Here is the opening of an early draft of an essay about teaching mathematics written years ago by a celebrated professor. As he himself has acknowledged, he is a less than gifted writer, and our goal is to maintain his ideas while presenting them more clearly and gracefully.

It is important to recognize the fact that every subject, given that its content is not totally reducible to some other subject area, presents a special set of pedagogic problems arising as a result of the distinctive character of their contents and their essential nature. The problems may be regarded as particularizations of the general pedagogical considerations which must be treated by any and all teachers who seek to seriously discharge his or her educational responsibilities in a highly efficacious manner.

Where to begin?

The opening construction (\”It is important to recognize the fact that …\”) is overwritten. Ninety-five percent of the time when you write \”the fact that,\” you can cut \”the fact.\” Let\’s do so here, and the phrase now reads: \”It is important to recognize that …\”

Better, but can we cut more? How about \”It is\”? When \”it\” has no antecedent, the combination is best avoided, and here is an ideal opportunity to excise it. But why not go further? After all, these opening words merely alert us to an \”important\” thought. Why not eliminate the warning and simply state that thought?

The sentence now begins: \”Every subject.\”

How about the next phrase: \”given that its content is not totally reducible to some other subject area\”?

First, the adverb \”totally\” can be cut with no damage. Either something is \”reducible\” or it isn\’t. How about \”area\”? Can we distinguish between a \”subject\” and a \”subject area\”? Not easily. Let\’s remove \”area.\”

Now think about the entire construction that begins \”given that ….\” What precisely does it mean? To be honest, we\’re not sure. After all, isn\’t every subject part of another subject? Therefore we ask whether the phrase \”given that its content is not reducible to some other subject\” contributes anything at all. Our answer: No. Our solution: Cut it.

Now just two words of text remain: \”Every subject.\”

Thus we move to the next phrase: \”presents a special set of pedagogic problems.\” The idea is important, but do we need \”special\”? Our author is discussing \”problems,\” and aren\’t all problems in some respect special? In other words, no problems are special. Therefore we need not say more.

Then how about \”a … set of\”? Why \”a set of pedagogic problems\” rather than just \”pedagogic problems\”? No reason.

Now only three words of this adjectival phrase remain: \”presents pedagogic problems.\” We place those after our opening two words, so that the first sentence reads: \”Every subject presents pedagogic problems.\”

Not bad, but it\’s flat, so we might want to add a modifier. Before we do, however, we note that the author has added a supplementary phrase of his own: \”arising as a result of the distinctive character of their contents and their essential nature.\”

In evaluating this thought, we see first that \”as a result of\” is wordy. Instead we could use \”from.\” Do we need the adjective \”distinctive\” before \”character\”? Probably not. To what word does the adjectival pronoun \”their\” refer? \”Subject.\” But \”subject\” is singular, and \”their\” is plural. Thus we must twice change \”their\” to \”its.\”

We move on. Do we need \”essential\” to modify \”nature\”? No. What is the difference between \”contents\” and \”nature\”? We\’re not sure, so one of the two words can go. \”Nature\” seems more interesting, so let\’s keep it.

Now the phrase reads: \”arising from the character of its nature.\”

Yet the construction remains redundant because \”character\” and \”nature\” are virtually synonymous. Of the two, \”nature\” is again the more interesting, so let\’s do away with \”the character of.\”

The phrase now reads: \”arising from its nature.\”

At this stage, having exerted so much effort to tighten the phrase, we must ask another painful question: What does the phrase \”arising from its nature\” contribute?

Reluctantly, we answer: Nothing. Why? Because everything has a \”nature.\” Why specify this particular \”nature\”? Thus the entire clause, even the pared-down version, may disappear.

Here, then, is the new opening sentence: \”Every subject presents pedagogic problems.\”

But as we noted earlier, that construction sounds flat. How about adding \”its own\” before \”problems\”? That insertion suggests that all subjects present challenges, but mathematics, the chief concern of this essay, has unique ones.

At last our tentative version of the opening sentence reads: \”Every subject presents its own pedagogic problems.\”

It\’s short, direct, and clear.

After drawing a breath, we move to the next sentence: \”The problems may be regarded as particularizations of the general pedagogical considerations which must be treated by any and all teachers who seek to seriously discharge his or her educational responsibilities in a highly efficacious manner.\”

Some potential alterations are obvious. \”Any and all\” can be reduced to \”all.\” The pronouns \”his or her\” refer to \”teachers\” and therefore should be \”their.\” The adverb \”seriously\” is superfluous and therefore can be omitted. Finally, the adverb-adjective combination \”highly efficacious\” could be simply \”efficacious.\” For that matter, \”in a highly efficacious manner\” could be \”efficaciously.\”

Still, we must wonder if these cosmetic changes help. Or do they avoid a larger problem? To decide, we ask: What does the author seek to say? His sentence implies that the special problems created by mathematics, those mentioned in the previous sentence, are related to general challenges encountered by all teachers, particularly those who are dedicated.

But aren\’t all teachers supposed to be dedicated? Certainly the ones who might bother to read our author\’s essay would be. And aren\’t those same teachers aware that their profession poses challenges? Absolutely.

Then what does that sentence add? Nothing. It merely reaffirms the concept that teaching offers many problems.

Thus we can cut the entire construction, a move that leaves us once again with nothing but our first sentence: \”Every subject presents its own pedagogic problems.\”

We may seem to have accomplished little, but in fact we\’ve made substantial progress: pruning excess verbiage, sharpening word choice, and clarifying exactly where we\’re headed.

Author Bio: Steven M. Cahn is a professor of philosophy at the Graduate Center of the City University of New York. Victor L. Cahn, his brother, is a professor of English at Skidmore College. Excerpted from their forthcoming book, \”Polishing Your Prose: How to Turn First Drafts Into Finished Work\” (Columbia University Press, 2013). Used by arrangement with the publisher. All rights reserved.