I am as critical about schools’ mindless bureaucratic methods as the next guy. But I am not sure I agree with this article when it comes to mathematics. This is one subject where grading might actually work perfectly well to serve as proxy for measuring understanding, provided exams contain the right kind of questions. These should perhaps then include both mechanical sorts of questions (calculate 4635 x 234) and insight questions (on a 2 x 3 meter roof, given a rainfall of 500 mm per year, how many litres of water can you collect off the roof in a year?)
With both of the above questions, there is only one correct answer, and in both cases there is no way in hell you can get to the correct answer without, in the first case, knowing how to do it, and in the second case, actually clearly understanding what you are doing.
Furthermore, with a subject like mathematics, I don’t see anything wrong with a certain amount of competition. I want the most mathematically adept students to go become scientists and engineers, not the ones who have most happiness or feel good about themselves. Those ones should go become psychologists, perhaps (a field in which a brilliant but nerdy and socially inept engineer may well be a walking disaster).
Trying to turn the mathematically dumb kids into engineers is as pointless as trying to turn the nerdy ones into football stars by forcing them to attend gym classes.
Now perhaps this is not what the article is implying we should do, and I have no objection to some of the suggestions made here. I like the idea of a mathematics portfolio, for example.