## Aggravating factorials

This week, fool that I am, I’ve been marking a homework question that involves factorials. The question was intended to identify weaknesses in the students’ understanding, and in that sense it’s been a success.

Q: “Factorise the expression $(n+1)!(n-1)!-n!n!$ and hence simplify it as much as possible.”

A: $n!(n-1)!$ (but you knew that, right?)

I was expecting a few students to be misled by the superficial resemblance to $(n+1)(n-1)-n^2 = -1$, and to try to convince me that the answer was $(-1)!$ . To my surprise, this didn’t happen. Instead, I got an awful lot of instances of the following, at various stages in the working:

(a) “$(n-1)! = (n-1)n!$” (very popular, though this may reflect collaboration);

(b) “$n!(n-1)! = n(n-1)! = n!$” ;

and of course our old friend

(c) “$(n+1)! = n!+1!$” .

What all these students seem to have fallen victim to is that perpetual temptation, reasoning by vague analogy. In (a), the analogy is presumably with a result they’d used on the previous line, $(n+1)! = (n+1)n!$ . In (b), the analogy is presumably with a rule of indices, $n^a(n-1)^a = [n(n-1)]^a$, combined with a touch of bracket-blindness. In (c), as ever, the analogy is with linear functions — the good old Freshman’s Dream — itself, I suspect, arising from a vague analogy between $(x+y)^2$ and $2(x+y)$.

I also wonder whether one problem with factorial notation is that it looks “silent”. My impression is that students are particularly prone to omit symbols that they don’t know how to pronounce or that are silent in normal reading. Brackets are the obvious example — and even those students who do use them are often blind / deaf to the distinctions between ( , { and [ , as setting questions on set and interval notation reveals. “Cup” and “cap” ($\cup$ and $\cap$) are another two that are frequently omitted altogether or replaced by the universal comma. I’m not sure to what extent students are reasoning verbally rather than symbolically when they make these mistakes, but it would certainly explain a few things if they were.

The factorial function was introduced to these students a couple of weeks ago, and although it’s been covered in at least one batch of tutorial work it’s still relatively new to them. I guess what these problems suggest is that when faced with an unfamiliar function with offputting notation, rather than going back to the definition a lot of students grasp about in the dark for something analogous and just hope it’ll work. In concept image / concept definition terms, rather than trying to build a new concept image for factorials, they’re trying to take an existing concept image for something else and relabel it.

This, in turn, suggests to me that if I could find a way to make my students realise when they’re reasoning by vague analogy then I could help them to dynamite a lot of their problems overnight. (No, just telling them about it doesn’t work: I’ve tried that.) Maybe when they get their assignments back, with these errors carefully pointed out, it’ll help in this instance — but I wish I knew how to drive the point home in general…