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

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

#21

Post by ghufran_syed »

rayuela wrote: post #10

Watch some lectures or pick up an undergrad book on Real Analysis. You don't have to become an expert in it by any means, but the motivation and logic underlying most of the math you will encounter will become much clearer. It's hard, but don't give up on it. It will be worth it. Before delving into analysis I felt like I was just memorizing and following formulas. It really is like the fundamentals of Math.

I know real analysis(or abstract algebra) is the traditional first “proofy” course in the US, but I disagree with this way of teaching proofs - think its better to learn one thing at a time. So instead of learning how to do proofs while also learning real analysis, I think its better to learn the basic proof skills in the context math you are fluent with i.e. basic algebra and basic number theory (e.g. what integers divide easily into another integer). The books by Hammack and Chartrand do that well I think, see elsewhere in this discussion for more info.

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

#22

Post by redvenom »

I am a mathematician.

I would say the absolute best thing you can do is take the problems you DO know how to solve and write out the solutions, using complete sentences. If you can't explain some part, then you will quickly discover which parts you are following by rote.

At that point you need to go back and understand the process. It will be a process of writing and writing, revising, and understanding. It's not an instantaneous process and may take a few months to a year, but it is a rewarding process.

You may find you will have to think about certain concepts you thought you knew, even basic ones. Don't worry about that. It took centuries for mathematicians to even write down precisely what a set is for example.

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

#23

Post by meken »

m11a wrote: post #3

The issue is here: "I have no problem solving most of algebra or calculus problems, but if you asked me to explain what I'm doing I wouldn't be able to tell you, at least not using mathematical constructs/words." This is slightly unclear; there are two possible problems here from what I can see, and they're totally different problems to have. Either you cannot do maths, and you're only good at solving repetitive prob…

I suffered from problem 1.

I was able to steamroll my way through calculus II. Then I hit a brick wall in calculus III and came to the harsh realization that I had simply rote memorized everything up to that point.

I resolved to actually understand the material, so I took the rather drastic step of going through all of khan academy, starting with arithmetic, all the way through calculus. This was easier back in 2010, when all khan academy videos were on a single page. Luckily, the internet archive can help us out

https://web.archive.org/web/20100206074742/http://www.khanac...

By the time I finished, I had built up a good mathematical intuition.

This was actually one of the most satisfying intellectual journies I have gone on.

As a shout out to circa-2010 khan academy, it’s still so beautiful to me that all the videos are there, on one page. You can see them all, unlike the layout of the current website, where you can easily go down rabbit holes and lose your sense of progression.

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

#24

Post by dalmo3 »

redvenom wrote: post #22

I am a mathematician. I would say the absolute best thing you can do is take the problems you DO know how to solve and write out the solutions, using complete sentences. If you can't explain some part, then you will quickly discover which parts you are following by rote. At that point you need to go back and understand the process. It will be a process of writing and writing, revising, and understanding. It's not an…

>It took centuries for mathematicians to even write down precisely what a set is for example.

I had a calculus professor who would hand us assignments and say things like "this one should take 3 to 4 lifetimes to solve".

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

#25

Post by sinuhe69 »

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 to apply thinking routines, like: - what are given?

- what are the unknowns?

- what do we know?

- what do we need, how i can get there? (strategies)

- choose a strategy and explore it

- rethink the strategy and optimize it

- connect and reflex

Once we practice thinking routines, we will find it increasingly easier to explain our thinking and even be able to debug our work. If your class also applies thinking routines, it’ll get even easier for everyone to learn and understand each other.

I recommend the following books:

- Polya, How to Solve It

- Velleman, How to Prove It: A Structured Approach

- Ritchhart: Make Thinking Visible

Mathematical work is basically a series of equivalent transformations and logical reasonings, starting from the given informations to get where we want, thus once we knew the why, it should not be too difficult to explain our work because most are self-explanatory.

Hope it can help.

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

#27

Post by porknubbins »

chmod775 wrote: post #14
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…

> 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. This is what mathematical notation looks like if you are lacking some of the fundamental knowledge to read it. I know what that's like because I've been there. Learning mathematics like you…

I understand needing to learn the basics from the ground up but I always get bogged down by going back too. I have started to wonder if its worth trying to self learn mathematics at all since even when I do make progress its so hard to remember. If there were a well categorized Leetcode for math problems some more might stick but I feel like thats not really the point since people always say higher math is about proving or understanding things not mechanically solving problems.

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

#28

Post by zmmmmm »

LudwigNagasena wrote: post #15
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…

> integrals and sums with no subscripts / unclear scoping, etc That’s just seems like bad writing. Out of math subjects I learned (discrete math, linear algebra, real analysis, statistics), only in the last one it may be unclear what’s going on, whether you are dealing with a random variable or its realization, and arguments of random variables are almost always omitted, so they don’t look like functions.

So here's an example on Wikipedia [1]

https://wikimedia.org/api/rest_v1/media/math/render/svg/6c66...

Does the sum include both terms or only the first? I happen to know what Covariance is, but if I didn't I wouldn't be clear. I'd probably guess right, but scroll down a bit to this:

https://wikimedia.org/api/rest_v1/media/math/render/svg/e634...

Is the +2 a multiplier on the second term or additive within the first sum? I try and read it from the spacing but I honestly can't tell.

[1] https://en.wikipedia.org/wiki/Covariance

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

#29

Post by j2kun »

I wrote a book that you may enjoy, "A Programmer's Introduction to Mathematics"

It spends a lot of time on the language and how to express things. The ebook is also "pay what you want," so you can see if you like it before deciding I deserve your money :)

https://pimbook.org

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