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Department of Mathematics

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London School of Economics

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These notes are intended to offer a few basic points about studying, particularly for mathematical, statistical, and other quantitative subjects. Much of the advice given is common sense, and will be unsurprising. But it is important for you to think about your approach to studying. Of course you’ve studied before, and have been highly successful, but there are some differences required in approaching university-level study.

**Creating time and space for study**

You worked hard to get here and our courses are demanding, so you have already studied hard and will continue to do so. But don’t work so hard that you don’t enjoy your time at LSE. You should have plenty of free time for leisure activities. Try to spend a few hours’ work based around each lecture, making sure you comprehend the notes, reading the text, working through the examples. You should also spend a few more hours each week, for each course, on any problems assigned for classes. (This is a very important part of the learning process.) In working through the problems, it will often be necessary to refer to your lecture notes, so your comprehension of the lectures will be enhanced too. The exact amounts of time you spend will vary from week to week or from course to course, but the most important thing is to keep a regular program of work going.

The type of environment you best work in will depend on you. You should try to use a location that you are comfortable in (but not so comfortable that you can doze off). Some like to study with music playing in the background, whereas others prefer complete silence. Try to reach agreement with your friends or family about your study environment in your accommodation. During the day, perhaps when you have a block of time between lectures or classes, you might like to use a quiet spot in the library, for instance. To and from LSE, journeys by train, tube or bus (if they are not too packed) provide an opportunity for reading and reflection.

*Planning study across the year*

It is a good idea to have a fairly regular pattern to your work. You could, for instance, set aside the same block of time each day for some work. To try to cram your work into short periods of time as the exams come close is a bad approach. It will be difficult and stressful. In many subjects, particularly mathematical and statistical subjects, cramming simply won’t work because lectures build upon each other. It is a good idea to keep up with the lectures by doing a little work, often.

*Managing a study session*

Try to have some realistic goals for each study period. Often, this is easy: you could set aside a slot during which you will review a particular lecture, read a specific chapter of a book, or complete a certain problem sheet. Having achieved your goal, you can then relax with a clear conscience. Try to organise your studying in such a way that you feel in control of it.

It is probably most effective to work for reasonably short, concentrated blocks of time, of, say 30 to 45 minutes, in between which you take short breaks. Research has shown that concentration can only be maintained at a high level for discrete blocks of time like this.

When you sit down to work, get stuck in. Don’t spend half an hour arranging your differently coloured pens (useful as these are). Give yourself small goals and rewards: ‘I’ll do this one problem, then I’ll have a cup of coffee,’ for example. Some students find making a checklist at the start of the session useful. Tick off items as you achieve them.

*Becoming an active learner*

Being an *active learner* means taking a pro-active, interrogative approach to your study. Above all, it means making your learning your responsibility. The role of teaching staff at university is not just to transmit information: it is to enable you to grapple with key concepts and ideas, and to apply these. This can’t be achieved through *passive* learning, where you simply try to remember what lecturers have said, agree with it, and commit it to memory. Until you know that you understand the key concepts, you should ask questions, do some more reading, talk to your friends, do whatever it takes until you are sure that you are on top of it.

*Active learning in a study session*

When you are studying, you should regularly review what you have learned. It’s very easy to get on with some reading and slip into ‘auto-pilot’, losing concentration. Stop and ask yourself just exactly what you’ve learned in the past few minutes.

Try to take some initiative with reading materials. It is a good idea to read more than you have to, if you can. Having a wider perspective or deeper level of knowledge than is minimally required certainly helps with understanding the material of the course, and it keeps you interested. There is a huge amount you could try to read on any given topic, and you could go much deeper than the lectures, so it’s important to be realistic in what you try to achieve: don’t be too hard on yourself. Consulting texts other than the core texts can also be very useful: a slightly different perspective on a given topic might help ideas slot into place.

As you study (for example as you review your lecture notes) you should make additional notes, either on fresh paper, or on your lecture notes, or your text. Highlighter pens are useful, and it’s good to use different colours for different things.

*Actively using textbooks and hand-outs*

Mathematics and statistics textbooks require a special form of interaction with the reader: active rather than passive. It’s too easy simply to agree with a mathematics or statistics book, without actually understanding it, or without being able to apply the concept you have been reading about. You should work through any calculations in the text by yourself. And you should certainly attempt some of the textbook problems that relate to your recent lectures and classes. When doing this, though, avoid the temptation of referring to the answers at the back of the book (if there are any): wait until you think you have solved the problem before doing this.

If you have hand-outs in advance of lectures, it’s worth skimming them to get some idea of what’s coming up.

*Lectures*

There are many different styles of lectures. Some will consist of a lecturer writing or speaking continuously. Taking notes in such lectures is a valuable skill, and one which requires practice. Try to note down the main points, and certainly definitions, theorems, proofs, examples, and references to any relevant reading in the textbooks. There is no need to write down everything the lecturer says. It can sometimes be difficult in such a lecture to concentrate enough to understand the material: note-taking may be taking up too much of your concentration. But don’t panic if this is the case. You can work on the details after the lectures, and follow it up with questions to fellow students, class teachers and the lecturer. (For specific information on note-taking, see the book by Northedge cited at the end of these notes.)

However, many lecturers (especially in first-year courses) will rely heavily on lecture notes they have handed out or posted on Moodle (the web-based platform we use for course materials), or on textbooks. Here, too, note-taking will be important, but will probably be easier. Hand-outs in such lectures are likely to take the form of summary notes, outlining the main ideas. You should aim to augment these hand-outs with any additional useful points the lecturer makes, and also with important details (examples or proofs, for instance) omitted from the hand-outs but discussed in the lectures.

The most important thing to realise about university lectures, which you have probably already noticed, is that the rate of delivery of new ideas is much faster than in high school or sixth-form college. A lot of work is required by you, outside of lectures, to make sure you understand the material. The lectures themselves are just a starting point and you should not be annoyed with yourself because you didn’t understand absolutely everything during the lecture itself.

*Classes*

In quantitative subjects, classes are immensely important. Subjects like mathematics and statistics are only really mastered by working through lots of problems. Classes are most useful if you have attempted the assigned work: even if you can’t complete the problems, you should at least try them, and locate exactly where it is you have difficulty. (The most common problem might be that you simply didn’t know where to begin, and we’ll discuss this further later.) Class teachers want to know what problems you’ve been having. They don’t particularly want to know the answers—they already know them! Don’t be afraid to learn from your mistakes, and don’t worry too much about the class grades: these are just for information, for you and the teacher. They do not contribute to final assessment. There’s no point in handing in to your class teacher a perfect set of answers that you obtained from someone else. Unless you’ve grappled with the problems yourself, you won’t have learned anything. Even when you see a solution presented in class, it won’t have much value if you haven’t thought about the problem for yourself.

You should ask questions in classes. You may be too daunted by the size of lecture groups to do so in lectures, but classes are the right sort of size for raising questions and having discussion.

Problem solving is a mixture of frustration and satisfaction. It is the most important skill you develop in studying mathematics and statistics, and will stay with you for the rest of your life, even if specific techniques fade from your memory.

*Types of problem*

Different types of problem are asked in quantitative subjects. Here are a few of the most common types.

*routine, ‘drill’ problems*, involving straightforward (or not so straightforward) application of a technique. These may be technically difficult, but at least you know how to approach them.

*modelling (or applications) problems*, requiring a translation of a descriptive problem (such as a statistical one) into mathematical language before solving using standard techniques. The translation can be very hard.

*proof problems*, requiring the use of formal definitions and proof techniques. It is not always clear how to approach these, and students often comment that they simply don’t know where to start.

*Approaching problems*

There is no recipe for successful problem solving, but Polya’s general four-point approach is useful[1]:

1. Understand the problem

2. Devise a plan

3. Implement the plan

4. Look back/check

The first step is hugely important and often not taken seriously enough. You need to know exactly what it is you need to establish. This is impossible if you don’t know what key concepts mean, exactly. For this reason, mathematical definitions are extremely important. This is particularly so for proof problems. Suppose you were asked to show that a given sequence of numbers has a limit. Unless you know exactly what is meant by a limit, there’s no way you can even begin to solve this problem.

Once you know what you need to show, do not be afraid to try something (steps 2 and 3 of Polya’s scheme). If that fails, then that’s OK: try something else. Or, try to solve a special case or a simpler version of the problem, in the hope that you can then get a feel for the problem and generalise to the required level. These are the ways in which real mathematics and statistics is often done.

The checking part of Polya’s approach is sometimes easy to do. For example, if you were asked to solve a system of linear equations, then to carry out the check you could simply put the supposed answers into each equation, verifying that each equation holds. Sometimes checking isn’t so simple, but you should always look back over your solution to make sure that it at least makes sense to you on a second look.

You should spend a lot of time on problems. Don’t worry about taking more time than you would have in an exam. Problem solving, and the speed of problem solving, improves with experience, so by the time the exam has approached, you will hopefully be proficient enough in solving problems to cope well.

*Help from your teachers*

Your teachers are there to help you. Remember to try to be an active learner. You should make full use of classes for asking questions. Your class teacher or lecturer will be available to see you during his or her office hour, and these opportunities are worth taking up. Staff are happy to help, but it will lead to a better discussion during an office hour if you can focus in on what exactly is causing you problems. To approach a teacher and say

‘I haven’t understood the last three lectures. Can you explain them to me?’

is pretty pointless. In an office hour, he or she is not going to be able to re-iterate the past three lectures in a way you will be able to understand (if you didn’t already understand them at the regular speed). So you would need to be more specific, as in, for example,

‘OK, so I know how to differentiate, in that I can use the product rule and so on, but I have difficulty when I try to understand what the derivative actually means. For example, in question 5 of exercise sheet 3, which is all about differentiation, I have to work out by how much a function changes if I increase the variable a little bit. What’s the connection with the derivative? I don’t get it.’

This is a great question, which the teacher can happily deal with. It’s also one which he or she will be able to know they have satisfactorily answered for you, and that makes them happy too!

*Help from your friends*

Your fellow students in a course might be having difficulties too, but these may be different difficulties. Together you might be able to overcome the various difficulties by working in a small group. You can discuss key ideas and concepts, and work through problems together. You can also initiate discussions through Moodle. Working with others is often very useful, provided everybody contributes, but bear in mind that in the end, you will be on your own in the exam.

Please also see the departmental webpage 'Examinations in Mathematics - guidance' for some more detailed information about exam preparation

*Exam revision*

Exam revision should be revision: you should not find yourself learning things for the first time! (See the discouraging words about ‘cramming’ earlier.)

It is a good idea to plan your revision, and to stick at least roughly to a timetable. Make full use of the vacations, not just Easter, but Christmas too.

When revising your courses, it is a good idea to summarise your lecture notes, and work through problems again. Focus in on areas you are having difficulty with, and talk to fellow students and academic staff about these. A particularly useful resource will be the recent exam papers, and it is perhaps a good idea to leave them until exam revision rather than to attempt them earlier in the session.

Some exams (such as those in pure mathematics) require you to reproduce some ‘bookwork’, meaning the statements of definitions from the lectures, and the proofs of key results. Proofs are enormously difficult to memorise. The only way successfully to be able to handle bookwork is to understand the definitions and proofs, and not simply to memorise them. (Precise notations used in proofs are usually not very important. The key thing to be remembered is the approach used.) You will, in any case, need to understand the key ideas and concepts in order to handle the bulk of the exam questions, which will be testing that you can solve unseen problems.

*Exam technique*

First of all, know when and where the exam is: don’t rely on friends for this information!

The most useful piece of advice for exams is: don’t panic. Try to relax and take control of your exam. The exam you sit will quite probably contain problems that are unlike others you have seen before, though there will usually also be some less surprising questions. This is part of the nature of exams, and you should not let yourself panic about it.

When you start the exam, make sure you understand the rubric. Then, it’s a good idea to skim the whole paper quickly just to size it up. You should answer questions in the order you want to. Why not bag a few easy questions for starters, to build up your confidence?

Don’t get bogged down with small bits of questions that might be worth only a few marks: you should move on and return to these later if you have time.

Remember that in many cases what examiners are testing is that you know how to solve a problem: that is, that you know what technique to use, and how it works. Some credit will be awarded for correct approaches, even if you mess up the subsequent calculations. If you don’t have time to finish a problem, but have enough time to explain how you would have finished it, then it might be worth doing so: some credit may be given for this.

Much of what I have written here draws on my personal experience, from observing my fellow students as an undergraduate, and from my experience as a lecturer and advisor. There are people at LSE who know much more than I do about study skills generally. The LSE’s Teaching and Learning Centre is an excellent resource and it runs many courses and advice sessions that could be of use. See its website: http://www2.lse.ac.uk/intranet/LSEServices/TLC/Home.aspx

There are many books on study skills, such as:

*The Good Study Guide*, Andrew Northedge, Open University 2005. (ISBN 0749259744)

This has good discussions of time-management, note-taking, and revision. It also covers essay writing extensively, which is of less relevance to mathematics and statistics but may be useful for some of your other courses. A version of this book has been written for science students, and may be more appropriate:

*The Sciences Good Study Guide*, Andrew Northedge, Jeff Thomas, Andrew Lane and Alice Peasgood, Open University 1997. (ISBN 0749234113)

**Martin Anthony****, September 2012.**

[1] From George Polya, *How to Solve it: a New Aspect of Mathematical Method* (Princeton University Press, 1945). Available in the library: QA36 P78.

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