Ask HN: How do I improve my command of mathematical language? | Hacker News Reader

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

First of all, sorry if my English sounds off, maybe weird; it's not my first language but I'm striving to improve.

Short story short: I went back to high school in my late 20s and next year I'll graduate. Over the years I went from hating math to really enjoying it, especially calculus, but there's still one thing that keeps bugging me, and that's my command of its language. 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. It doesn't help that I can't find any resources like you could for any natural language.

When I asked my teacher she said to practice over and over, but that's all I've been doing these years and it didn't do much to improve my oral presentation. So, I'm asking here in the hope that someone who went through the same struggle or understands the mental process very well and knows what it takes to "fix the computation-presentation bridge", could help me or guide me in the right direction, to overcome this obstacle.

Thank y'all for your time.

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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.

+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.

So a twitch.tv for mathematicians might help.

I'm not a mathematician and not good at it either, but something that I see a lot is that mathematics in general are very keen on sharing the conclusions in concise, if not cryptic, forms.

A lot of metamathematics are lost in the process.. but I think that people that are really good in math like this way of life. But it kinda hides a part of what doing mathematics is. It should be written as books preludes.

My favorite math professor was great because any time he would forget the statement of a theorem or the outline of a proof he wouldn't just go to his notes to remember it, he would out-loud reason his way to the correct version. This would include the minor missteps he made on the way.

Best math lecturing experience I have had by far.

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.

>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".

Mathematician too. I would say we still don't really know what a set is. ZFC "explains" sets in terms of... axioms and models of them which are also just sets? Should I worry about this recursion? Who knows, we just pretend things work.

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 problems of the same genre from school textbooks. Or you can do maths, and you struggle to follow the generally accepted set of notation.

If your problem is the first, you need to explore a wider range of mathematical problems, which will fundamentally require you to think, apply your skills to the problem in various new ways. Maths is a subject where learning to think and learning to solve a specific set of problems are two very different things. Good resources for this are the Art of Problem Solving, and various Olympiad questions. You can also just try to apply your mathematical knowledge to real life examples.

If your problem is the second, I don't know a solution, but this is a far better problem to have than the first. I'd rather have the problem solving ability and use my own set of notation that nobody else understands, than have less problem solving ability and understand the notation. But, in short, that'll just come down to practice, seeing the general notation, and somehow forcing yourself to use it. Alternatively, just try to define your own set of notation at the start of your solutions. It may depend on what kind of professor you have as to how many credits you'll get, though. A good professor will not penalise you that much for making up your own notation, if your solutions are correct, but may try to encourage you to use standard notation (as this is required for any kind of collaborative mathematics).

The problems in Spivak are pretty good I think - they'll show if you can do maths or not.

Some also have answers (and an answer book for the rest), so there's examples of explaining what was done, for harder ones. The explanations are sometimes too concise, but at least it gives some guide for how you can explain your answers.

At a much easier level, Khan Academy videos mostly are explaining how a specific problem was solved - another resource.

Meh. Everyone pushes their favorite calculus book but I think better is to just try to push yourself to gain an intuition for your subject. Try to visualize what it is the text is saying, try to apply the problem etc.

When I took calculus there were three things that really helped me with this:

1) A study partner that's competitive. when you have an intuition you're more likely to get the correct answer first because you can check what's going on quickly without having to recalculate things. Also when they get stuck you'll be expected to explain how you figured it out. Studying with someone like this will force you to grow an intuition.

2) Writing programs that use what you've learned. At least for me this helped a lot. I learned to program before I learned any interesting math so being able to rewrite what are often implicit ideas in an explicit language can really help you think things through. Like many people here I gained an intuition for integration early on by writing physics engines for games.

3) Just meditate on the ideas.

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.

Instruct. Explaining it to other people is helpful for developing your ability to express mathematical ideas (any idea really). You'll learn where you're limited in expressing your understanding and be forced to develop the language.

I was thinking of how to answer the OP, but when I read your post, it made the most sense.

OP - even if you don’t teach others, record videos to yourself trying to explain concepts and then slowly improve from there

This is the right answer. If you think you understand something, pretend that you have to teach it to a beginner and write a short lecture/presentation to give them. Explain what they'll be doing, why, and why it is important. Do it in which ever language is most comfortable. If you can prepare that lesson then great. If you cannot, then your initial assumption that you understood the material was incorrect. Mathematicians write material like this all the time (they like to call it "expository material").

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

If you want to improve your oral presentation, you need to practice your oral presentation. Doing problems on paper and hoping your verbal description will improve is not going to work.

I suggest you find some maths videos on Youtube which explain maths you already know with lots of verbal detail narrating each step. Pay attention to the words and description (you already know the maths), then try talking through your own working in the same way. You could record yourself and watch it back (camera pointing straight down at paper is good, and phone camera held up on a ruler and stack of books is easy). As others have suggested, once you can do that to some extent, explaining things to others can be good practice, but if you're stumped as to how to describe anything, that's not going to be a good first step.

I teach university maths, and when I'm doing a problem in a lecture, writing down the steps to demonstrate, I am constantly talking, narrating everything I'm doing and why. E.g. "Okay, we need to cancel out that 3, so let's divide both sides by 3, and then we can use log rules to bring that 2 on the right inside the log as a power. Now to cancel out these logs we need to use an exponential on both sides, and if we move these terms over we'll have a quadratic to solve. We know how to do that! Hm, it doesn't seem easy to factorise, so let's complete the square. We need half of this coefficient to go inside the brackets, and don't forget the minus is part of it! Now if we think about expanding this out we'll get the x squared and minus 4 x that we want, but also a plus 4, so let's take that away and add on the 3 we need instead ..."

I'll also be referring to the context of the problem ("that makes sense, because ..." etc.) and earlier examples to compare and contrast with.

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.edu/~rhammack/BookOfProof/

[0] Obviously, I am not talking about the theory of algos where you design algos and consider questions of optimality and correctness.

I strongly agree with this, you already have enough math knowledge to learn how to do basic proofs. Hammack’s book is great, and it’s great that the author made it freely available. I personally used Chartrand’s “mathematical proofs” and thought it was very helpful, I like it a little better than Hammack’s book, but they are both great for people getting started, particularly for self-study.

You also need to talk to other humans about math on a regular basis - feel free to contact me, I love talking math! Contact info is in my profile.

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).

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 read these and I even know people who will ignore all the text in a paper and just look for the equations. I wish I could master it because in some areas it is a genuine barrier to me achieving my goals.

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.

> 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 would programming doesn't really work - because often you can't just look up something you don't understand (how can you Google a mathematical symbol when you don't even know it's name?).

Pretty much the only way to learn mathematics is from the ground up.

> and I even know people who will ignore all the text in a paper and just look for the equations

Mathematical notation is mostly a substitute for words and that's why those equations can be embedded in natural language and "read". Ignoring the words in a mathematical paper and just looking at the "funny symbols" doesn't make any sense. You're only seeing tiny glimpses of the actual reasoning and are probably missing most of it.

A mathematician may chose to write "B contains all elements of A, and A contains the element x.", or they may chose to write "A ⊆ B and x ∈ A". Same thing. Also note how the second example still had an english word in the middle of it. If that "and" was an "or" the sentence would have an entirely different meaning. Can't just ignore that stuff.

Sure you could replace that "and" with yet another symbol, and in something as simple as that nobody would really care, but in something more complex the mathematician would just be trying to produce an unreadable mess on purpose.

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.

> 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

I absolutely do find infuriating when math is written down vaguely/confusingly/poorly, and ML papers are a great example of this happening way too often.

BUT, for your specific examples (in case that is an actual question):

The first one can be parsed using the fact that y_i in the second term wouldn't make sense if it was outside the summation, since the i index is only defined inside

The second (and actually, the first one, too) uses the very well established convention that multiplication always has precedence over summation, in every kind of expression. So the 2 multiplies the second sigma, and then the product is summed to the first sigma. This rule should make a lot of cases much clearer.

There is a reason Wikipedia is never in a syllabus in universities.

Don't feel too bad about it, there are far too many notations for the same concepts - completely dependent on the background of the author.

This becomes especially noticeable in hodgepodge fields like Machine Learning / Deep Learning, where you essentially have scientists from all walks of life/sciences, trying to convey the same ideas, but using their own standards of notation.

Check out this Math Overflow answer from none other than Terry Tao, on inner product notation: https://mathoverflow.net/questions/366070/what-are-the-benef...

Point is - he listed 18 different notations, on the spot, all "belonging" to different fields. Now imagine 18 different scientist, from their respective fields, writing research papers using their preferred notation.

Sure, there's going to be a lot of overlap - but there's still room for confusion, even for seasoned engineers and scientists.

(I just used ML and DL as an example, because that's the field where I've noticed the most of this - because of the nature of those fields)

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.

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 ;)

If the issue is a lack of way to structure the explanation or missing the justification to go from one step to the next, having a structure that forces one to be explicit helps.

Other than sympy, a technique for formal proof might help.

One of my buddies showed me this talk from Leslie Lamport that I found interesting [1], albeit in french. I think this one [2] is an english version. And the paper behind is here [3], linked from Lamport's website [4].

Lamport makes a structured way to write proof so that all details can be written down. It seemed to me that a lot of knowledge and steps stay implicit in proofs. So if you're not familiar with the subject, you easily get lost.

[1] https://www.youtube.com/watch?v=k-i7y0R_-KE

[2] https://www.youtube.com/watch?v=uBiJpip9NVc

[3] http://lamport.azurewebsites.net/pubs/proof.pdf

[4] http://lamport.org/

edit: fixed links and text structure

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.

> 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

That can come later, the more important point is to explain "why" you are doing a specific step and not "what" and you are free to use just English words.

For example, say you want to find the maximum of f(x)= -x^2 + 4 x + 10. A bad explanation would be some mumbling about taking down the 2, ... , f'(x) =-2x +4, so x=2, f''(x)=-2, is negative, done.

Better: We know that at a maximum the slope of f has to be zero (otherwise we could go left or right to get larger), so let's find such points first. We thus compute the derivative function as f'(x)=-2x+4 and find that it is zero if x=2. But that does not give an answer yet, because it could still be a min/max or saddle point. .. draw a small picture of each. We thus compute the second derivative to see in which of these three cases we might be. f'(x)=-2 is negative so we locally look like a parabola opened downward, so this is indeed a maximum.

Here in Germany early college math classes were mostly a rehash of everything from school, but with much more rigor and sometimes more detail.

So I'd recommend some college textbooks. If you live near a college, their library should have a ton of them to browse through and lend. You should also be able to find accompanying lectures on YouTube or college web sites.

So, bear with me. I think I've got a similar problem with music. I have a good ability to improvise, and play tunes that come into my head. And I've got a decent grasp of music theory. But I find relating improv to the theory(and making deeper sense of music that's written down) really hard.

I've got a collection of texts that I can go through, and I've found lately that starting on a random key of the day I roll a dice - has really helped me on both sides - in that I've got a bunch of kinds of exercises I do for practice, but the piece of randomness helps me with finding things to practice, that help me broaden my existing knowledge and practice.

What does this mean for you? Well maybe you need to pick a random page of a suitable book each day, find the start of the section for that page, go through it and then see if you can think of any exercises or things you need to do to help you with your knowledge consolidation from that.

> It doesn't help that I can't find any resources like you could for any natural language.

Try Playing with Infinity by Rózsa Péter. https://www.maa.org/press/maa-reviews/playing-with-infinity

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.

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!

Read Dijkstra’s essay on notation[1]. Not only is it very sensible, his reasons for using it provide general insight.

[1] https://www.cs.utexas.edu/users/EWD/transcriptions/EWD13xx/E...

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.

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

> First of all, sorry if my English sounds off, maybe weird; it's not my first language but I'm striving to improve.

Language is a medium to communicate. Actual message is more important than medium.

Can you give an example of some problem that you can't explain?

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