This document is intended for anyone taking an undergraduate examination in a Mathematics course at LSE, especially those who have not taken such an exam before - some of what is said will also be of use in preparing for our graduate examinations. Its purpose is to advise you on: how you might go about revising for an exam, how you can get help from the Department, what is expected of you on the exam, and how we go about marking and grading the exam.
To a large extent, you should treat a Mathematics examination just like any other. However, there are a few differences, and this document concentrates on those. You should also be aware that there are some differences in approach between various Mathematics courses.
The first piece of advice is to make sure that you go to all the lectures and classes on all your courses, submit your weekly homework to get feedback on your progress, and hear everything the lecturer has to tell you about the particular exam that you will be taking. There are revision lectures and/or classes in all Mathematics courses in the summer term; these are all timetabled formally, so that details can be found on the School's website, but there is some possibility that changes will be made, or that the sessions will take place in a different room from usual.
In general, it is wise to assume that all elements of the course you have been taught are examinable. However, you are likely to be told, by the lecturer, which pieces of the course (if any) are not examinable. Similarly, if there is something new in the course that hasn't been examinable before, the lecturer will probably tell you so. (If a large section of the course is new, it is up to the lecturer to give you some indication of what type of question might be set, perhaps by providing "sample" exam questions for that section.) Conversely, if something has been dropped from the course, you should be told this (so you'll know that there are some past exam questions you can't be expected to do). Of course, you will be told most of this at the appropriate point of the course, but it should be said again during the revision lectures. You should also be told if there is any change to, for instance, the number of questions you are required to answer. If in doubt, ask!
Apart from this, what happens in a revision lecture varies from course to course. Perhaps the most common is for the lecturer to go through some questions from a past examination.
Tip: Attempt these questions yourself before the lecture, and look again at your homework exercises.
Often the lecturer will ask students whether there are particular parts of the course they would like to see again. Even if they don't ask explicitly, if there is something you want to see covered, don't be afraid to ask for it; other students almost certainly want the same as you do.
Past Examination Papers
Some past exam papers are available from the relevant course Moodle page, or alternatively from the Library's website if not on the Moodle page. For some courses, the examiners will also provide a copy of the solutions to some of the past papers on the course Moodle page (though be aware that some lecturers do not do so).
So why look at past exam papers? First of all, you can see what kind of thing is typically asked on the exam. Also, you can use them to test your knowledge of the material and your ability to solve the problems. When you've tried the question (and not before), look at the model answer and compare it with yours. Don't despair if your answer isn't exactly the same as the model; it might well still be correct, or at least worth quite a bit of credit - why not ask your lecturer or class teacher about such differences in their office hours?
However, whilst past papers can be extremely useful as an aid to revision, they are not intended to be a definitive guide as to what may appear in future exams. If there is a type of question that has appeared in each of the three previous years, it is certainly a good idea to be prepared to answer such a question, but there is no guarantee whatsoever that one will appear.
Conversely, if you look at the most recent past paper for a course, you will see some questions, or parts of questions, that bear little resemblance to any questions from previous years. Examiners deliberately set questions that will be unfamiliar to students, requiring an understanding of the material. The main aim of university study is to acquire insight and understanding, and therefore the exam is designed to test this. Students who prepare for exams solely by rehearsing answers to earlier papers cannot expect a high mark.
A student revising with the help of past exam questions and exercises should always aim to understand the relationship to the respective underlying general concept as taught in the lectures, in addition to acquiring proficiency in the solution method.
Some people get through mathematics exams at school with hardly any revision. Hopefully you've realised by now that you will have to revise for university exams if you hope to pass them, let alone achieve a good mark.
The School runs "study skills" sessions covering revision for examinations; have a look at the website of LSE's Teaching and Learning Centre for more information. Some general tips follow.
Go back and try homework exercises again; hopefully they should be easier the second time around. Try past exam questions. Read your notes thoroughly and make sure you understand each topic properly before going on to the next. If you can't understand something, perhaps a friend taking the same course will be able to help - and explaining it to you ought to help their understanding too. Or you could ask the lecturer or class teacher.
Early on in your revision, concentrate on understanding rather than learning. If you understand the concepts properly, often you'll find that just reading through a proof once or twice enables you to reproduce it.
There are certain things that it is advisable to be able to reproduce more-or-less exactly in the form given in the lecture notes or course book. Obvious examples include formal definitions. Even there, understanding what is going on helps enormously. It's not normally a good idea to try and learn a proof by rote; concentrate on remembering what the structure of the proof is, and hopefully most of the small steps will become easy.
Members of the Department hold office hours as usual in the lead-up to exams, and will be keen to help you, but you still shouldn't expect them always to be available exactly when you want! Class teachers who aren't members of staff don't all have office hours after classes are over; if your class teacher doesn't hold office hours then you should go to the lecturer.
Not surprisingly, office hours can get extremely busy, and you might have to timeshare with other students - especially if you want help at the last minute. In fact, it's not a bad idea to get together with one or two other students on the course, agree what you want to ask about, and go along together. In any case, it helps to make a list beforehand of what you would like explained. When you go to an office hour, be prepared to think; not just to soak up information.
A typical examination question in Mathematics will have several parts to it. Some parts (most usually at the beginning of the question) test your knowledge, by asking you to reproduce "bookwork", i.e., material presented in the lectures. Really, these parts test how well you've revised. Occasionally, especially in certain more advanced courses, there are entire questions that are bookwork. In some courses, some pieces of bookwork come up in the exam almost every year. In other courses, hardly any bookwork is set explicitly.
Tip: Figure out which pieces of bookwork come up most frequently, and make sure you can answer those questions easily and quickly.
You would be surprised how many poor attempts at routine bookwork questions we see every year. These are the parts of the questions that we expect students to be able to do.
Other parts of examination questions involve a "problem". In a "Methods" course, this will typically involve you applying a known technique from the course, and again this is something we expect you to be able to do. In a Pure Mathematics course, you might be asked to prove a result, or to apply a result in a particular setting.
Tip: Sometimes (but certainly not always!), the first part of the question is intended as a big hint as to how you should approach the second part.
Many exam questions, especially (ideally) those that are otherwise very routine, have a last part (a "rider") which is more challenging than the rest of the question. This is quite deliberate, and the intention is to test whether you've really understood the material.
Tip: Do try all the riders (they're not always so hard after all!) but don't waste too much time on them in an exam if there are other things you can tackle instead.
Students sometimes seem to be very annoyed that they have to do something clever to get 100% on a question. Don't forget, in many other subjects it's practically impossible to score 100% on a question!
Tip: Make sure you've answered all the parts of the question. Sometimes you're asked to do seven or eight things, and it's easy to overlook one.
Marking the Examination
Your exam script comes to us without your name on it. We really don't have any way of finding out which numbers correspond to which names; we might recognise your writing, but this is really far less common than you might think.
Individual questions are marked out of 25. In all courses, full marks are extremely rare. A score of, say, 23/25, on a question is outstanding. A score of 17/25 is very respectable, and if you do that well on all the questions you ought to be somewhere near the borderline for a first class mark on the paper. Bear this in mind before you panic because you haven't been able to do the very last part of a question! Conversely, don't turn up your nose at the thought of answering just the very first part of some question. Maybe you'll only score 2/25, but that might make all the difference. And sometimes it only takes a minute to score those 2 marks: this is very cost-effective!
The exam is set by the lecturer(s), and they will normally also be the person who first marks your exam script. The marking is then checked by a second examiner, who will verify that everything has been done properly. Their job includes making sure that all your work really has been marked, and that the marks have been added up correctly. On large courses, things are necessarily done a little differently, with several people involved in the marking.
Once the marking is complete, the two examiners then agree a formula for converting raw script marks to final marks (a "curve"). We regard it as undesirable for the final percentage mark to differ by more than a few points from the raw percentage, and we set the exams with this in mind; however, we sometimes do need to apply a curve, in order to assign marks fairly should the exam turn out to have been more difficult, or easier, than intended.
Besides the two internal examiners, there is an external examiner, who is an experienced academic from outside the LSE, responsible for each exam paper. Their duties include looking at the paper after it is set, to ensure that the questions are of an appropriate standard. After the exam is sat and marked, the external examiner looks over the scripts, and verifies whether the marking has been carried out properly, and whether the curve is appropriate. Another part of their job is to look at all the scripts deemed to be failures, and to confirm (or not) that these deserve to be marked as a fail. Scripts falling near borderlines between degree classes will be looked at carefully, and examiners may decide to alter a mark (almost always upwards) if they feel the script overall merits it.
This is the technical term for the instructions to candidates that appear on the cover of the examination. In the Mathematics Department, rubrics all look much like this (though there may be some variations on this):
This examination paper contains 6 questions. You may attempt as many questions as you wish, but only your best 4 questions will count towards the final mark. All questions carry equal numbers of marks.
Answers should be justified by showing your work.
Calculators are not allowed in this exam.
For a whole unit paper, you would be required to answer 6 questions out of (about) 8, rather than 4. Selecting which questions you answer is, by accident or design, part of what is being tested, as is your ability to manage your time so that you can make a serious attempt at the right number of questions.
Tip: Read the rubric and make sure you answer the appropriate number of questions.
What the Examiner is looking for
It is a (rather irritating) myth that marking Mathematics exams is easy because answers are either right or wrong! It is true that there are some answers that are undeniably "right", and these score all the available marks. Also, there are some answers that deserve, and get, no marks. But most answers to complex questions come somewhere in between.
If you are asked to solve a problem using a standard method, we are looking primarily to see whether you know "how to do it". The best way to convince the examiner that you know the method is of course to solve the problem correctly, but you should get almost all the marks even if you make an arithmetic slip or two along the way. If you get part way, and then forget what to do or do the wrong thing, you should get some marks.
If you are asked to reproduce material from the course, we are looking for both knowledge and understanding. (If you learn your notes word-for-word, you might in principle be able to disguise a complete lack of understanding, but this is not a good strategy.) You will get marks for knowing how a proof goes, and marks for showing that you understand what is going on. Usually, there are marks allocated to specific parts of the material.
If you are asked to solve a problem, examiners usually have in mind the way that they would tackle it, and there will be marks available for going a certain way down that road. However, examiners are very willing to give marks to students who have reasonable ideas for alternative approaches, and pursue them sensibly, even if the method actually is never going to succeed. In extremis, examiners have been known to give a few marks for something that is vaguely relevant and makes sense.
There are not many places in Mathematics exams where you can score marks for "waffle", i.e., very wordy answers that sort of suggest why something might be true without ever getting down to detail. We are looking for understanding above all, but also for precision, detail and accuracy.
Undergraduate courses and degrees in the Mathematics Department conform to the description in the 'Subject Benchmark Statement for Mathematics, Statistics and Operational Research' -http://www.qaa.ac.uk/Publications/InformationAndGuidance/Documents/Maths07.pdf - which sets out the expectations for any undergraduate degree course in the UK in these subjects. Section 5, in particular paragraphs 5.12-5.15, sets out what is required for a student to achieve a "threshold standard" (i.e., to pass a course) and a "typical standard" (roughly sufficient for a mark of 60) in a course in mathematics.
The Mathematics department has also produced some more detailed general guidelines on assessment criteria, which provide broad descriptions of the meaning of marks awarded on our examinations.
For undergraduate courses, these can be found here.
For MSc courses, these can be found here.
Graham Brightwell updated 07/01/2011