mind sharing a little more about where you started, that is what your beginning was, and what sorts of resources you used? I'm thinking of doing something similar and am looking for avenues to take.

Well funnily enough I started back at what a matrix really is. What does it mean to multiply them, add them, invert them and so on. The first good resource I used was Better Explained! I also took out some Linear Algebra text books from the library. First was Linear Algebra for Dummies (don't knock it!) and then an undergraduate text book. I've still a ways to go but it was a good start.

I agree, and would say that teaching others something forces you to look for weak points in your own understanding and firm them up. Teaching, for me, makes me think about the "why" more, which is something I always struggle with. Going through the motions is the easy part.

hehe...

Funny thing is, if people who learn about software engineering ask me "why" I often know the answers. But I wouldn't know how to start teaching someone this topic...

Linear algebra was one of those parts of math that I understood rather fast.

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