How to Communicate Mathematical Ideas Clearly

If everyone followed these tips, the world would be a better place. Unfortunately, many people, including many mathematicians (and in particular, many senior textbook authors), do not. However, when you see a poorly explained solution or an exceptionally badly written text, you should not think to yourself, "Ah, this means I can be just as obfuscatory and obscure as this author is in my solutions." Rather, think to yourself, "I hate reading this, and I would never want to inflict this experience on anyone." Strive to do better.

• Be clear with what you are trying to solve.

State the problem clearly. In English (not just in math). Draw a picture. State your givens. Word problems always need "Let.." and "Therefore..." statements.
 
• Don't skip steps.

Textbook authors, you may have noticed, throw words like "clearly," "trivial," and "obvious" around with gleeful abandon. This does not give you license to do the same. First of all, some things may seem "clear," but are often actually quite tricky. Some things may seem "clear," but are, indeed, actually false. Always check your reasoning, even the "trivial" steps (especially the "trivial" steps). Also, consider that your reader might not be as inspired and enlightened as you are when you write the solution (and that reader might be you, six months from now), so don't hesitate to explain things in pedantic, intelligence-insulting detail.

There is another reason you shouldn't skip steps, even "obvious" ones: the marker might not believe you. The logical extreme of skipping steps is to simply write down, "Proof: Obvious. QED." as an answer to every homework question, and no one is going to give you marks for that. When in doubt, show your work, even if you think it's obvious, trivial, boring, etc.
 

• Learn the proper mathematical notation and when to use it.

Believe it or not, the reason all this confusing mathematical terminology and cryptic notation was invented was to make it easier to communicate mathematical ideas clearly. Clear communication of mathematics is always the most important goal. When used properly, in the appropriate situation, this cryptic notation can facilitate communication. When used unnecessarily or wrongly, it often serves only to obfuscate.

Many students get in the habit of trying to compress their work into mathematical symbolic shorthand as much as possible, because they think that this somehow makes it more "mathematical." It doesn't; it just makes it impossible to read. There is nothing wrong with expressing an argument in plain old English; far from it, this often makes it clearer and easier to understand.

Furthermore, don't "invent" new mathematical notation. If a variable is introduced in the question, use the same variable. Use standard variables for physics problems. It's all very well to use """ when you mean "p", but no one (including you, six months later) will know what you're talking about (and the other mathmaticians will laugh at you).
 

Finally, remember that math is a language. Math sentences have subjects and predicates, just like spoken language. In fact, you should be able to read your solution out loud and have it make sense in English/French/Mandarin. Don't leave mathematical expressions, such as (x + 1) lying around unattached or unexplained. (See What about the cat? in the Marking Key section.)

• Break down solutions into modules as much as possible.

A good solution can be broken down into a series of seperate pieces; like subroutines in a computer program; like parts in an automobile engine; like acts in a play; chapters in a book; or moves in a chess game. The written version of the solution should clearly reflect this logical structure. Often, you will find you will have to solve mini problems in order to find the ultimate solution. You should indicate when you are digressing from the main point.

For example, a sketch of a solution might look like this:

Problem:

ABCDEFG

Solution:

Sketch

Givens

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In order to solve ABCDEFG, I'm going to have to solve problem XYZ:

Solution:

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Therefore the answer to XYZ is ...

Subbing this into the equation...

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To continue solving ABCDEFG, I have to solve problem LMNOP:

Solution:

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Oops, to find LMNOP, I need to solve QRST:

Solution:

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Therefore, the answer to QRST is ...

Continuing on with problem LMNOP:

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Oops, now I need to solve yet another pesky problem UVW...

Solution:

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Therefore, the answer to UVW is ...

Are we there yet?

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I'm glad that's done.

Therefore, the answer to LMNOP is...

Now we can finish the main solution....

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Therefore, the answer to ABCDEFG is ...

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