Show Your Work.

One of my close friends tutors a couple of kids in math, and recently they were complaining to her about needing to show their work. She told them that she had a friend who could solve very difficult differential equations in his head, and that they had to show their work until they were right as often as he was.

There is a general viewpoint in academia that it is necessary to write down work because it helps avoid errors. Students have to show their work on tests so that professors can tell they aren't cheating. Not only do I respectfully disagree, but I believe they are completely missing the point. Encouraging students to avoid doing work in their head is actually doing them a significant disservice; fast mental math is a very valuable skill in the real world. Richard P. Feynman himself was very proud of being able to complete difficult arithmetic computations faster than most abacus users. Especially in engineering, the ability to ballpark estimate performance figures or material requires quickly without needing to reach for pen and paper is a very useful one indeed. Imagine being part of an aircraft design team and having your boss ask "what if we crammed 50 more seats into our new airliner?" You want to be the guy who can come right back with "the extra weight would decrease our range by about 5%." Aside from being a useful skill, doing work in your head does wonders for your mental sharpness and memory. Forcing yourself to keep track of all those variables without writing them down expands your mind and helps you think differently. It will help you remember what you're doing better and make you a better engineer in general. Both professors and students place far to little value on mental math abilities, and it's the students that eventually suffer for it.

That being said, I don't condone never showing work, especially on tests. Writing down every step helps you double check that your mental math is correct; being fast and wrong is worse than useless. More than that though, showing your work is a communications skill. What's the use of being able to do all the analysis in the world if your boss and colleagues can't understand it? Writing down and explaining everything you do is worthwhile because being able to quickly convince people you're right is as valuable a skill as being right fast, if not more so.

So where does that leave us? Do all of your problems completely in your head first, even on tests. Then write down your solutions in terms even a freshman could understand. This will help you get the most from your classes and be the most successful in life. One of my favorite professors put it this way: "There are three kinds of students: the smart ones who will start a problem immediately and slow down when they get a bit stuck, the stupid ones who will have no idea how to do it, and the true geniuses who will write nothing down until they've solved it then write down their whole solution in a minute or two."