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.
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 a library, or you could make use
of the Mathematics and Statistics students’ study room (room A504). To and from
LSE, journeys by train, tube or bus (if they are not too packed) provide an
opportunity for reading and reflection.
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.
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.
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.
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.
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.
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.)
Some lecturers will rely heavily on lecture notes they
have handed out, 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.
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.
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.
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.
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 hopeless if you don’t know what key concepts mean, exactly. That is,
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.
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!
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. This is
often useful, provided everybody contributes, but bear in mind that in the end,
you will be on your own in the exam.
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 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.
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 personal tutor. There are people at LSE who know much more than I
do about study skills generally. Dr Liz Barnett, the Teaching and Learning
Development Officer, has organised a series of lectures and practical workshops
on study skills. The slides for these sessions are available on the Outlook
Public Folders, and you should check the web-site http://learning.lse.ac.uk/studyskills.asp.
There are many books on study skills, such as:
The Good Study Guide, Andrew Northedge, Open University 1990. (LSE
library LB2395 N87, ISBN 0749200448)
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)
There are also some web-sites devoted to study skills.
The following two are specifically concerned with mathematical sciences:
http://www.maths.soton.ac.uk/teaching/studyskills.htm
http://euler.slu.edu/Dept/SuccessinMath.html
Also
of possible interest is the more general study-skills site
Martin
Anthony, October 2002.
(with thanks to Liz Barnett, Graham Brightwell and
Colleen McKenna)
[1] From George Polya, How to Solve it: a New Aspect of Mathematical Method (Princeton University Press, 1945). Available in the library: QA36 P78.