Undergraduate mathematics is quite different from high school math, at least here in Canada. Study habits that give good grades in high school don’t translate well to university, especially for mathematics courses.

I did my bachelor’s degree in physics, with a math minor, and am now a PhD student in math. Over the first few years as an undergraduate, I learned how to study for math (and math-heavy courses like physics) in a way that let me do pretty well. Now I have another perspective, having taught and graded math courses at McGill and UW. These experiences inform the method that I am laying out here.

As many students have asked me for help learning how to learn and study in their math courses, I have had to formalize my learned method more and more, and have decided it makes sense to simply share it online for anyone to try.

Even the most noble students care about getting good grades. The goal of this study programme is to find a balance between optimizing for the highest grade, and optimizing for long-term, deep understanding of the mathematics. There is a happy correlation between understanding the material and getting good grades!

Finally, this method aims to front-load the work that gives you the best payoff for your effort. If you’re reading this the night before an important exam, getting through only the first step or two will put you much better off. The later steps have diminishing returns - often the hardest problems on an exam are worth only 10-20% of the grade.

Step 1: Definitions

The easiest and most important step to studying mathematics is to understand definitions. Many problems on quizzes and exams simply boil down to writing the definition of everything in the problem, and then performing some very simple algebraic manipulation. Although memorization is anathema to mathematics students, memorizing definitions is a highly efficient way to begin studying for a math course.

Begin by skimming through your course or lecture notes, and making a list of all the definitions in the course. For example, in a first calculus class, this might include

convergent sequence, cauchy sequence, continuous at x, derivative f’(x), even function, odd function,…

Then, for each item in your list, find the definition and write it down, in full. For example, you should be able to write “A sequence ${x_i}$ in $\mathbb{R}$ is convergent if there exists a $L\in\mathbb{R}$ such that, for every $\epsilon>0$, there exists a natural number $N$ such that, for all $n>N$, we have $|x_n -L| < \epsilon$.”

You should write these in sequential order of the course, as many later definitions will depend on earlier definitions.

One problem I see students fall into, is forgetting the original definition of something, in favour of a simpler characterization given by some theorem. For example, the definition of f is increasing is “for all $y>x$, $f(y)>f(x)$”, but many students will say f is increasing is “$f^\prime(x)>0$”. This latter statement is a theorem which is usually true, but not always. At least at the University of Waterloo, there are often questions on assessments designed to fit into the edge cases, testing if students know the proper definitions.

Once you have written down definitions for all the ideas in your course, try to memorize them. You don’t have to memorize them completely before moving on to step 2, as you’ll be using them in the later steps, which will help them stay in your memory.

Step 2: Key Theorems and Key Problem Types

After definitions, the second most important things in a math course are the theorems. However unlike definitions, there are typically too many little theorems to fully memorize – we’re trying to get the most out of our time here. Thus, we need some method to determine the key theorems in the course.

To do this, we’re going to make another pass over the course material. This time, you want to make a list of theorems. Theorems with names, like Intermediate Value Theorem are clearly important, but many key theorems do not have names, such as If f is differentiable at a, then f is continuous at a.

After making the list of theorems, now go over the assignment questions, practice questions, and any quiz or midterm problems that you have access to. Categorize the problems into types, and for each type of problem, write down what theorems are needed in the solution.

For example, one type of question would be find the derivative of a function, and the theorems used include chain rule, product rule, linearity of the derivative and so on.

Now, the key theorems are the ones that appear in many types of question. Just like with the definitions, you should now go and write down all the key theorems in full. Make sure you write down the hypotheses of the theorem, and not just the conclusion. For example, when I ask students to tell me the Mean Value Theorem, many say “$f^\prime(c) = \frac{f(b)-f(a)}{b-a}$”. This is only the conclusion; the full theorem is

“Let $f$ be continuous on $[a,b]$ and differentiable on $(a,b)$, $a<b$. Then there exists $c\in(a,b)$ such that $f^\prime(c)=\frac{f(b)-f(a)}{b-a}$.”

Knowing the hypothesis is critical for true and false or multiple choice questions – often the answer boils down to checking the hypotheses for a suitable theorem.

If you’ve made it this far, I am very confident that you will pass the course, and probably with a decent grade.

Step 3: Grinding Problems

This step is very straight-forward. Just get homework problems and practice problems, and do them. When you work out these problems, try to do it in a test-like environment. If your exam is not open book, don’t do practice problems with your notes on hand. Don’t do them in a group, and preferably do them without music or anything, just like you’re writing the exam.

Many students jump straight into problems when they’re studying for an exam. However, if you don’t know the course’s theorems, and especially if you don’t know the definitions, many problems will be exceedingly difficult. This leads to students frequently needing to look up solutions, and exhausting the pool of questions they can study “fresh”.

However, now that you know your theorems and definitions, your success rate on the problems should be much better. If you don’t know how to approach a given problem, begin by writing down the definitions in the question. Often, that will make you see the correct approach. If not, start thinking about all the theorems that apply to that problem type, and if their hypotheses hold in your problem.

Let me show you one example of this approach.

Question: Let $f(x)$ be differentiable on $\mathbb{R}$. Furthermore, let $f(x)$ be positive and concave up. Then show that $g=(f(x))^2$ is concave up.

To solve this, we begin by writing all the definitions:

We know:

$f$ diff’able - $f^{\prime}(x)$ exists for all $x\in\mathbb{R}$

$f$ positive - $f(x) > 0$

$f$ concave up - $f^{\prime\prime}(x) > 0$

$g = f^2$

and we need to prove:

$g$ concave up - $g^{\prime\prime}(x) >0$.

Now its clear that the question is asking us to show $g’‘(x) >0$. So far, we may not see how to do the problem, or why we need all these conditions on $f$, but we have a clear starting point – compute $g’‘(x)$.

\[g'' = (f^2)'' = (2f(x)f'(x))' = 2(f'(x))^2 + 2f(x)f''(x)\]

We suddenly have $f^\prime(x)$, $f^\prime(x)$ and $f^{\prime\prime}(x)$ showing up! Now it is obvious how we need to use the conditions in the question. We use $f^{\prime\prime}(x)>0$ and $f(x)>0$ to say $2f(x)f^{\prime\prime}(x)>0$, and we know any real number squared is positive. Thus we can conclude $g^{\prime\prime}(x) >0$, finishing the problem.

Bonus Step 4: Understanding Proofs

With the first three steps completed, you’re probably more prepared for the exam then most of your classmates, and have nothing to fear from the exam. However, if you want to ace the test, or if you just want to deeply understand the course material, there is another step to take. To be completely clear, you’re now in the realm of diminishing returns for your grades. However, this is the richest territory for understanding the mathematics.

For each of the key theorems you learned in class, there should be an accompanying proof. In this step, you are going to analyze these proofs, to understand why the theorem is true. For each theorem, I suggest the following approach to understanding the proof.

a) Read the proof, and make sure you understand why each step follows from the last.

b) Try to pick out a key idea behind the proof. Often the bulk of the proof is just simple algebra, but there is one or two clever steps that you probably wouldn’t think of on the spot.

For example, for the proof of f differentiable at a implies f continuous at a, the key idea would be “multiply the definition of the derivative by $(x-a)$”. The rest of the steps are straightforward to come up with if you know that key step.

c) Next, you’ll analyze the hypotheses of the theorem. For each condition in the hypothesis, identify where it is being used in the proof. Try to come up with an counter example that shows why that hypothesis is necessary. Doing this will help you understand exactly the role of each piece of the proof.

Conclusion

If you implement this method for your math course, I would love to hear how it went! It worked well for me in my math and physics education, but perhaps it doesn’t work for everybody. Any feedback is welcome here.