What Is the Difference Between School Math and Research Math?

In high school, our teacher used to like, solve all the problems out there. And then in the exam, he really tried very, very hard to put a problem that we've never seen before. But even in our class exams, the problems that we see in exams and tests would have to mabetententially be related to what we did in class. Solving a quadratic equation was never a problem. For example, here in montreal, in my college, we don't allow calculators because they calculate driveres. So proving something about a driverev, for example, finding a difficult limit with using lopita rol. I mean, it it's useful