What makes a good mathematical justification?

One of the biggest challenges a lot of students face when learning college-level mathematics is justifying one’s work. It’s not enough in a true college-level math course to compute a right answer; you must be able — at least in theory and most of the time in practice — to explain to an appropriate audience how you got your answer and why you followed the solution method you did. Students have an awfully hard time seeing past the supposed all-importance of The Right Answer to the true importance of the right solution, justified at each critical step by math, English, or (most of the time) a linear combination of the two.

In my estimation, it’s the process of explaining oneself that separates college math from high school math. A justification in a mathematical setting teaches an important writing style that I have never actually seen taught in composition classes: Writing a brief, information-packed statement that gives the reader a solid reason to believe your conclusion, without wasting the reader’s time on superfluous items on the one hand while not making the reader work hard on the other. In other words, the act (art?) of stating your conclusion; getting to the point in explaining it; and then getting out of the way.

Here is a question from the most recent test in calculus that has to do with identifying the graph of a derivative from among a lineup of three possible choices. But more than that, it’s a problem that assesses students’ abilities to write effectively in a technical setting. (Click to enlarge.)

If you know the concept of the derivative, it’s very easy to answer this question in a succinct style like I mentioned above. For instance:

The leftmost graph is the derivative. If you look at x=0 on the original, we see that the graph of f(x) is increasing there. That means that the derivative must be positive at x = 0. The leftmost graph is the only one that is positive at x = 0.

We state the answer and then give just enough explanation to explain why the answer must be right — and no more. There are other correct ways to identify the right graph. But here are some ways that I found in the student responses that work against a high-quality justification:

• The graph on the left is the correct one because the slope of the original tells you the height of the derivative.” This is, of course, a correct statement. The problem is that there is no demonstrated connection between the answer and the explanation. I had several responses that said the graph in the middle, or on the right, was the correct one — but which also had the exact same justification (“…because the slope of the original tells you the height of the derivative”). Just being able to state a definition or concept correctly does not imply that you know how to draw the right conclusion. There has to be a specific point of contact between the general idea (stated correctly here) and the specific instance (not mentioned here).
• Responses that identify EVERY POSSIBLE THING that could tell us that the left-hand graph is the derivaitve. This is the most common student error in written work — saying too much. Most students feel that quantity is quality, and that any piece of work that goes on for a long time or a large amount of space must be good, in proportion to its length. In this case, the students said more or less what I gave above as an example — but then listed Exhibits B, C, …, Z that also show that the left graph is correct. This is not actually wrong, but it’s not good form; justifying an answer isn’t like trying a court case, where one has to build a mountain of evidence to win. One logical argument, no matter how short, will force the reader to either accept your conclusion or question your assumptions.
• The leftmost graph is the derivative because it is crossing the x-axis at the appropriate places” or “...where it should be.” The subjunctive mood is the killer of all technical justifications. Couldn’t I give an incorrect answer and the same justification? What exactly does “appropriate” mean? Where “should” the derivative be negative, and does your answer fit that description? True justifications must get right at the heart of the matter — using specific, straightforward reasoning and NO WEASEL WORDS.

It’s certainly not automatic for students coming out of high school to write like this. English departments tend not to focus on short, tight writing styles. Math departments don’t often focus on writing at all.