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Ask HN: How did you learn math notation?

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Re: Ask HN: How did you learn math notation?

#31

Post by swframe2 »

Naively, I would say the following:

1) Search youtube for multiple videos by different people on the topic you want to learn. Watch them without expecting to understand them at first. There is a delayed effect. Each content creator will explain it slightly differently and you will find that it will make sense once you've heard it explained several different times and ways.

I will read the chapter summary for a 1k page math book repeatedly until I understand the big picture. Then I will repeated skim the chapters I least understand until I understand its big picture. I need to know the terms and concepts before I try to understand the formulas. I will do this until I get too confused to read more then I will take a break for a few hours/days and start again.

2) You have to rewrite the formulas in your own language. At first you will use a lot of long descriptions but quickly you will get tired and you will start to abbreviate. Eventually, you get the point where you will prefer the terse math notation because it is just too tedious to write it out in longer words.

3) You might have to pause the current topic you are struggling with and learn the math that underlies it. This means a topic that should take 1 month to learn might actually take 1 year because you need to understand all that it is based on.

4) Try to find an applied implementation. For example photogrammetry applies a lot of linear algebra. It is easer to learn linear algebra if you find an implementation of photogrammetry and try to rewrite it. This forces you to completely understand how the math works. You should read the parts of the math books that you need.

Re: Ask HN: How did you learn math notation?

#32

Post by gspr »

I hear this question asked quite often, particularly on HN. I think the question is quite backwards. There is little value alone in learning "math notation", even ignoring what many people point out (there is no one "math notation"). "Math notation", at best, translates into mathematical concepts. Words, if you will, but words with very specific meaning. Understanding those concepts is the crux of the matter! That is what takes effort – and the effort needed is that of learning mathematics. After that, one may still struggle with bad (or "original", or "different", or "overloaded", or "idiotic", or…) notation, of course, but there is little use in learning said notation(s) on their own.

I've been repeatedly called a gatekeeper for this stance here on HN, but really: notation is a red herring. To understand math written in "math notation", you first have to understand the math at hand. After that, notation is less of an issue (even though it may still be present). Of course the same applies to other fields, but I suspect that the question crops up more often regarding mathematics because it has a level of precision not seen in any other field. Therefore a lot more precision tends to hide behind each symbol than the casual observer may be aware of.

Re: Ask HN: How did you learn math notation?

#33

Post by NamTaf »

ColinWright wrote: post #27
NamTaf wrote: post #24

Earlier quoted context omitted.

I feel you're coming at this without appreciating your body of prior knowledge. Intended or not, your statment "But that can't be it, because just two lines above it we have..." assumes a whole lot of knowledge. You and I both know that it reads as one term, but for someone unfamiliar with calculus but exposed to algebra they are drilled to understand separate graphemes as separate items, because the algebraic 'multi…

I agree completely that I'm coming at this with a lot of background knowledge, but if I'm reading in an unfamiliar field and I see a symbol I don't recognise, I look in the surrounding text to see if the symbol appears nearby. As I say, "Δt" appears immediately above ... that's a clue. As you say, it's drilled in at school that everything is represented by a single glyph, and if these are juxtaposed then it means mul…

I absolutely agree with your comments regarding teaching the underlying approach to digesting a paper. You definitely raise good points, especially the 'one true meaning' comment. I should state that I'm not discounting the value of your point, especially given this clarification, however I guess that when I reflect on my experience in my time learning this, the time I best learnt was via initial expalnation, then worked example, then customary warning of corner-cases and here-be-dragons.

e: I also think, on reflection, that a signfigicant part of your ability to grok a new paper per your comments is your comfort in approaching these concepts due to your familiarity. Think of learning a new language - once you have a feel for it, you're likely more comfortable exploring new concepts within it, however when you're faced with it from the start you probably feel very lost and apprehensive.

I feel that understanding calculus is a fairly fundamental step in the 'language of maths', teaching that symbols don't necessarily represent numbers but can represent concepts (e.g. delta being change). This isn't something you encounter until then, but once you do you begin to understand the characters associated iwth integrals, matricies, etc. in a way that you may not have previously with algebra alone.

Re: Ask HN: How did you learn math notation?

#34

Post by gspr »

rullopat wrote: post #19
ColinWright wrote: post #14

Earlier quoted context omitted.

You say "There's a formula with a triangle ..." without telling me where. That's not real helpful, and you're making me do the work to find out what you're talking about. If you want assistance to get started, you need to be more explicit. However, I have done that work, so I've looked, and in the second column of page 210 there's a "formula with a triangle": t_c = 5 \middot 10^{-5} \sqrt( V / Dt ) ... where the "D"…

Sorry I didn't mean to make you work for me, but it's a PDF and I didn't know how to explain better the position (maybe I should have told you the first formula on page X). For you it was a D, for me it was a triangle and I didn't get the meaning of that Dt. Maybe it's just a too advanced paper for my knowledge.

I'll use this as an example for the point I'm trying to make in my comment https://news.ycombinator.com/item?id=29341727

Please don't take this the wrong way. It is not meant to be demeaning, and it is not meant to be gatekeeping (quite the contrary!). But: If you do not know what a derivative is, then learning that that symbol means derivative (assuming that it does, I have not actually looked at what you link to) will help you next to nothing. OK, you'll have something to google, but if you don't already have some idea what that is, there is no way you will get through the paper that way.

I hope you take this as motivation to take the time to properly learn the fundamentals of mathematics (such as for example calculus for the topic of derivatives).

Re: Ask HN: How did you learn math notation?

#35

Post by db48x »

solmag wrote: post #5
solmag wrote: post #4

Practice, just like you learned programming. "The Context" gives you the meaning for the notation, sadly. You have to kind of know it to understand the notation properly.

You can also get sufficiently angry and just write out linear algebra books and what not in Agda / Coq / Lean if it pisses you off so much (I've done a bunch of exercises in Coq)

I like the approach they took in Structure and Interpretation of Classical Mechanics, where the whole book is done in Scheme:

    (define ((Lagrange-equations Lagrangian) q)
      (- (D (compose ((partial 2) Lagrangian) (Gamma q)))
         (compose ((partial 1) Lagrangian) (Gamma q))))

Re: Ask HN: How did you learn math notation?

#36

Post by wrycoder »

rullopat wrote: post #19
ColinWright wrote: post #14

Earlier quoted context omitted.

You say "There's a formula with a triangle ..." without telling me where. That's not real helpful, and you're making me do the work to find out what you're talking about. If you want assistance to get started, you need to be more explicit. However, I have done that work, so I've looked, and in the second column of page 210 there's a "formula with a triangle": t_c = 5 \middot 10^{-5} \sqrt( V / Dt ) ... where the "D"…

Sorry I didn't mean to make you work for me, but it's a PDF and I didn't know how to explain better the position (maybe I should have told you the first formula on page X). For you it was a D, for me it was a triangle and I didn't get the meaning of that Dt. Maybe it's just a too advanced paper for my knowledge.

The triangle, or “delta”, is used to indicate a tiny change in the following variable.

Let’s say you go on a journey, and the distance you’ve travelled so far is “x” and the time so far is “t”.

Then your average velocity since the beginning is x / t .

But, if you want to know your current velocity, that would be delta x divided by delta t .

The delta is usually used in a “limiting” sense - you can get a more accurate measurement of your velocity by measuring the change in x during a tiny time interval. The tinier the interval, the more accurate the estimate of current velocity.

What I’m talking about here is the first steps in learning differential calculus. You could look for that at kahnacademy.com. You might also benefit by looking at their “precalculus” courses.

Just keep plugging away at it, the concepts take awhile to seep in. Attaining mathematical maturity takes years.

Re: Ask HN: How did you learn math notation?

#38

Post by ColinWright »

rullopat wrote: post #19
ColinWright wrote: post #14

Earlier quoted context omitted.

You say "There's a formula with a triangle ..." without telling me where. That's not real helpful, and you're making me do the work to find out what you're talking about. If you want assistance to get started, you need to be more explicit. However, I have done that work, so I've looked, and in the second column of page 210 there's a "formula with a triangle": t_c = 5 \middot 10^{-5} \sqrt( V / Dt ) ... where the "D"…

Sorry I didn't mean to make you work for me, but it's a PDF and I didn't know how to explain better the position (maybe I should have told you the first formula on page X). For you it was a D, for me it was a triangle and I didn't get the meaning of that Dt. Maybe it's just a too advanced paper for my knowledge.

BTW ... you say:

> Maybe it's just a too advanced paper for my knowledge.

Maybe it is for now ... the point being that if you start at the beginning, chip away at it, search for terms on the 'net, read multiple times, try to work through it, and then ask people when you're really stuck, that's one way of making progress.

You can, instead, enroll in an on-line course, or night-school, and learn all this stuff from the ground up, but it will almost certainly take longer. Your knowledge would be better grounded and more secure, but learning how to read, investigate, search, work, then ask, is a far greater skill that "taking a course".

Others have answered your specific question about the delta symbol, but there are deeper processes/problems/questions here:

* Not all concepts or values or represented by a single glyph, sometimes there are multi-glyph "symbols", such as "Δt" in your example.

* When you see a symbol you don't recognise, read the surrounding text. The symbol will almost always be referenced or described.

* The notation isn't universal. Often it's an aid to your memory, to write in a succinct form the thing that has been described elsewhere.

* In these senses, it's very much a language more akin to natural languages than computer languages. The formulas are things used to express a meaning, not things to be executed.

* Specific questions about specific notation can be answered more directly, but to really get along with mathematical notation you need to "read like math" and not "read like a novel".

* None of this is correct, all of it is intended to give you a sense of how to make progress.

Re: Ask HN: How did you learn math notation?

#39

Post by ColinWright »

NamTaf wrote: post #33
ColinWright wrote: post #27

Earlier quoted context omitted.

I agree completely that I'm coming at this with a lot of background knowledge, but if I'm reading in an unfamiliar field and I see a symbol I don't recognise, I look in the surrounding text to see if the symbol appears nearby. As I say, "Δt" appears immediately above ... that's a clue. As you say, it's drilled in at school that everything is represented by a single glyph, and if these are juxtaposed then it means mul…

I absolutely agree with your comments regarding teaching the underlying approach to digesting a paper. You definitely raise good points, especially the 'one true meaning' comment. I should state that I'm not discounting the value of your point, especially given this clarification, however I guess that when I reflect on my experience in my time learning this, the time I best learnt was via initial expalnation, then wo…

Agreed ... I think we're pulling in the same direction ...

Re: Ask HN: How did you learn math notation?

#40

Post by dqpb »

solmag wrote: post #5
solmag wrote: post #4

Practice, just like you learned programming. "The Context" gives you the meaning for the notation, sadly. You have to kind of know it to understand the notation properly.

You can also get sufficiently angry and just write out linear algebra books and what not in Agda / Coq / Lean if it pisses you off so much (I've done a bunch of exercises in Coq)

Ultimately I think this is the right answer.
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