UNIVERSITY OF CAMBRIDGE Faculty of Mathematics STUDY SKILLS IN MATHEMATICS This guide is intended for first-year students. Faculty documents for students taking the Mathematical Tripos are available from http://www.maths.cam.ac.uk/undergrad. Revised 12th September 2017
UNIVERSITY OF CAMBRIDGE Faculty of Mathematics STUDY SKILLS IN MATHEMATICS This guide is intended for first-year students. Faculty documents for students taking the Mathematical Tripos are available from http://www.maths.cam.ac.uk/undergrad. Revised 12th September 2017
1 Introduction Learning mathematics is very different from learning other subjects as anyone who tries to read a mathematics text book soon finds out And learning mathematics at university is very different from learn- ing mathematics at school,as anyone who sits through a university lecture soon finds out.Intense is the word that springs to mind when trying to describe the difference in style between school and university mathematics. Lectures are intense compared with lessons because there are comparatively few of them;supervisions are intense compared with lessons because you go over a week's work in a single hour;work is intense because terms are much shorter (the 8 weeks looming ahead of you may seem an eternity,but at the end of term you will be wondering what happened to the time);and examinations are intense because you have to cram a year's work into 4 three-hour papers taken in the space of a few days. This booklet is intended specifically for first-year mathemati- cians:it was written to help you make the most of your time at Cambridge. There is also a booklet that is specifically intended for supervi- sors.You may well find it interesting and even useful to read the ad- vice we give them on how to supervise.The pdf version can be found at www.maths.cam.ac.uk/facultyoffice/supervisorsguide. There's even a booklet intended for lecturers which you may find in- teresting:www.maths.cam.ac.uk/facultyoffice/lecturersguide. Faculty of Mathematics, University of Cambridge September 2017
1 Introduction Learning mathematics is very different from learning other subjects, as anyone who tries to read a mathematics text book soon finds out. And learning mathematics at university is very different from learning mathematics at school, as anyone who sits through a university lecture soon finds out. Intense is the word that springs to mind when trying to describe the difference in style between school and university mathematics. Lectures are intense compared with lessons because there are comparatively few of them; supervisions are intense compared with lessons because you go over a week’s work in a single hour; work is intense because terms are much shorter (the 8 weeks looming ahead of you may seem an eternity, but at the end of term you will be wondering what happened to the time); and examinations are intense because you have to cram a year’s work into 4 three-hour papers taken in the space of a few days. This booklet is intended specifically for first-year mathematicians: it was written to help you make the most of your time at Cambridge. There is also a booklet that is specifically intended for supervisors. You may well find it interesting and even useful to read the advice we give them on how to supervise. The pdf version can be found at www.maths.cam.ac.uk/facultyoffice/supervisorsguide. There’s even a booklet intended for lecturers which you may find interesting: www.maths.cam.ac.uk/facultyoffice/lecturersguide. Faculty of Mathematics, University of Cambridge. September 2017 1
2 Lectures The really big difference between school and university is in lectures The lecturers have only 12 lectures a week in which to give you enough material to keep you occupied for the other 156 hours.Therefore,the material comes at you pretty fast. It follows that,whereas at school you probably expected to un- derstand what the teacher said as it was said,here there will be great chunks of the notes which you will not understand until you have worked on them later:line by line,if necessary.Even then,there may be some parts of the course that only really become clear when you come to revise the material.1 Nevertheless,it is very important to try to understand as much as possible of what is being said as it is said.Apart from saving time later,you may otherwise miss vital explanations and insights:the lecturer's commentary on what he or she is writing is what makes lectures more instructive than simply reading a textbook.So: Do make the effort to concentrate.2 We have all heard that,in a mathematics lecture.what the lecturer writes on the black- board goes straight into the student's notebook without passing through the brain of either.You should do everything in your power to prevent this happening:sit near the front if you find 50 minutes of mathematics a strain;don't let your thoughts wander;don't keep checking your phone (it should be turned off!);and remember that concentration is just a matter of self- discipline and practice. Do ask questions during the lecture rather than let some diffi- culty pass by.If the lecturer is writing too fast,or too illegibly or is speaking too quietly for you,it is likely that others are having the same difficulty Reread this last paragraph!It is easy to become discouraged if you do not ratartwin te momet oew mc pe7产阅mt1 von get into the letare theaire.Rober 2
2 Lectures The really big difference between school and university is in lectures. The lecturers have only 12 lectures a week in which to give you enough material to keep you occupied for the other 156 hours. Therefore, the material comes at you pretty fast. It follows that, whereas at school you probably expected to understand what the teacher said as it was said, here there will be great chunks of the notes which you will not understand until you have worked on them later: line by line, if necessary. Even then, there may be some parts of the course that only really become clear when you come to revise the material.1 Nevertheless, it is very important to try to understand as much as possible of what is being said as it is said. Apart from saving time later, you may otherwise miss vital explanations and insights: the lecturer’s commentary on what he or she is writing is what makes lectures more instructive than simply reading a textbook. So: • Do make the effort to concentrate.2 We have all heard that, in a mathematics lecture, what the lecturer writes on the blackboard goes straight into the student’s notebook without passing through the brain of either. You should do everything in your power to prevent this happening: sit near the front if you find 50 minutes of mathematics a strain; don’t let your thoughts wander; don’t keep checking your phone (it should be turned off!); and remember that concentration is just a matter of selfdiscipline and practice. • Do ask questions during the lecture rather than let some diffi- culty pass by. If the lecturer is writing too fast, or too illegibly, or is speaking too quietly for you, it is likely that others are having the same difficulty. 1Reread this last paragraph! It is easy to become discouraged if you do not fully appreciate this message. 2The brain is a wonderful organ; it starts working the moment you get up in the morning and does not stop until you get into the lecture theatre. Robert Frost (American poet 1874 – 1963). 2
Don't be afraid to ask what you may think is a silly question. Nine times out of ten,most of the rest of the audience will be impressed (if only with your bravery)and many of them will also want to know the answer.And it is just possible that the lecturer has made a mistake Do try to appear responsive:look up when you have finished writing and are ready for more (this helps the lecturer pace the lecture);look puzzled when you are puzzled(so that the lecturer knows when more explanation is required):allow a gleam of recognition to surface if you suddenly realise what is going on (Bob Hope,the American comedian,used to say that he liked British audiences,because even if they don't feel like laughing at one of his jokes,they nod their heads to show that they have understood it.)） Don't ever miss lectures and rely on getting notes from a friend. You will understand the material much better if you were there to hear the explanation yourself. The usual convention in lectures is that you should write down exactly what the lecturer writes on the board or overhead projector.3 You may find that you can supplement this a little during the lecture but often there will not be time.Here are some important trivialities: Write the page number and lecture number on each sheet;if you drop your notes or get into a muddle photocopying them, you will find that one page of mathematics looks very like any other.This may seem a ludicrously trivial point.A few years ago when there was a severe water shortage the water boards put out an advertisement telling everyone to turn off the tap while brushing teeth.The purpose was not just to save the cupful of water,but to put people in the right frame of mind for making other more significant savings.You will find that 3In some lectures,you will receive hand-outs ranging from a summary to something resembling a text book.You may still want to tak e notes in the lecture, annotate the handout,or make your own summary notes after the lecture. 2
• Don’t be afraid to ask what you may think is a silly question. Nine times out of ten, most of the rest of the audience will be impressed (if only with your bravery) and many of them will also want to know the answer. And it is just possible that the lecturer has made a mistake. • Do try to appear responsive: look up when you have finished writing and are ready for more (this helps the lecturer pace the lecture); look puzzled when you are puzzled (so that the lecturer knows when more explanation is required); allow a gleam of recognition to surface if you suddenly realise what is going on. (Bob Hope, the American comedian, used to say that he liked British audiences, because even if they don’t feel like laughing at one of his jokes, they nod their heads to show that they have understood it.) • Don’t ever miss lectures and rely on getting notes from a friend. You will understand the material much better if you were there to hear the explanation yourself. The usual convention in lectures is that you should write down exactly what the lecturer writes on the board or overhead projector.3 You may find that you can supplement this a little during the lecture but often there will not be time. Here are some important trivialities: • Write the page number and lecture number on each sheet; if you drop your notes or get into a muddle photocopying them, you will find that one page of mathematics looks very like any other. This may seem a ludicrously trivial point. A few years ago when there was a severe water shortage the water boards put out an advertisement telling everyone to turn off the tap while brushing teeth. The purpose was not just to save the cupful of water, but to put people in the right frame of mind for making other more significant savings. You will find that 3 In some lectures, you will receive hand-outs ranging from a summary to something resembling a text book. You may still want to take notes in the lecture, annotate the handout, or make your own summary notes after the lecture. 3
numbering your pages will help with the overall orderliness that can be a great time saver. Leave wide margins;you will certainly need to annotate when you go through later. File your work (annotated lecture notes and supervision mate- rial)in an orderly way;this will save a lot of time when you come to revise. Here is the most important tip:you will save an immense amount of time if you always get to grips with one lecture before go- ing to the next.This way,you will get much more out of the lectures,which will in turn save time when you go through your notes later.You should therefore set aside a slot each day for going through your lecture notes-not just reading them,but working through them line by line.This is easy to say but hard to do;as soon as you fall behind it requires an enormous effort to catch up again. One final point.You may think that the lecturer is talking to you as a big group,but the lecturer actually sees a large number of indi- viduals.You should extend to the lecturer the normal courtesies of an individual conversation;behave as if the lecturer is talking to you personally.Don't,for example,spend the lecture chatting to your neighbour or reading the newspaper.This is most distracting for the rest of the audience and also for the lecturer,and is a sure recipe for a poor lecture.And please remember to turn off your mobile phone in lectures;or,better still,leave it in your room. Lecturing styles You will find that lecturers adopt a range of strategies for conveying to you the material listed in the schedules. For example,some lecturers work entirely on the blackboard or on overhead projectors;some give out a complete set of printed notes; some cover theory(say)on the blackboard and give out the examples 4
numbering your pages will help with the overall orderliness that can be a great time saver. • Leave wide margins; you will certainly need to annotate when you go through later. • File your work (annotated lecture notes and supervision material) in an orderly way; this will save a lot of time when you come to revise. Here is the most important tip: you will save an immense amount of time if you always get to grips with one lecture before going to the next. This way, you will get much more out of the lectures, which will in turn save time when you go through your notes later. You should therefore set aside a slot each day for going through your lecture notes — not just reading them, but working through them line by line. This is easy to say but hard to do; as soon as you fall behind it requires an enormous effort to catch up again. One final point. You may think that the lecturer is talking to you as a big group, but the lecturer actually sees a large number of individuals. You should extend to the lecturer the normal courtesies of an individual conversation; behave as if the lecturer is talking to you personally. Don’t, for example, spend the lecture chatting to your neighbour or reading the newspaper. This is most distracting for the rest of the audience and also for the lecturer, and is a sure recipe for a poor lecture. And please remember to turn off your mobile phone in lectures; or, better still, leave it in your room. 3 Lecturing styles You will find that lecturers adopt a range of strategies for conveying to you the material listed in the schedules. For example, some lecturers work entirely on the blackboard or on overhead projectors; some give out a complete set of printed notes; some cover theory (say) on the blackboard and give out the examples 4
(say)on hand-outs;some give out notes with gaps for diagrams or equations to be filled in by you.Some styles will suit you better than others,but it is very much a personal matter;do not assume that others will agree with you about what is best. Often,students especially first-year students who are used to A-level learning methods-want complete printed lecture notes, thinking that this is what they need to learn the material,as from a text book.That may be so,but the aim is to understand the material. which is a very different matter.For this,it may be much more useful to have a carefully distilled set of notes that brings out all the main ideas;the work you do in fleshing out the details will serve you far better in the long run than reading a complete set of printed notes. Supervisions There is apparently not much scope for heated discussion of topical issues in mathematics supervisions.In fact,any debate at all can be difficult (at least at first),since your opinion seems not to count for much when your supervisor knows all the right answers (having been told them by his or her supervisors when they did the course). Nevertheless,a supervision should not be a mini-lecture:if it turns into one,then that is a valuable opportunity wasted. Whereas lectures are to some extent interactive,this is very much the case with supervisions.A good part of the responsibility for mak- ing the supervision useful and interesting lies with you.Remember that most supervisors are human beings too:they like you to talk to them and show an interest (e.g.by asking questions)in what they are telling you. Generally,in a mathematics supervision,you sit at a desk with your supervisor who will write out solutions to exercises or explana- tions of pieces of mathematics on paper (not on a blackboard).You should not take notes yourself;leave your mind completely free to concentrate on understanding everything your supervisor says.At the end of the supervision,you take away what he or she has written and (best)use it to annotate,correct or complete your own super-
(say) on hand-outs; some give out notes with gaps for diagrams or equations to be filled in by you. Some styles will suit you better than others, but it is very much a personal matter; do not assume that others will agree with you about what is best. Often, students — especially first-year students who are used to A-level learning methods — want complete printed lecture notes, thinking that this is what they need to learn the material, as from a text book. That may be so, but the aim is to understand the material, which is a very different matter. For this, it may be much more useful to have a carefully distilled set of notes that brings out all the main ideas; the work you do in fleshing out the details will serve you far better in the long run than reading a complete set of printed notes. 4 Supervisions There is apparently not much scope for heated discussion of topical issues in mathematics supervisions. In fact, any debate at all can be difficult (at least at first), since your opinion seems not to count for much when your supervisor knows all the right answers (having been told them by his or her supervisors when they did the course). Nevertheless, a supervision should not be a mini-lecture; if it turns into one, then that is a valuable opportunity wasted. Whereas lectures are to some extent interactive, this is very much the case with supervisions. A good part of the responsibility for making the supervision useful and interesting lies with you. Remember that most supervisors are human beings too: they like you to talk to them and show an interest (e.g. by asking questions) in what they are telling you. Generally, in a mathematics supervision, you sit at a desk with your supervisor who will write out solutions to exercises or explanations of pieces of mathematics on paper (not on a blackboard). You should not take notes yourself; leave your mind completely free to concentrate on understanding everything your supervisor says. At the end of the supervision, you take away what he or she has written and (best) use it to annotate, correct or complete your own super- 5
vision work or lecture notes,or (second best)file it with your own work.Your supervision partner can use the notes after you,or save them digitally in some way. If you find that you come away from the supervision without adequate notes.you should discuss the matter with the supervisor either by e-mail or at the beginning of the next supervision.It is your supervision,so the supervisor should try to fit in with what you want. You must hand your work in well before the supervision,and certainly by the time specified by your supervisor. Your work should be neatly and clearly presented.If your work is scribbled and scruffy,then you should rewrite it.4 You will not regret time spent on this:clarity of presentation comes from and leads to clarity of mathematical thought It should be logically and carefully argued otherwise it is not mathematics.You may think you can do a problem before you even set pen to paper,but you don't know that you can do a problem until you write out all the details.Also,unless you are in the habit of writing careful solutions,you will come unstuck in the examinations when you do not have the opportunity to explain what you really meant. If you do not make good use of supervisions,then you will squan- der one of the most important(and expensive)assets that Cambridge has to offer.Experience has shown that to make best use of your su- pervisions,you should: bring your lecture notes to the supervision,having marked in the bits you don't follow; hand your work in on time,so that it can be marked thoroughly; mark your own work:make a note in the margin wherever there is a step you are not sure of,or which you have missed out.This 4Please read that sentence again.It is really disrespectful to hand in scruffy 6
vision work or lecture notes, or (second best) file it with your own work. Your supervision partner can use the notes after you, or save them digitally in some way. If you find that you come away from the supervision without adequate notes, you should discuss the matter with the supervisor either by e-mail or at the beginning of the next supervision. It is your supervision, so the supervisor should try to fit in with what you want. You must hand your work in well before the supervision, and certainly by the time specified by your supervisor. Your work should be neatly and clearly presented. If your work is scribbled and scruffy, then you should rewrite it.4 You will not regret time spent on this: clarity of presentation comes from and leads to clarity of mathematical thought. It should be logically and carefully argued otherwise it is not mathematics. You may think you can do a problem before you even set pen to paper, but you don’t know that you can do a problem until you write out all the details. Also, unless you are in the habit of writing careful solutions, you will come unstuck in the examinations when you do not have the opportunity to explain what you really meant. If you do not make good use of supervisions, then you will squander one of the most important (and expensive) assets that Cambridge has to offer. Experience has shown that to make best use of your supervisions, you should: • bring your lecture notes to the supervision, having marked in the bits you don’t follow; • hand your work in on time, so that it can be marked thoroughly; • mark your own work: make a note in the margin wherever there is a step you are not sure of, or which you have missed out. This 4Please read that sentence again. It is really disrespectful to hand in scruffy work; why expect the supervisor to spend time trying to make something of it when you can’t be bothered to present it nicely? 6
is not just to help your supervisors,though it will undoubtedly make their lives easier:a self-critical approach to your own work will prove invaluable when it comes to exam time; tell your supervisor (if appropriate and polite)exactly what you would like him or her to do-remember that sometimes your supervisor will have much less experience of supervisions than you,and will be glad of your advice; don't be afraid to say you don't understand something or couldn't do something -your supervisor is there to help you not to judge you,and this is your big chance to fill any gaps in under- standing and learn how to do the things you were stuck on; similarly,don't be afraid to hand in partial solutions to ques- tions you couldn't complete;your supervisor will then be able to see where the sticking point was; make sure your supervisor writes down enough on each example for you to reconstruct the solution afterwards; have an intelligent question ready in case the supervision is grinding to a halt with time to spare; review the supervision as soon as possible afterwards,while it is still fresh in your mind. Last of all,here is the most important tip:do not be lazy.It is very easy to let what the supervisor is saying just wash over you,perhaps hoping that all will become clear later.If you don't understand what the supervisor has done,say so. 7
is not just to help your supervisors, though it will undoubtedly make their lives easier: a self-critical approach to your own work will prove invaluable when it comes to exam time; • tell your supervisor (if appropriate and polite) exactly what you would like him or her to do — remember that sometimes your supervisor will have much less experience of supervisions than you, and will be glad of your advice; • don’t be afraid to say you don’t understand something or couldn’t do something — your supervisor is there to help you not to judge you, and this is your big chance to fill any gaps in understanding and learn how to do the things you were stuck on; • similarly, don’t be afraid to hand in partial solutions to questions you couldn’t complete; your supervisor will then be able to see where the sticking point was; • make sure your supervisor writes down enough on each example for you to reconstruct the solution afterwards; • have an intelligent question ready in case the supervision is grinding to a halt with time to spare; • review the supervision as soon as possible afterwards, while it is still fresh in your mind. Last of all, here is the most important tip: do not be lazy. It is very easy to let what the supervisor is saying just wash over you, perhaps hoping that all will become clear later. If you don’t understand what the supervisor has done, say so. 7
5 Supervision reports At the end of each term,each of your supervisors will lodge a report for you on CamCORS5.It has to be released by your Director of Studies before you can read it:this is just a precaution,to cope with the extremely unlikely event of a completely inappropriate report,or a muddling of names,or some other disaster.You will receive an automatic e-mail as soon as a supervision report is available.It is important that you read your reports,and if they don't appear to be there,ask your Director of Studies to investigate.Usually,they will appear(because supervisors are not paid until they file a report),but they might be late in which case they may arrive at a time when your Director of Studies is not expecting them and will not release them without a prod from you. 6 Guide for supervisors If you would like to see how the process of supervision looks from the 'other side',you can see our (rather good)guide for supervisors: http://www.maths.cam.ac.uk/facultyoffice/supervisorsguide/ 7 Methods of working Each of you will have to decide individually what method of working suits you:where to work,when to work,how to work.Only you can decide what is best for you.Some like to work in their rooms and some like to work in libraries away from temptations.Some like to work late into the night and some prefer sleep at night.Clearly it is best to pace your work over the week or fortnight between supervisions rather than to leave it to the last moment,especially as you cannot be sure how long the work will take. 5Cambridge Colleges Online Reporting System;your Director of Studies will explain about this
5 Supervision reports At the end of each term, each of your supervisors will lodge a report for you on CamCORS5 . It has to be released by your Director of Studies before you can read it: this is just a precaution, to cope with the extremely unlikely event of a completely inappropriate report, or a muddling of names, or some other disaster. You will receive an automatic e-mail as soon as a supervision report is available. It is important that you read your reports, and if they don’t appear to be there, ask your Director of Studies to investigate. Usually, they will appear (because supervisors are not paid until they file a report), but they might be late in which case they may arrive at a time when your Director of Studies is not expecting them and will not release them without a prod from you. 6 Guide for supervisors If you would like to see how the process of supervision looks from the ‘other side’, you can see our (rather good) guide for supervisors: http://www.maths.cam.ac.uk/facultyoffice/supervisorsguide/ 7 Methods of working Each of you will have to decide individually what method of working suits you: where to work, when to work, how to work. Only you can decide what is best for you. Some like to work in their rooms and some like to work in libraries away from temptations. Some like to work late into the night and some prefer sleep at night. Clearly it is best to pace your work over the week or fortnight between supervisions rather than to leave it to the last moment, especially as you cannot be sure how long the work will take. 5Cambridge Colleges Online Reporting System; your Director of Studies will explain about this. 8
An important question to decide early on is whether you want to work alone or sometimes to get together with a friend.The essential thing to remember if you work with someone else is that collaboration must not mean copying or you will lose all the benefit:it is fine to discuss the work but you must have had a good go at it first and it should be an equal partnership with the aim of going further in the time available rather than halving the amount of time you need to spend on the work. However you work,you should remember that university math ematics should not be regarded as a competitive sport.No one is interested in how quickly you managed to do the problems or your other achievements,and making a big deal out of it can be very dis- couraging for others not as inwardly confident as you appear.Surveys show that unhelpful comparisons with peers easily become a source of stress and anxiety-so try not to add to the problem(or succumb to it). 8 Writing mathematics Most mathematicians can write accurate grammatical prose,they understand (for example)why the comma in this sentence should have been a full stop or a semi-colon.There is a grammar to writing mathematics as well.Symbols such as/,→，3，etc,should be used in a way that makes grammatical sense if read out in full.If you are careless about this,then you will certainly find yourself using sloppy logic as well as sloppy mathematical grammar You should try to write in full sentences,using normal punctua- tion:full stop at the end of a sentence even if it ends in an equation, commas normally in pairs,etc.Sentences should be short and as sim- ple as possible.If you find yourself writing paragraphs of text,then vou should consider whether vou are writing more than is necessary to explain what you are doing.6 Note:in this e.if is follo you should a to thi en;it sounds a bit stilted here but ematical sentence
An important question to decide early on is whether you want to work alone or sometimes to get together with a friend. The essential thing to remember if you work with someone else is that collaboration must not mean copying or you will lose all the benefit; it is fine to discuss the work but you must have had a good go at it first and it should be an equal partnership with the aim of going further in the time available rather than halving the amount of time you need to spend on the work. However you work, you should remember that university mathematics should not be regarded as a competitive sport. No one is interested in how quickly you managed to do the problems or your other achievements, and making a big deal out of it can be very discouraging for others not as inwardly confident as you appear. Surveys show that unhelpful comparisons with peers easily become a source of stress and anxiety — so try not to add to the problem (or succumb to it). 8 Writing mathematics Most mathematicians can write accurate grammatical prose, they understand (for example) why the comma in this sentence should have been a full stop or a semi-colon. There is a grammar to writing mathematics as well. Symbols such as ∀, ⇒, ∃, etc, should be used in a way that makes grammatical sense if read out in full. If you are careless about this, then you will certainly find yourself using sloppy logic as well as sloppy mathematical grammar. You should try to write in full sentences, using normal punctuation: full stop at the end of a sentence even if it ends in an equation, commas normally in pairs, etc. Sentences should be short and as simple as possible. If you find yourself writing paragraphs of text, then you should consider whether you are writing more than is necessary to explain what you are doing.6 6Note: in this sentence, if is followed by then; it sounds a bit stilted here but you should always adhere to this rule in a mathematical sentence. 9