It’s that time of the year. Classes are starting up. You’re nervous and excited to be taking some of your first “real” math classes called things like “Abstract Algebra” or “Real Anaylsis” or “Topology.”
It goes well for the first few weeks as the professor reviews some stuff and gets everyone on the same page. You do the homework and seem to be understanding.
Then, all of a sudden, you find yourself sitting there, watching an hour-long proof of a theorem you can’t even remember the statement of, using techniques you’ve never heard of.
You panic. Is this going to be on the test?
We’ve all been there.
I’ve been that teacher, I’m sad to say, where it’s perfectly clear in my head that the students are not supposed to regurgitate any of this. The proof is merely there for rigor and exposure to some ideas. It’s clear in my head which ideas are the key ones, though I maybe forgot to point it out carefully.
It’s a daunting situation for the best students in the class and a downright nightmare for the weaker ones.
Then it gets worse. Once your eyes glaze over that first time, it seems the class gets more and more abstract as the weeks go by, filled with more and more of these insanely long proofs and no examples to illuminate the ideas.
Here’s some advice for surviving these upper division math classes. I’m sure people told me this dozens of times, but I tended to ignore it. I only learned how effective it was when I got to grad school.
Disclaimer: Everyone is different. Do what works for you. This worked for me and may only end up frustrating someone with a different learning style.
Tip Summary: Examples, examples, examples!
I used to think examples were something given in a textbook to help me work the problems. They gave me a model of how to do things.
What I didn’t realize was that examples are how you’re going to remember everything: proofs, theorems, concepts, problems, and so on.
Every time you come to a major theorem, write out the converse, inverse, switch some quantifiers, remove hypotheses, weaken hyphotheses, strengthen conclusions, and whatever you can think of to mess it up.
When you do this you’ll produce a bunch of propositions that are false! Now come up with examples to show they’re false (and get away from that textbook when you do this!). Maybe some rearrangement of the theorem turns out to be true, and so you can’t figure out a counterexample.
This is good, too! I cannot overstate how much you will drill into your memory by merely trying unsuccessfully to find a counterexample to a true statement. You’ll start to understand and see why it’s probably true, which will help you follow along to the proof.
As someone who has taught these classes, I assure you that a huge amount of problems students have on a test would be solved by doing this. Students try to memorize too much, and then when they get to a test, they start to question: was that a “for every” or “there exists?” Does the theorem go this way or that?
You must make up your own examples, so when you have a question like that, the answer comes immediately. It’s so easy to forget the tiniest little hypothesis under pressure.
It’s astounding the number of times I’ve seen someone get to a point in a proof where it looks like everything is in place, but it’s not. Say you’re at a step where is a continuous map of topological spaces, and is connected. You realize you can finish the proof if is connected.
You “remember” this is a theorem from the book! You’re done!
Woops. It turns out that has to be surjective to make that true.
But now imagine, before the test, you read that theorem and you thought: what’s a counterexample if I remove the surjective hypothesis?
The example you came up with was so easy and took no time at all. It’s given by . This example being in your head saves you from bombing that question.
If you just try to memorize the examples in the book or that the professor gives you, that’s just more memorization, and you could run into trouble. By going through the effort of making your own examples, you’ll have the confidence and understanding to do it again in a difficult situation.
A lesser talked about benefit is that having a bunch of examples that you understand gives you something concrete to think about when watching these proofs. So when the epsilons and deltas and neighborhoods of functions and uniform convergence and on and on start to make your eyes glaze over, you can picture the examples you’ve already constructed.
Instead of thinking in abstract generality, you can think: why does that step of the proof work or not work if ?
Lastly, half the problems on undergraduate exams are going to be examples. So, if you already know them, you can spend all your time on the “harder” problems.
Other Tip: Partial credit is riskier than in lower division classes.
There’s this thing that a professor will never tell you, but it’s true: saying wrong things on a test is worse than saying nothing at all.
Let me disclaimer again. Being wrong and confused is soooo important to the process of learning math. You have to be unafraid to try things out on homework and quizzes and tests and office hours and on your own.
Then you have to learn why you were wrong. When you’re wrong, make more examples!
Knowing a bunch of examples will make it almost impossible for you to say something wrong.
Here’s the thing. There comes a point every semester where the professor has to make a judgment call on how much you understand. If they know what they’re doing, they’ll wait until the final exam.
The student that spews out a bunch of stuff in the hopes of partial credit is likely to say something wrong. When we’re grading and see something wrong (like misremembering that theorem above), a red flag goes off: this student doesn’t understand that concept.
A student that writes nothing on a problem or only a very small amount that is totally correct will be seen as superior. This is because it’s okay to not be able to do a problem if you understand you didn’t know how to do it. That’s a way to demonstrate you’re understanding. In other words: know what you don’t know.
Now, you shouldn’t be afraid to try, and this is why the first tip is so much more important than this other tip (and will often depend on the instructor/class).
And the best way to avoid using a “theorem” that’s “obviously wrong” is to test any theorem you quote against your arsenal of examples. As you practice this, it will become second-nature and make all of these classes far, far easier.