Ask HN: How do I improve my command of mathematical language? • Hacker News Board

Re: Ask HN: How do I improve my command of mathematical language?

#31

Post by nbaksalyar » Sun, Oct 25, 2020, 3:05 AM UTC

mxmhrmn wrote: post #6

Have you done (or seen) math proofs? If not, the following FREE book can serve as a great starting point for learning how to get your point across in the most clear, complete and concise way. Algorithms you see in elementary algebra (as opposed to, say, abstract algebra) and calculus are just that -- algos[0]. They teach you neither math, nor critical thinking. Book Of Proof by Richard Hammack https://www.people.vcu.…

I can also highly recommend starting with proofs.

There's a similar book, "How to Prove It: A Structured Approach" by Daniel Velleman. It's one of the best math books I've ever read: it's concise, has excellent examples and exercises, and it teaches the most essential skills (including the mathematical language in question, e.g. the basics of set theory and mathematical logic).

Re: Ask HN: How do I improve my command of mathematical language?

#32

Post by quantified » Sun, Oct 25, 2020, 3:11 AM UTC

sinuhe69 wrote: post #25

One problem with mathematical difficulty is its “invisibility”. We can’t see how other people are thinking, how they solve the problem, how they struggle. And sometimes, we cannot even explain our thinking! That’s why making thinking visible is so important! Because only then we can learn from the others, they can learn from us and we can see where it’s right and where is wrong. One way to making thinking visible is…

+1. Across a set of rote problems that you know how to solve: take the problem and write down in any language the steps you take to solve it, then also write down why you took those steps at the points you did. When you’ve covered the set, review them all. If you tried to record a tutorial for someone else at that point to follow in your footsteps, what would you say? One of the great ways to learn is by teaching, you can simulate for yourself. But it has to come from your understanding, not just reciting the words of your textbooks and problem sets.

Re: Ask HN: How do I improve my command of mathematical language?

#33

Post by baking » Sun, Oct 25, 2020, 3:25 AM UTC

I'm surprised no one has mentioned this. Get a rubber duck (or any toy-like object that can represent a listener) and explain problems to it.

You may also find doing mathematical proofs to help your thinking, although it is a higher level than explanation. It might help with clarifying the thought process.

Re: Ask HN: How do I improve my command of mathematical language?

#34

Post by mettamage » Sun, Oct 25, 2020, 4:10 AM UTC

sinuhe69 wrote: post #25

One problem with mathematical difficulty is its “invisibility”. We can’t see how other people are thinking, how they solve the problem, how they struggle. And sometimes, we cannot even explain our thinking! That’s why making thinking visible is so important! Because only then we can learn from the others, they can learn from us and we can see where it’s right and where is wrong. One way to making thinking visible is…

So a twitch.tv for mathematicians might help.

Re: Ask HN: How do I improve my command of mathematical language?

#35

Post by nelsonenzo » Sun, Oct 25, 2020, 4:16 AM UTC

Try watching/listening to youtube videos of someone else explaining a problem you can already solve. Then write out thier words, like if you were writing the script for the video, as well as the math. I found that focusing like this really helped me learn the specific math vocabulary of both the symbols, and constructs. Good luck!

Re: Ask HN: How do I improve my command of mathematical language?

#36

Post by whimsicalism » Sun, Oct 25, 2020, 4:57 AM UTC

jes5199 wrote: post #9

zmmmmm wrote: post #7

I feel like I have some kind of mathematics dyslexia. I can understand all the concepts, but I find formal mathematical notation almost completely incomprehensible. Expressions with all kinds of strange symbols, often variables introduced that are either completely undefined or defined ad hoc locations remote from the usage, integrals and sums with no subscripts / unclear scoping, etc etc. Clearly other people just r…

yeah I have a similar thing - I’m trying to read a computational geometry textbook (for a code project) and I end up skipping the equations and trying to glean the meaning from the prose... it’s a slow process. There must be a resource out there for decoding unfamiliar equations but I don’t know what it is

> There must be a resource out there for decoding unfamiliar equations but I don’t know what it is

Honestly? To me this has always been an indication that I am being over-optimistic/reading ahead of my paygrade. If I can't understand the equations, then I'm not really grasping the fundamental concepts and I'm lying to myself.

This is from my experience in physics.

Re: Ask HN: How do I improve my command of mathematical language?

#37

Post by tikej » Sun, Oct 25, 2020, 5:31 AM UTC

Maybe slightly radical idea for beginner, but since we’re on hackernews.

Try to write down solutions to known, quite easy problems (that you are able to solve easily by hand) step-by-step (line by line) using SymPy (https://www.sympy.org/) library (or any other computer algebra system; but SymPy is most well documented, free and most beginner friendly IMO).

It may seem counterintuitive at first (you want to do learn mathematics by heart and I’m proposing to offload it to computer...) but carry on.

Contrarily to hand written math, it’s harder to abuse notation using computer algebra system. Don’t use pre-made functions (such as Integrate, Simplify, Solve) right at the start, but manually work your way (via simple operations like expand, substitute, and your hand-written functions working on expression parts) to the point when expression is in form you would put it by hand.

Each step, substitution, transformation etc has to be stated explicitly in order to be understood by SymPy! That way if you are able to “explain” it to a computer (that has no intuition about objects that it’s working with, maybe beyond some basic algebra and fact that they are symbols) it will be much easier to explain and present it more clearly.

That said the obvious con of this method is learning SymPy, but believe me it will pay back both by understanding math more deeply, as well as in ability to solve problems that are too large/tedious to do by hand in the future.

Btw I believe this is basically the way the applied math (computational, non-proof part of math) should be presented nowadays. When I (hopefully) finish writing my phd thesis I hope to to write CAS dedicated to this purpose (doing math in by-hand like way, but with help form computer) from scratch. Until then SymPy is IMO your best bet for that ;)

Re: Ask HN: How do I improve my command of mathematical language?

#38

Post by lironmo » Sun, Oct 25, 2020, 6:23 AM UTC

My partner and I are mathematicians, from my experience mathematical understanding is acquired through practice and learning. Try to study the following 3 courses (can be found for free): Differential and Integral Calculus 1, Introduction to Group Theory, Topology

Re: Ask HN: How do I improve my command of mathematical language?

#39

Post by jl6 » Sun, Oct 25, 2020, 7:13 AM UTC

Mathematics is English.

OK, not English specifically - what I mean is that mathematics is not a bunch of abstract squiggle notation on paper, it’s a series of sentences written in a natural language. The abstract squiggles are shorthand for those sentences.

Some might say the squiggles are more precise than the natural language words. Not true. The notation really is just a shorthand for the words, and mathematics is the study of words that have precise meanings.

Now, don’t get me wrong, notation is great and can help make connections that would otherwise be obscured by wordy prose - but it is not primal.

So, to become better at expressing mathematics, you need to rediscover the connection between the notation and the underlying sentences that they represent.

Take the following “statement”:

a = b & b = c => a = c

Translate it into:

“If a number a is equal to another number b, and b is equal to another number c, then a is equal to c”

Note that this wasn’t even a rote translation - I added in the premise that the variables are numbers. This is the kind of information which can get forgotten or assumed in notation, which could make it harder for an uninitiated reader to follow. Of course, you may also need to remind the reader what a number is and what the “equals” relation means... knowing what to assume about your audience’s level of knowledge is part of the art.

The “connective tissue” between the symbols is the language of logic. Words like “if”, “then”, “and”, “or”, “not”, and “implies”. These are the words that allow you to form statements with precise meaning, and you can improve your use of them by improving your language skills, not just your mathematical skills.

Now, I’m not advocating for writing everything out in long form, because some notation is just too good and too expressive to pass up. The point is to remember that mathematics is a form of writing, intended to be read by other humans, that they will be reading it through the lens of natural language, and that all the rules of good writing still apply to mathematics exposition.

Introduce your topic. Motivate your definitions. Clearly and precisely lay out your reasoning. State your assumptions. Emphasise important points. Give illuminating examples. Explain your thinking. Avoid trying to demonstrate your own cleverness. Summarize your conclusions.

And all of this applies doubly for oral presentation instead of written.

Re: Ask HN: How do I improve my command of mathematical language?

#40

Post by hyperpallium2 » Sun, Oct 25, 2020, 7:43 AM UTC

I had an inkling about your question, and now I can articulate it:

  You don't have to be able to explain how you got the answer, but you do have to be able to explain *a* way to get the answer.

Minsky said "You don't understand anything until you learn it more than one way."

In proving theorems in mathematics, there's often an intuition that it's true, and even why it's true, but still it needs to be put into a formal proof.

But it's very hard to arrive at a proof, by thinking in formal terms, because the search space grows exponentially. Intuition is a shortcut. But you still need to go back and work out the formal proof, to show you're right - to yourself and also to communicate it to others.

And, once you know the answer, and have an idea of why it's true, this guidance makes it much easier to find your way.

To come back to your question: you can learn another way to get the answer by watching videos, where they explain what they are doing as they go. So, this won't explain how you do it, but you can learn another way of doing it that you can explain.

BTW IMHO the most insidious difficulty in learning mathematics is gaps - an incomplete or subtly incorrect understanding. It's insidious because the gap is not apparent at later material, because it assumed, so you won't know what you're missing, and therefore can't diagnose or fix it. You need either an excellent teacher/tutor/mentor/coach to pick it up (but they often don't), or to methodically go through all the previous work. The sooner you do it, the easier the easier it is.