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Since I started university, almost every exam I've had in mathematics have ended up as failed mostly due to my lack of using proper "syntax" on paper, written by hand.

Where can I learn the standard way of writing mathematics, beginning with the basics? Does this make sense? Is there a standard "uniform" syntax, or does it depend on locale - country from country?

EDIT: With syntax I mean the way you write down your logic on paper - for instance, how you should put up a solution for display; what format to use, when to use deduction/implication arrows, how your answer should be represented etc. When to tell whether you are you allowed to skip mental steps and so on?

Very simple example:

x(x + 3)(x + 7) =                     
= x(x^2 + 10x + 21) =                              
= x^3 + 10x^2 + 21x         
Answer: x^3 + 10x^2 + 21x


x(x + 3)(x + 7) = x^3 + 10x^2 + 21x

and so on... Hope this makes it a bit more clear?

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How many mental steps you are allowed to skip really depends upon your level. The more basic your level, the more mental steps you should include. –  Fredrik Meyer Dec 6 '10 at 19:24
My basic rule of thumb as I convey to my students is: "If you have to stop and think about it before writing down the final answer, then you need to write down those mental steps." –  Arturo Magidin Dec 6 '10 at 19:28
Your question is about syntax and your example is about style. The syntax part is interesting. Traditional math is always a mixture of prose (non comment prose I mean) and concise syntax. I find this insufficient and once in a while I search for an answer to your question but without success unfortunately. Did you find something in the mean time? –  Patrick Fromberg Dec 21 '14 at 19:05
@PatrickFromberg Hm.. I am a bit ambivalent to say yes, I came across a book called How to think like a mathematician; amazon.com/How-Think-Like-Mathematician-Undergraduate/dp/…, a few months ago and it has been very helpful, however I feel like the origin of all help it provides would be nice to know, where or how they came to be. –  Zolomon Dec 21 '14 at 22:55

4 Answers 4

up vote 13 down vote accepted

Learning to write mathematics properly is much like learning to write any language properly, only more so because such a premium is placed on logical organization and clarity.

Mathematics, both the part that is written in words and the part that is written in symbols, is attempting to convey meaning. When you write and read symbols, think about what they say when you read them out loud. Every time you see the symbol $=$, remember that the symbol has a pronunciation when you read it, and it says "is equal to". So if you write things like $$2x = 4 = \frac{4}{2}=2,$$ (which I see far too often on exams) then you are saying "twice $x$ is equal to four, which is equal to four halves, which is equal to $2$", which of course is false and liable to cost you points, even though you probably don't actually think that $4$ and $\frac{4}{2}$ are equal. Remember, first and foremost, that every symbol has a meaning and a pronunciation. Unless you recognize that, you won't be able to get very far.

So first you need to be clear on what you want to say, and then make sure that what you wrote actually conveys that meaning and not other meanings. If you can do that, even if it is with "plain English", you will have gotten over more than half the problem.

That said, mathematics is also a technical language with a number of conventions and jargon. The very best thing you can do to become familiar with, and good at using those conventions, is to read a lot of mathematics, with an eye towards understanding what is written and how the language helps that understanding (just like doing a lot of reading is one of the best ways to improve one's spelling).

A close second is to read books that are meant to help introduce you to proofs and logical arguments, usually with subtitles like "first course in advanced mathematics" or "introduction ot abstract/advanced mathematics". Find out what the "Intro to proofs" course is at your school, and look at the textbook they use.

One important thing is not to simply try to mimic the language you see: that will result in the mathematical equivalent of saying "Buenos días. Yo quiero estación de tren ser, por favor" when trying to ask for directions to the train station ("Good morning. I want be train station, please.") You want to keep an eye on the meaning that the words are conveying, and how that particular choice of words (and even the order of the words) matters. Notice, for instance, that saying "For every $x$ there is a $y$" is not the same thing as saying "There is a $y$ such that for every $x$...", even though they may seem very similar when thought of in English.

So: always think first about what you are trying to say and make sure you say that. Read what you've written, pronouncing every symbol to yourself to make sure you aren't saying that you "want to be train station". And read your books and professors' notes to see how the language works and become familiar with it.

Added: It seems I rather badly misinterpret the true thrust of your question (despite the fact that you seem to have "accepted" my answer). In so far as what steps to add or what steps to skip, as has been pointed out, it depends on your level. I would not object to a student in my graduate abstract algebra class going from $x(x+1)(x+2)$ directly to $x^3+3x^2+2x$ (or even not writing out the computations before using it!) but I would definitely request a precalculus student to write out how he got to that final answer. If you have to stop and think carefully about what the answer is, then you should not skip the step and write it down. For a particular class, look at the textbook and which steps it works out explicitly and which steps it skips. Look at what steps your professor works out explicitly and which steps he skips. You'll want to not stray too far from them as far as skipping more steps (though of course you can always skip fewer steps if you are unsure about a calculation/argument).

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Lovely answer, and I believe what you say is true! I'll try and scour through my math books to see if I can find the hidden answer to the arts of mathematical representation. :) Thanks! –  Zolomon Dec 6 '10 at 19:58
IMHO, as far as "what steps to include", I think an even better rule of thumb is to imagine that you are explaining your solution to a classmate, maybe one who's on the slow side. If the classmate wouldn't see right away what you are doing, then you should explain it. –  Nate Eldredge Dec 7 '10 at 1:01
when you decide to write a book, please make sure to let me know :) –  KerxPhilo Apr 23 '11 at 8:12
Since I asked this question, I have relentlessly continued studying mathematics. I'm waiting for the grades on my most recent calculus exam. The most important thing I've learnt since I asked this question is that the main point isn't to find the correct answer -- it's to prove that the answer is correct. Once I understood this, I started making major advancements in my comprehension and confidence in writing proofs. –  Zolomon Mar 13 '14 at 21:15

Read your textbook and then use the syntax in your textbook when you take the exam.

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It is not clear what you mean with "syntax".

Assuming you by "syntax" mean your way of reasoning, i.e. your "path" via logical connectives to the result, my best tip is to carefully read all proofs in your math book. Not only read through them, but understand them.

You must be able to prove things correctly, i.e. using only valid deductions. They way they do it in your math book is not a "local" thing, but "global" (no pun!). As long as your reasoning is understandable by anyone else reading it, it should be fine.

If you could be more spesific with what you mean by "syntax", I'm sure somebody could give a clearer answer.

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The level of allowed ellipsis depends very much of your reader. It is part of the style, not syntax.

Some mathematicians like Laplace, are very boring to read because they spend so much time on tiny details. Some others are very difficult to follow because some of the most critical piece of reasoning are left as an exercise to the reader.

A famous example was written by Fermat in 1637: «cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet» [I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.] It took 358 years to rebuilt the missing steps. In facts, almost every work of Fermat was first rejected by the mathematical community,... before he had a chance to work out and publish the details.

So, don't lose hope, even master mathematicians fail the exam.

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It's worth noting here that Fermat's proof is still a mystery. Wiles' proof used contemporary ideas and techniques which Fermat did not know. In light of that, it seems very unlikely that Fermat had a valid proof. –  Adam Saltz Nov 30 '11 at 22:54

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