A question arised while I was reading "A Guide to Write Mathematics" (by Dr. Kevin P. Lee, p. 5) about writing explanation of steps, in equations calculations or listing some equations, without using two columns like this:
I don't think is bad or hard to read, actually I believe is easier to follow up the calculations. Do you have any arguments for not to do it?.
I attach some examples of how do I apply it.
I'd appreciate your comments.
On
In my experience teaching university mathematics (admittedly in the US) for close to 40 years, the "two-column proof" style that grew out of high school geometry classes seems to have been responsible for making 99% of mathematics students hate proofs (and, in some cases, mathematics). Trying to belabor minutiae, including (often numerical) references to every little fact or proposition does mathematics a disservice. People should be learning to write mathematics as prose, and to justify significant statements, not every pedantic step of algebra. My university students were relieved to move on to a more mature, holistic way of thinking and writing about mathematics.
On
The author of the guide is trying to teach some elements of compositional style, emphasizing the idea that mathematics should be presented in a way that is readable as sensible English sentences. From this point of view, the two-column display reads as an alternation of declaratives and imperatives (i.e., statements and commands); substituting "expression" and "unknown" for the precise formulas and writing out the equal sign as the verbal "is equal to," we have the following piece of prose:
A complicated expression is equal to negative one. Solve this equation. The complicated expression plus one is equal to zero. Collect the terms on one side. The square of a simpler expression is equal to zero. Factor. A very simple exponential is equal to one. Use the Zero Factor Property. The unknown is equal to one. Solve for the unknown.
Compare this with the guide's recommended presentation excerpted in Matthew Daly's answer. Among the defects of the two-column presentation, most of the commands (Collect! Factor! Use! Solve!) come after the command has been carried out; only "Solve this equation" precedes the steps that do what that sentence commands. One can, of course, observe that that's not how the imperatives are meant to be understood, but that's sort of the author's point: good, readable, mathematical writing does not contravene the rules of good English (or whatever language is being written in).
For people who avoided clicking through to the article (which is well written and mostly uncontroversial and very appropriate for newbie math majors), this is what Dr. Lee suggested as a preferred style for the above proof (even though it has a mistake in the factored line):
I think it's largely a matter of opinion and the other proof isn't drowning-kittens bad by any means, but I think Lee is making a useful point here. The weakness of the two-column style is that one feels indebted to give explanations for all of the shown steps even when they are trivial things like "Use the Zero Factor property". In his preferred style, all of the non-trivial steps are explained as part of the narrative flow while the trivial steps are appropriately demoted to in-line formulas with no justification.
Certainly there are times when the judgement falls in favor of justifying every line of the proof. Axiomatic set theory, first order logic, and high school geometry come immediately to mind here. But for the most part, mature mathematical writing is much closer to a persuasive essay in content and style than a grid. Undergrads would be well served to practice this mixed style of switching between ordinary language and the symbolic language whenever it is appropriate to do so.