Exercises are Written with Intention
Date Written: June 19, 2024; Last Modified: June 19, 2024During lectures, definitions, theorems, and propositions are often put in a larger mathematical context, perhaps by providing motivation from a more grounded field or by describing a theoretical problem that one wants to solve. Similarly, homework problems are often (but not always) chosen to supplement lecture content, for instance to give an unintuitive example of a mathematical phenomenon or to fill in certain steps of a techincal proof.
I think almost every mathematical exercise is written with some larger context in mind, though it’s really easy to neglect this. I think being aware of these broader intentions (though not always possible) is important for identifying which parts of the exercise are most important. These intentions could include:
- A computation that illustrates a certain technique, e.g. contour integration or classifying groups using Sylow’s theorems.
- Demonstrating an unintuitive property of a certain mathematical object, typically to give an example qualifying a claim or a counterexample demonstrating the importance of an assumption.
- Showcasing a surprising application of a theorem or a particular kind of argument.
- Exploring to what extent a certain theorem or statement holds or generalises when certain assumptions are removed, added, or modified.
On exam day, have this meta-awareness is useful for identifying what sorts of techniques and theorems one should have in mind. For instance, one might find the following classic exercise in a section of a textbook about the Stone-Weierstrass theorem:
Problem 1.
Let \(f\in C \left( \left[ 0,1 \right] \right)\) satisfy \[\int_0^1 f(x) x^n dx = 0\] for all integers \(n \geq 0\). Show that \(f\) is the zero function.
It might take a while, but one will eventually realise that the intention is to somehow apply Stone-Weierstrass, either organically or by understanding its placement in the textbook (thereby deducing the author’s intention). On exam day, one might then come across the problem
Problem 2.
Let \(f\in C \left( \left[ 0,1 \right] \right)\) satisfy \[\int _{0}^{1}f(x) e ^{ax} dx = 0\] for all \(a \in \mathbb{R}\). Show that \(f\) is the zero function.
First, it’s invaluable to just know immediately that this problem is some application of Stone-Weierstrass, and sometimes this is enough to just piece together the solution directly.
Better than that, one might realise, “Hey, the exam writer probably wants to examine my understanding of Stone-Weierstrass. I may not have the time to work this problem out in full detail, but maybe I can sketch a rough solution and get partial credit.”
At least when I grade exams, having the right idea will usually net a good chunk of partial credit, even if a lot of details are missing or misguided. A lot of work in the completely incorrect direction is usually worth less than a tiny bit of work in exactly the correct direction. Understanding the exam writer’s perspective helps with orienting oneself.