Many mathematicians are very fond of saying that, and it is only too true! You cannot learn maths without doing maths. In this post I’m going to explain some different ways you can find and try of doing maths for different learning purposes.
In my lectures, as in many lectures, there are now and then “exercises”. These do not appear on the Example Sheets. So why are they there? Am I just to lazy to prove these things in the lecture? No!
Exercises are intended to be straightforward bits of proofs or checks, which are an excellent opportunity for you to check your understanding. “Straightforward” does not necessarily mean “easy”. “Easy” is a subjective judgement, and how easy or hard you find a given exercise depends on how well you understand the concepts, definitions, objects etc which the exercise involves. So in that way, doing exercises helps you check your understanding. But they also help you to gain understanding: for some topic you are finding hard to get into, the more straightforward exercises are a very good place to start exploring, practising, thinking about the new topic.
Example Sheet questions are (certainly in Cambridge) meant to be harder, they are designed to teach you new ideas or extend ideas mentioned in lectures. So you will need to do more “problem solving” and more creative thinking on the example sheets. This is where you start understanding topics in much more depth. The example sheet questions and your solutions to them are discussed in supervisions with your supervisor (if you’re a Cambridge undergraduate; in other universities I expect you have something like examples classes where these questions are discussed).
As the exercises are not something which is discussed in supervisions, how can you know whether you have done them correctly and really understood? Well, many students can tell on these more straightforward exercises whether they have got it right or don’t know what to do. But just in case, I have put solutions to these exercises onto the course Moodle page, so you can check your answers.
How should you use such provided solutions responsibly? If you check a solution before having tried the question sufficiently, you rob yourself of many learning opportunities. As maths is not a spectator sport, just reading the solution (and even understanding the solution) does not teach you anywhere near as much as if you try to do the exercise yourself. You need to work through the steps of thinking what you might try to do, doing it, seeing if it works, and if it didn’t work reflecting on why not, what else you could try, and so on. Even if you try very hard and have 4 or 5 different wrong approaches to a problem, you will learn much much more than if you just look up a solution at the beginning.
So, responsible use of any solutions or “cribs” is: use them only once you have worked seriously on the problem/exercise/question. Then use them as hints: if you’re stuck, read only far enough until you have another idea of how to proceed. Then go back to trying yourself. When you are done with a question, think over what you have done and why it worked, and then you can also check the answer.
It is very important to keep in mind: if you use solutions irresponsibly, the only person you are harming is yourself. You’re responsible for your own learning at university, you are encouraged and required to develop into an independent learner. We (try our best to) give you help along the way.
What if some lecture doesn’t have any or enough exercises of the straightforward kind for you to try? Well, most lectures have “hidden exercises” which you can turn into exercises for yourself:
- Examples which are just stated rather than shown in full: fill in the details, check why it really is an example.
- “Bigger” jumps in proofs: fill in the details, make sure you can get from one step to the next, understanding entirely everything that is going on.
- Find your own examples and counterexamples.
So, in summary: Maths is not a spectator sport. Take all the opportunities you can to do maths yourself.