So what could be so useful about depression? Depressed people often think intensely about their problems. These thoughts are called ruminations; they are persistent and depressed people have difficulty thinking about anything else. Numerous studies have also shown that this thinking style is often highly analytical. They dwell on a complex problem, breaking it down into smaller components, which are considered one at a time.
Basically, the idea is that sad bastards sit and home and think about their lives over and over again, and that this in turn causes their brain to function in a more intensely focused, analytical manner, which in turn makes them smarter, or at least better at critical thinking and problem solving.
These people who wrote the article are scientists, whereas I host a weekly web show in which I make sarcastic remarks about cat videos and Shane Dawson, so you decide who to trust...but I'm not sure I buy this idea. It implies that the thinking you do when you're depressed is, in some ways, analytical, and even productive. And it's not. Repetition alone doesn't count as critical thinking...and that's all depression is...the repetition in the mind of negative, hateful thoughts without an "off" switch.
It's not like depressed people are really looking at their lives analytically and breaking the shit down in an organized, strategic manner. "Well, let's see, on this side, I hate my father, but I love gelato! And I'm not happy at my job, but puppies! And I'm uncertain as to whether I'll ever find true love, and deep down, I suspect that it's all meaningless and I could die at any moment and no one would ever care...but, hey, that new Final Destination movie's coming out soon!"
I don't think there's anything productive about being depressed, and the idea that it might make you better at math strikes me as counter-intuitive. Sure, you're focused when you're depressed, but only on one thing - feeling depressed.