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Studying mathematics and statistics at university

Every student has a personal learning style. At the same time, there are common threads in learning in revising. The collection of topics and thoughts below is intended to provide some support for your transition from school to university and beyond.

What are MATHEMATICS and STATISTICS?
SCHOOL mathematics versus UNIVERSITY mathematics
How can I solve PROBLEMS?
Why THEOREMS and PROOFS? Should I worry about DEFINITIONS?
List of STUPID QUESTIONS by first year students
Time to work ALONE and time to work in GROUPS
What about LECTURE NOTES?
Carrying BOOKS in the 21st century
Now we EASILY see, by a CANONICAL CHOICE, that the next step is TRIVIAL - or is it not?
About working TOO MUCH and working NOT ENOUGH
You forgot to mention PAST EXAM PAPERS

What are MATHEMATICS and STATISTICS?

Mathematics has been called the language of science, and statistics the language of empirical science. These definitions focus on the roles mathematics and statistics play for natural and social scientists. A common way of defining mathematics is to say mathematics is the science of patterns. Probability, in particular, is the science of random patterns, and it provides the mathematical structures used in statistics. Statistics is the theory of translating data into information.

SCHOOL mathematics versus UNIVERSITY mathematics

To start with the commonalities, most of the mathematics is centred around problem solving. We learn to apply methods to problems given to us by someone more experienced. Graduate students (to some extent) and researchers will generate questions themselves. In applied research the questions are often triggered by what mathematicians like to call "the real world".

An important difference is that in school you learn just a few concepts and methods at a time, and subsequently work through many questions that can be solved with exactly these few concepts and methods. In university you learn a huge amount of theory in a short time, along with just a few examples. In the exercises you find yourself struggling to identify which concepts and methods they actually relate to.

Unless you had exceptional school teachers, you will experience a fundamental change in the style and emphasis when you come to university. School mathematics puts a huge emphasis on obtaining the answer to a question quickly and accurately, whereas university mathematics is mostly concerned with how such an answer could be obtained and whether or not there is one to start with. That includes reasoning about how many answers are there, in how many ways they can be found, and which of them is the most elegant. A mathematician won't stop there, but will move on to generate variations of the question and revisit all the answers, to check whether they still make sense.

University mathematics starts with axioms that define mathematical objects in a very abstract manner. A few weeks into the term you find yourself buried in facts that mathematicians have been deriving about these objects over the course of centuries. The only way to arrive at your own understanding is to work through a mountain of problems yourself.

You probably chose to pursue a mathematical degree because you were good at mathematics, and we hope that you will use your enthusiasm and energy to master the transition to university and fall in love with problem solving. Along the way you will probably experience many hours of frustration, maybe even feel like you are failing completely, and that you will be puzzled by this new experience. The good news is that it will never be boring or repetitive.

How can I solve PROBLEMS?

Following lectures and reading textbooks can be misleading, in terms of the difficulty of tackling problems like the proof of a theorem, a complicated computation, a search for good examples or counterexamples, or the contruction of a link to an application in the real world. The lecturer typically shows you just the most elegant solution. What they do not show are the many detours and dead-ends taken by many people attempting to solve the problem before that elegant solution was found. In other words, most of the work that was invested in solving the problem in the first place remains invisible to you. Now that you are solving a problem yourself you find yourself actually doing all this work.

Do not look at the solution. Solving problems does require a good portion of luck. You may first give it a go by just diving into the problem and exploring it, following your intuition, just enjoying the ride. If that doesn't lead to a solution you may go about it more systematically:

  • Write down a list of all the assumptions and a list of what you're trying to calculate or find.
  • Write down definitions of properties or objects that are referred to in the question. Do you really understand what that all means? Work through examples for a while before returning to your problem.
  • Write down all the facts you know about the properties.
  • If you're working on a proof, explore the feasibility of a number of strategies (e.g. induction, indirect proof, counterexample).
  • If you have to proof a statment of the form "Object alpha has property A implies that object beta has property B" (*) do a few things: (i) write down what it means for alpha to have property A as explicitely as you can using the definition, (ii) do that same for object beta and property B, (iii) look carefully at the relationship between alpha and beta to tackle the proof of the implication as requested (e.g. alpha is defined as some funtion of beta or the other way round). Examples for such types of statement: "Let N be a positive integer. If N^2 is even then N is even." "If the sequence (a_n) is bounded than the sequence (a_n/n) converges to 0."
  • If it doesn't work with using the definition, is there an equivilent property you could use instead?
  • Sometimes you are in a situation with alpha=beta: Is property B actually weaker then property A? Is there some fact implied by A that would imply B? In the general situation you may still be able to work along those lines.
  • Sometimes you are in a situation that where A and B refer to the same property. While that means you only have to look up one definition, you may get mixed up more easily, because once you have to fill it in for alpha and once for beta defined as some function of alpha. So just take you time.
  • Look at special cases and examples. If you are not sure how to proof this for whatever general alpha and beta are given to you, just pick a couple of actual numbers, plug in the statement and see if you can prove it for those particular numbers. Note that if alpha and beta are in a functional relationship you need to pick the numbers accordingly. Look at how you showed the statement for this special case and try to generalize it.
  • Sometimes you have to prove something along the lines in (*) but involving several objects at once. For example: "If f and g are continuous functions than f+g is also a continuous function." Or this example for a false(!) statement: "If fxg is continuous then f and g are continuous." You can adapt the strategies above to this situation.
  • If there's a calculation or reasoning that ends in something already given to you, can you work from the bottom up?
  • Try to work through similar problems you have seen in the lecture, tutorials, exercise sheets, textbooks etc.
  • Find a fellow student to listen to your attempts and thoughts about why they failed.
  • "Ever tried. Ever failed. No matter. Try again. Fail again. Fail better." (Samuel Beckett)

With more experience you will do a lot of this automatically and you will gather more and more shortcuts and tricks that speed up your attempts. If you ended up looking at a solution for the problem, try to understand it in full. A few days later, attempt the problem again without looking at the solution. That will show whether you really understood the solution.

Why THEOREMS and PROOFS? Should I worry about DEFINITIONS?

Modern mathematics writing and lecturing typically adopts a standardised style of presenting knowledge by dividing it up into three different parts all of which are important:

  • Axioms and definitions of mathematical structures and objects. You need to know definitions by heart and you also need to really understand them. In fact, once you really understand them it is much easier to memorise them. Look at examples and counterexamples. Change them a bit and see what property they define instead, if at all. For example, exchange "for all" and "there is" in definitions about convergence and continuity.
  • Statements (theorems, lemmata, propositions etc.) about mathematical object. Find examples. Change the assumptions and figure out whether the statement is still true.
  • Reasoning about why these statements are true (proofs). Try to understand a proof by extracting the most important points. Identify where the assumption are being used. (If you can't find that you typically have not yet understood it.)

List of STUPID QUESTIONS by first year students

This is list empty. In the first year you are a complete beginner at university mathematics and you are allowed to ask any question about the study material without being judged.

Time to work ALONE and time to work in GROUPS

Make time for both. It has been observed that working with a team leads to better results than exclusively studying alone. Why is that? Bouncing your thoughts off fellow students is highly effective for recognising what you did understand completely and where you still have gaps. It shows you that others also have gaps. It may be satisfying to explain something you already understood to others. It may be helpful to get a clue from others where you are stuck. It can be fun. It can help to put things into perspective. It can help overcome anxiety. Outside term, email and social networking sites may actually help you to keep up some level of group activity, and prevent isolation.

At the same time, you have to practice hanging in there, solving problems, and working through notes all by yourself, because that is how you are going to take the exam. You can practice definitions, theorems and proof sketches with flashcards. You can tackle problems without consulting the solutions. You can write up your own collection of examples and counterexamples.

What about LECTURE NOTES?

Take notes. Get used to not worrying about what they look like at this stage; neat handwriting is not going to get you a First. By the end of the third week you may have learned to write without looking at your sheet of paper. Get as much information as possible both from the writing on the blackboard and from what the lecturer says. Scribble down keywords of explanations you don't understand in real time to be able to reconstruct the argument.

Later, by yourself or in study-group meetings, you go through your (possibly messy) notes and transform them into something readable and understandable. You will probably add explanatory comments and extra examples. This step is a great opportunity to digest the lecture material. If the lecturer provides notes, you may combine those with your own to get a more comprehensive picture of the material. Books may help you to close any remaining gaps.

Carrying books in the 21st century

If the module comes with a recommended list of textbooks, they are a good place to start. In addition, you may want to find out what other books on the topic are available in the library. Books can help you put the lecture in a wider context and are very useful for finding extra examples. If you have trouble connecting to the style of the lecturer, books may provide you with an alternative. At the same time, each book uses a (slightly) different notation, which may take some time to get used to.

You would want to avoid hopping from book to book in search of an explanation that you would better derive yourself by just thinking very hard about it. This is even more tempting with websites. Be careful when using websites such as wikipedia for stuying. They are of some use for readers who already know the subject, but are less suitable for learning it in the first place. Reasons for this include inconsistent notation, repetition, misleading emphasis, lack of actual explanation, and substantial amount of errors on low quality sites.

Now we EASILY see, by a CANONICAL CHOICE, that the next step is TRIVIAL - or not?

These terms unfortunately sneak into all sorts of mathematical communication: in books, in lectures, in discussions. They typically pop up when someone tries to categorise a step of some longer mathematical explanation as less difficult to understand than other steps of that same reasoning. It helps to structure a long proof, by directing the attention to the crucial difficulties.

But, we claim the use of labels such as "trivial" has destructive potential, when talking to less experienced people.

Proof: Since even experienced people may make mistakes we need to distinguish two cases.
Case 1: Assume the step is actually not trival. Then it is obviously silly to say so.
Case 2: Assume the step is trivial. Then the recipient of that information has been led to believe that the argument can be spotted instantly, without thinking hard. However, since we assumed that the recipient is less experienced than the speaker, this is likely to be not the case for the recipient. The recipient has therefore two choices: (i) not think hard and not understand the step, or (ii) think hard and eventually understand the step at the price of first admitting to be stupid.

About working TOO MUCH and working NOT ENOUGH

They are listed together, because they are two sides of the same coin. Exams trigger some level of anxiety in most students, and as long as it does not become overwelming, it keeps you going. However, if concerns about the exam interfere with your sleep, there may be something wrong. If you find yourself not studying enough, because so many other things happen in a day, you may be setting the wrong priorities. While this has an appearance of being inefficient, it may point to an underlying fear of the exam. You may try to deal with that anxiety in a more direct way, such as finding ways to relax and to cut the load of study materials into smaller units. You may further identify what distracts you and find ways to keep that out of larger parts of the days. Typical candidates for distractions are social networking sites or other internet or mobile phone activities, all of which can actually be shut off.

If you feel overwelmed by exam anxiety or other stress, drink enough water or tea, eat and sleep regularly and go for walks frequently. We have beautiful walking areas around campus. If that doesn't help seek support from the student counselling service sooner rather than later. Further information can also be found on our departmental page about exam anxiety and related issues.

You forgot to mention PAST EXAM PAPERS!

Actually, it turns out that on this particular topic students seem to know more than lecturers.


Books

K Houston, "How to Think Like a Mathematician: A Companion to Undergraduate Mathematics", CUP, 2009. The book explains how write mathematics correctly and how to create your own examples. Of particular interest are the chapters on logical thinking, on definitions, theorems and proofs, and on proof techniques. See also K. Houston's paper "How to write Mathematics" and his slides "10 Ways to Think Like a Mathematician"

T Gowers, "Mathematics, a very short introduction", Oxford University Press, 2002. Explains the differences between advanced, research-level mathematics, and the sort of mathematics we learn at school. Of particular interest are the chapters about numbers and abstraction, proofs, and limits and infinity.

G Pólya, "How to Solve It: A New Aspect of Mathematical Method", Princeton University Press 1945. (Available in various versions today, including Paperback.) This is the classic reference on the matter of problem solving. Read more about it here...