Exercises play a central role in mathematics. You don't really do mathematics before you start to solve a problem. To do this, one has to analyse the problem and play with it in order to finally solve it with imagination and a sense of elegance and symmetry. Problem assignments are the natural way to acquire these skills.
Consider each exercise as an intellectual adventure. The harder the problem, the bigger the adventure. You do not learn mathematics from books or lectures, but only by doing it yourself. This is exactly what the exercises are meant to achieve. A single self-contained exercise replaces ten comprehensible examples in a textbook! There is no other way to learn maths. Hence, solve exercises.
It is not about preparing for the final exam. Conversely, the final exam is meant to assess whether you have learned to solve problems.
Among the exercises, there will certainly be some which practise the material and methods discussed in the lecture. Beware of just solving those. Such exercises are only meant as a warm-up. The actual learning effect arises only when you stretch your mind a little beyond the already known. Problem assignments are therefore different from the usual homework assignments in school. In particular, copying doesn't make much sense.
Use the available time!
You should use the available time before the submission deadline from the very first moment. If you take your first look at the exercise sheet only two days before the deadline, you have wasted valuable time.
Only some of the problems are designed to be solved effortlessly. Some exercises can be solved mechanically, and these are meant to practice a certain calculation process. But other problems will require careful thinking. Problem solving is a creative process. You cannot expect to come up with a brilliant idea after staring for five or ten minutes at the problem sheet. Many ideas must ferment and mature in the subconscious before they come to light as a solution. You should try to think about the problems in your spare minutes, in the shower or on the bus or when queuing in the supermarket, or at least give your subconscious the opportunity to do so.
This is only possible if you are at least aware of the problem and understand it. You should therefore start to think about the problems on the day you receive the exercise sheet.
A hard problem needs constant attention. You might have to return to the problem
again and again up to the point of submission. It is certainly the wrong
strategy to say:
I solve exercises only Tuesdays from 4 to 6 pm. Make
full use of the period before the submission deadline: even if you already
solved the problem, it may be worthwhile thinking about whether you can
simplify the solution or make it more elegant, or whether there are
alternative solutions altogether.
It might seem to be an obvious fact, but one first needs to know and understand the problem before one will be able to solve it. In order to give yourself the opportunity to ponder about a problem whenever it comes to your mind you need to be able to formulate the problem without looking at the exercise sheet. This does not mean that you have to memorize the problem as written, word-for-word, but you should at least understand the problem. Hence, when have your first look at a problem you should think about it at least until you are able to explain the posed problem in your own words, i.e. you must be able to explain the problem to a fellow student at any time. Try to formulate the problem in your own words without using the exercise sheet.
Always attempt every problem, not just those that seem easy or those that happen to appear at the top of the problem sheet. You will learn more from harder problems which take a long time to solve. The sense of achievement coming from solving a hard problem is one of the most rewarding experiences a mathematics degree has to offer. Don't miss out!
Analysing the problem
First, you need to make sure to understand all the terms, which appear in the formulation of the problem. Recall, if necessary, their definitions. In any case, you have to make sure that you associate with these terms not just vague ideas, but precise definitions. On the other hand, knowing only the wording of a definition doesn't give much insight into the underlying concept. The real meaning of a concept is only given by the set of statements that can be made about it. Hence, recall the essential properties of the concepts and how they relate to each other.
The next question could be: In which theorems from the lecture do the terms of the exercise appear? For example, is the exercise a special case of an already proven theorem from the lecture? Or does the exercise generalise a proposition from the lecture?
If you have to prove a general fact, consider some simple examples (= special cases) first, to check that the assertion can indeed be true. This is also important, to make sure that you really understand what the assertion actually claims. Knowing plenty of examples, or better, knowing the most relevant examples, often helps to develop an understanding of why an assertion is true, i.e. to find a proof for it.
Try to visualise the problem. For a given real function, it might help to draw its graph. When asked about geometric configurations, you first have to draw them. Even in purely set theoretic constructions schematic pictures are helpful.
Consider the proof methods which were used in the lecture in connection with the terms which appear in the formulation of the problem. Can you apply these methods for the given problem as well? Rarely you will be expected to come up with completely new and brilliant idea. Trust your intuition. Can you think of situations that remind you of the given problem?
Another trick is to try to disprove the assertion. For this it would be enough to construct a counterexample. Hence, if you cannot find a proof for the claim, try to invent a counterexample. Of course we know that this is impossible (unless there is a mistake in the exercise). But the crucial question is why this is impossible. If you then see that your best efforts of constructing a counterexample are always doomed to fail due to the constraints of reality, maybe a structure will gradually emerge which will eventually lead to a proof. This could be called the dialectical approach to the problem.
If you get stuck with one approach, you might have to use trial and error to find a suitable one.
Talk about the problem!
Generally, one should talk about mathematics as much as possible. Talking to other people helps to organise and clarify your own thoughts. You can talk with your fellow students or with your tutor about the formulation of the problem and possible approaches.
But talking about the problem is only beneficial if you have something to talk about, i.e. if you have spent some time thinking about the problem before entering the conversation. Likewise, it only makes sense to talk about approaches to a problem if you have tried several of them, but maybe got stuck every time. Then you can discuss such approaches or incomplete solutions with fellow students. Otherwise, you will miss the light bulb moment and therefore the whole purpose of the exercise.
Collaboration can be beneficial for everyone, if all collaborators are contributing more or less equally to the solution, but ultimately, you will be assessed by your own abilities. In particular, you shouldn't just have the solutions explained to you instead of finding it on your own. Of course, you can do that, but note that then at least half of the learning effect gets lost in this case. On the other hand, if you found a solution on your own, it can be very instructive to expose your own solution to the criticisms of fellow students or see how others tackled the same problem.
If you think you have found a solution then you should also be able to
explain this solution to someone else. If you are lacking the right
words or if you slip into phrases like
well, something along
these lines…, then this indicates that there is still a small gap
in your understanding.
Writing up the solution
Writing up a solution is a critical step of the problem-solving process. Here is where you find out whether the solution, which you have found in your mind's eye or at least anticipated, can be turned into a formal, written argument. Any correct solution can also be formalised and written down. If you have difficulties in putting your thoughts on paper, then the reason might be that your thoughts about the problem are not yet well-organised. Stop writing for a moment and start thinking again. In any case don't leave it to the reader/marker to make sense of your shreds of thought.
There are two extreme ways of presenting a solution, both of which are unsatisfactory. Either you give only pure calculations without any arguments and annotating text, or you write a novel without ever getting to the point. The truth lies somewhere in between.
The real subject of the argument will be certain defined objects,
logical or mathematical relationships between them or calculations.
However, surrounding text is used to clarify the logical meaning of
these mathematical building blocks. One and the same mathematical
phrase, say $x < n$, can have very different meanings, depending on
whether it is preceded by
Hence, we can assume without restriction that…
From this we conclude that… or
Assume that…. Only the descriptive text
assigns a meaning to the formulae by putting them into an overall context.
A solution to a problem consists of a text written in plain English. English is not meant to be in contrast to Russian or German, but in contrast to Mathsspeak. Your argument should be formal, not your language. Write good prose. Hence, your text should consist of complete sentences. Each sentence contains a subject and a predicate. Avoid writing just chains of logical symbols. But also avoid cumbersome verbal paraphrases, if there is a concise symbolism for it. Here the lecture is not always a good example! But a lecture is essentially an oral event. Communicating mathematics on a blackboard fulfils a different purpose, and therefore also follows different rules.
If you engage with the problems and treat your solutions as texts in their own right (although containing formulae and mathematical symbols), you will gradually develop your own mathematical style. Your solution has to be comprehensible for someone who only knows the problem, and hasn't solved it, yet. Indeed, your aim is not to convince a knowing reader by obscure hints that you have understood the solution as well, but to make an ignorant reader understand your solution.
Your solution should be written in readable handwriting (or LaTeX), this also applies for formulae and symbols. Help the reader to understand your formulae, e.g. by making it easy to recognise the pairing of parentheses.
A good marker will not only tick off correct solutions, but will also rigorously correct your style. One aim of the assigned problems is to teach you how to read and write mathematics.
Never submit the first version of your transcript. Make at least a clean copy of your solution! A text in which multiple corrections are made, in which entire passages are crossed out, in which the reader is invited to insert supplements from the last page, such a solution shouldn't be presented to anyone. Review your text from the following perspective: does the written argument actually convince you? Honestly? If not, start over again. Writing up your solution can be a hard and painstaking process. However, the reward of an equally correct and beautiful solution will compensate you.