Imagine that you are giving directions to someone you don't know, and that your life depends on it. You would speak clearly, or write in block letters, use basic terms, and you'd rely on well-defined, universal concepts rather than on notions specific to your culture or environment.

It is much the same with mathematics. When you use words, think about their meaning. When you use symbols, make sure they will mean the same to both you and the professor. This is easy in math, because almost everything you will be writing about has a precise, agreed upon definition. A "continuous function", a "Riemann integral", or an "open interval" mean the same thing to everyone with mathematical training.

Let's end with my favorite quote from a posting by Herman Rubin: "Mathematics must be written so that it is impossible to misunderstand, not merely so that it is possible to understand."

For example, "1/(*a* - 4) is positive since *a* is
a real number greater than 4" is a perfectly valid statement;
it follows from basic properties of inequalities (we must have
*a* - 4 > 0, and reciprocals of positive numbers are
positive). There is not much of a chance that the professor will
mark this with a big red **WHY?**, unless you are taking an
upper level number theory course in which you are supposed to
derive such simple properties from even more basic principles.

But in a Calculus course a statement "1/(*a* - 4) is
positive because *a* is a real number" will be rejected
as a `non sequitur': the conclusion does not follow from the
premise. It may very well be that in the context of the problem
*a* is indeed greater than four, and that the conclusion
is true, but you have not explained this clearly!

For example, a relation such as "=" can only be placed between two quantities which have a chance of being equal (e.g. when solving an equation), or when you are asserting that the quantities are in fact equal. One of the most common problems we see in student papers is the lack of respect for equality and other relations.

Consider this example: "*x*^{2} - 1 = 0 = *x* =
1, -1" and look closely at all the equalities separately.
This is unintelligible babbling. It can be fixed by using symbols,
or simply short English words and phrases to indicate what is
happening: "if *x*^{2} - 1 = 0, then
*x*^{2} = 1, so that
*x* = 1 or *x* = -1".

We often use relations "chained together" to make a point; for
example: *x* > *y* - 1 > 5 - 1 > 4, so
that *x* must be greater than 4. This is perfectly OK as
long as the relation is `transitive' (as > happens to be).
But mixing disparate relations in this fashion can lead to
nonsense: *x* > *y* - 1 = 4 < *x*
+ 2 > 3 doesn't really say anything and resembles a banana
split with an anchovy topping.

So if you write *x* = 1.4, you'd better *mean*
1.4, and not the square root of two! When you are told to find
an integral of a function and you compute its derivative instead,
don't be surprised if you don't get any credit. If you write
about an isosceles triangle but your reasoning relies on it
being equilateral, you won't get many points...

This insistence on accuracy is not some type of malice, or the
professor's way of having fun; mathematics simply cannot exist
without the rigor of meaning exactly what you say, and your
papers cannot be fairly judged in any other manner. So don't
get frustrated or angry, don't say that "I had it almost right".
Just make sure you get it *absolutely right* on the next test.

If you can, write only as little as necessary to make the meaning
very clear. It isn't easy to judge exactly "how much is enough",
but you can try the following experiment: imagine that you have
a little brother in the class, that he is a few weeks behind in
his studies, that you are explaining the problem to him, and that
he wants to know *why* you are doing whatever you are doing.

But if you have doubts, it is generally better to say more than to say too little. You'll make your teacher's life a bit harder, and you might come across as a stickler for unnecessary detail, but you aren't likely to get marked down for that.

Please send comments, suggestions, corrections to behr@math.niu.edu.