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Pretty simple question here but it does relate to math. I ask this as my writing is quite messy, possibly a cause of silly mistakes.

How important is neatness in math?

Does having messy writing put you at a disadvantage?

What is the tradeoff of speed vs. neatness?

Should I look to increase the neatness of my writing?

What are some simple tips to prevent mistakes that can be solved by simple organization / tidyness/structure?

In timed settings (exams), neat vs. speed?

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I would recommend to type things using LaTex (look at the beautifully formatted questions/answers on this site). Also, recommend reading: math.stackexchange.com/questions/266279/…. In industry, one of the biggest (maybe the biggest) is a lack of communications skills (writing, presentations in all forms). –  Amzoti May 16 '13 at 16:53
    
Yeah, I have been slowly learning latex, definitely useful for typing up notes. My primary concern though is hand written, occasions (tests, study) in which access to a computer is not possible. –  user78101 May 16 '13 at 16:56
    
I would spend the time to be neat because it matters in the real word in order to effectively communicate with others (at whatever level). –  Amzoti May 16 '13 at 16:59
    
It's hard to answer this question without knowing your intented goals -- are you turning in math homework, or a career in mathematics? –  Cam McLeman May 16 '13 at 17:01
    
I always find it interesting as some teachers have very neat writing while others have very messy writing. Is neatness purely a matter of better communication or does it make a difference, particularly in the long run to the person writing also? Curious to hear opinions from those with "messy" writing and those with "neat" writing. –  user78101 May 16 '13 at 17:04

3 Answers 3

It may just be my fanatical opinion, but I think that clarity is one of the most important attributes of performing mathematics, regardless of level.

For your purposes, let me present an example: Regardless of what course I am teaching, whether it be first year calculus or fourth year topology, I always have students who submit messy, poorly structured, incomprehensible nonsense on their assignments. They typically receive a very poor mark as a result, even if their answer/solution is entirely correct. They of course come and complain that because their answer is correct, they should receive full marks. This is my response:

"A mathematical solution is not just a correct number at the end of a computation: it is a logical sequence of events which is clearly motivated, explained, and justified at every step. If you present non-sense that happens to give you the right number, then while your answer may be correct, your solution is wrong."

Of course, this idea should transcend undergraduate mathematics and manifest in research mathematics as well. A proof is not a proof unless the community can verify your result.

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I realize my answer has not directly responded to your question. In essence, I would say that you should always favour neatness over speed, but obviously within reason. Even if you do not subscribe to my rant above, there is a purely "mark-related" reason to be neat. When a marker has to mark 2000 papers, the amount of time spent on each individual paper will be minuscule. A clear solution will almost always receive a better mark than a correct but messy solution, since the marker does not want to hunt through garbage to find your work. –  Tyler Holden May 16 '13 at 17:19
    
Interesting response! –  user78101 May 16 '13 at 17:19
    
Your paragraph that begins with A mathematical solution is not just a correct number at the end of a computation ... parallels very well with what I used to tell students. (I no longer teach.) In lower level classes (precalculus and below) I would tell them that do look at their work. Nontrivial correct mathematics, even if the answer is wrong, will definitely get partial credit. I sometimes awarded bonus points for something especially good, even if the answer was wrong (in a minor way), so that someone could get more points than the problem was worth without getting it entirely correct. –  Dave L. Renfro May 16 '13 at 20:10
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At one time I was teaching a course I designed involving the basics of writing proofs (in elementary set theory and number theory) to some very strong high school students (one took a number of upper level undergraduate classes while in high school, another now has a Caltech physics Ph.D. and is well known in his field, etc.), and I was quite surpised (and frustrated) at how weak their homework write-ups were, so I came up with the idea of turning the tables on them, as this example will illustrate. –  Dave L. Renfro May 16 '13 at 20:21

My handwriting is pretty bad, I love $\LaTeX$, I've lectured on chalkboards/blackboards a couple of times, and given computerized presentations.

My general feeling is that you should make the general direction of everything you write in exams crystal clear, and keep letters/symbols distinct, but otherwise don't waste too much time on lovely handwriting. Structure and layout is far more important. Brief, simple sentences saying what you're about to do, bullet points, headings are all sensible measures.

Lecturing and teaching, however, are different. People should never even think about your handwriting. It should be completely out of their minds - completely normal, legible, consistent, boring.

The magic is always in the maths, and its flow through structured discussion, nowhere else.

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I find this interesting myself as I also have messy writing.

For me I always try to keep equal signs aligned and leave equal spacing. I also prefer using paper landscape as opposed to portrait but this is all just personal preference.

As for speed vs. neatness, I think it really is just about finding a fine line between them, neatness is important but you don't want to jeopardize leaving things incomplete.

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