Math problems on standardized tests have short, simple answers that fit in a multiple-choice format. But college-level math problems require more detailed explanations. In How to Write a Math Solution, Richard Rusczyk and Mathew Crawford of Art of Problem Solving present a detailed checklist for ensuring that your proofs and solutions communicate your thinking as clearly as possible. Here’s some advice from the article that I found especially useful.
Don’t Make Me Think
Your goal in writing a clear solution is to prevent the reader from having to think. … The experienced reader should never have to wonder where you are headed, or why any claim you make is true.
In Why are Textbook Proofs so Short?, I pointed out how different audiences require different types of proofs. Textbook authors write short proofs to encourage their readers to fill in the gaps and to save space in a printed book. But in most other cases, you don’t want to skimp on details in your proofs. As Rusczyk and Crawford put it, “It’s better to prove too much than too little.” A reader who finds a proof section obvious can always skip it. But if your reader can’t figure out how you got from step 10 to step 11, they may wonder if you even know. Or they may get frustrated and give up.
The reader may even be a future version of you. Even if a proof is obvious to you when you write it, it may be incomprehensible when you come back to it in a year. Programmers run into this when they try to fix bugs in code they haven’t looked at in a while.
Since you don’t have unlimited time to write, you can’t derive everything from first principles. So you have to consider which details will best serve your audience. Rusczyk and Crawford provide a set of six guidelines for deciding when you need to prove a theorem you use in your argument, and when you can just cite it. The easy cases are famous named theorems (e.g., the Pythagorean Theorem). You can assume that everyone knows those, so you can just refer to them by name. For other theorems, it’s a judgement call. If there’s any doubt, fall back on the general principle that more explanation is better than less.
Once Upon a Time…
Proofs are a lot like stories. When writing a solution your job is tell a math story in a way your audience will understand and enjoy.
You can’t read a proof like you read a novel, and the enjoyment you get from a well-written mathematical argument differs from what you get from fiction. But some elements of good writing apply to both novels and proofs. Rusczyk and Crawford promote the value of naming your characters (with names like $x$ and $n$) at the beginning of the story. Introducing the characters in this way prepares the reader to follow their involvement in the story, making it clear who is doing what. Including a cast of characters at the beginning of your proof is the first step in writing an engaging account of their escapades.
Proofs can also benefit from nonfiction writing guidelines. Public speaking experts advise their students to “Tell the audience what you’re going to say, say it; then tell them what you’ve said.” Rusczyk and Crawford suggest using the same structure for proofs: Tell the reader what you’re going to prove and how you’re going to prove it, prove it, and then remind them that you proved what you said you would prove. As in a speech, you don’t want to lose your audience if their attention wanders, so help them remember what you’re up to.
Since proofs are nonfiction writing, nonfiction advice may be more practical for proof writers than advice for novelists. But I still like the idea of drawing the reader in with engaging characters and exciting plot structure. If the goal of a proof is to convince (for example, to convince a grader that your proof is correct), then you’re more likely to do that with an interesting piece of writing than with a dry list of mathematical statements.
“Think Backwards, Write Forwards”
Think of solutions as recipes. Start at the beginning and move forward. List the ingredients and explain how and when to add them to the pot.
Rusczyk and Crawford also compare mathematical solutions to recipes. In this analogy, a good solution helps the reader reproduce your argument as if they were cooking a dish from a cookbook. If you do your job, they can follow your instructions easily and end up with a tasty dish: understanding why a mathematical claim is true.
Just as cooks need clear instructions in the kitchen, scientists and engineers appreciate advice on solving problems step by step. The description on the back of each volume of The Art of Computer Programming claims that “practicing programmers have successfully applied [Knuth’s] ‘cookbook’ solutions to their day-to-day problems.” In more recent years, programmers feel a warm glow when they find a Stack Overflow answer with actionable advice to solve their problem. If you write a math solution in a form that you could post on a Q&A site for someone to use in their own work, your readers will thank you.
To produce a recipe-like math solution, Rusczyk and Crawford tell you to “Think backwards, write forwards.” This aphorism relates to how people typically solve math problems. When you first approach a difficult problem, you don’t proceed in a linear fashion through a set of solution steps. Instead, you try out different ideas to see what works, gradually building up pieces of the solution. This is the “think backwards” step. Sometimes, you might literally solve the problem backwards, starting with the desired result and working towards the givens.
But the person reading your solution doesn’t want to follow you through the twists and turns you took to discover a solution. They just want the cookbook instructions to get from the beginning to the end. That’s the “write forwards” step. Once you solve a problem, you have the required steps, but those steps probably aren’t in an optimal order. Think about how to order the steps for the reader’s convenience.
The Purpose of Mathematical Writing
In his commentary on the Rusczyk and Crawford article on Quora, Alon Amit identifies an exception to their recommendation. When he writes math answers on Quora, he often structures them in a roundabout way, ignoring the “think backwards, write forwards” advice. The reason for this: his readers are trying to learn mathematical thinking in general, not just the answer to a specific problem. By documenting his thought process rather than just his solution, he can illustrate how an experienced problem-solver approaches a problem.
But in most cases, it’s still best to introduce your characters, tell a story in order from beginning to end, fill in as many details as you have time for, and follow the rest of Rusczyk and Crawford’s advice. That will give you the best chance of communicating your thinking to your readers and clarifying it for yourself.
(Image credit: romana klee)