MTH6109 |
Combinatorics |
| Course Material | Autumn 2010 |
| For information |
|---|
| No revision lecture has been scheduled for this course during the revision "week". If you want help with your revision, you are welcome to visit me during my office hours, or email for an appointment. |
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Administrative information
Your working week
Each module is designed to fill one-quarter of a full-time working week. That is, you are expected to work roughly ten hours a week on each module. Only four hours are timetabled. The rest is your responsibility. It should include, at the very least:Your career
Employers value mathematics graduates for one quality above all others: they can think. In a recession, this is more important than ever. A 2(i) Honours degree on its own is not a passport into a good job. At interview you will be expected to show you can think for yourself as well. A module like this one (Combinatorics) is designed in particular to train you to think, at a high level. Make sure you use this opportunity well.Why mathematics?
Mathematics is the most empowering of all disciplines. In many disciplines, one has to appeal to higher authority to decide what is right and what is wrong. But in mathematics, you are given the tools to decide for yourself, using logic alone. As Richard Hamming said"In science and mathematics we do not appeal to authority, but rather you are responsible for what you believe."
Lectures
What is the purpose of lectures, if the notes are available online? Experience has shown that it is actually quite difficult to learn mathematics by reading notes or books, even if you are conscientious about doing the exercises (which most people are not). If you are studying music, it is much better to hear it in real time, rather than read the score. A mathematical proof is like a piece of music: there is really no substitute for seeing it develop in real time. A good lecturer tells you not only the facts, but also why they have to be that way, and provides explanations tailored to the class that you could never find in a book.Notes
I will not be preparing my own notes for this page. You are recommended to look at Prof. Cameron's Notes for the previous version of the course Notes on combinatorics. These are however not a substitute for taking your own notes during lectures.Reading list
The recommended course text is
P. J. Cameron, Combinatorics: topics, techniques, algorithms. Cambridge University Press (1994).
Other books you may find useful:
Prerequisites
Combinatorics is often thought of as an `easy option', as it has little in the way of formal pre-requisites. However, this is a mistake. The lack of formal pre-requisites means that you have to rely instead on native intelligence, imagination, pure thought and logic. This makes it potentially a hard subject. But the problem-solving aspect of it can be very satisfying, especially if you're good at it.
Results versus methods
Learning results ("know-what", also called theorems) is inefficient compared to learning methods ("know-how", also called proofs), because a result is a fixed thing with limited applicability, whereas a method is more general and may be used to work out many different results. And if you know the method, then you can work out the result; but if you only know the result, how can you work out the method?
Exercises and tutorials
These should be regarded as compulsory. Mathematics is not about learning facts ("know-what"), it is about learning methods ("know-how"). Methods and techniques cannot be learnt without practice. As Confucius (551-479BCE) said:"I hear and I forget;
I see and I remember;
I do and I understand."
Exercises. Also available as pdf
Joining a gym
If you paid hundreds of pounds a year for membership of a gym, would you skip the exercise classes organised for you? Would you expect to get fit if you stayed at home instead of going to the gym? Would you expect to get fit if you watched your personal trainer doing the exercises instead of doing them yourself?
No? So why do so many of you treat membership of a university in this way? There is no short cut to training your mind, just as there is no short cut to training your body. Exercise until it hurts: that is the only way. In the exercise classes organised for your benefit, your personal trainer will show you how to use the equipment, and get you started: then you need to put in the hard work yourself.
Continuous assessment and feedback
The purpose of continuous assessment is not for you to accumulate a handful of marks by copying someone else's work. This strategy gives you a short-term payoff, for which you pay dearly in the long run. By copying one piece of continuous assessment, you may, (if you do not get caught) gain 1% (which you do not deserve) towards your final module mark. By doing the exercises yourself, and using the feedback provided to see where you went wrong, you may learn enough to get an additional 10% (or even more) in the final exam. Which of these two strategies do you think is the more sensible?Working together
Working together can be a very useful way of learning from each other. However, it can also be an excuse for laziness, or, worse, plagiarism. Most perniciously, it can lull you into a false sense of security, thinking that you can do something when in fact somebody else is always doing it for you. Make sure that you are working together in a positive way. At the end of the day, when it comes to the exam, you have to be able to think for yourself.Solutions to selected exercises
The solutions given here are not "model solutions" which can be memorised and trotted out in an exam. They are designed to show you how to do something if you have not managed to do it yourself. They are often sketchy, and it is your job to fill in the details once the overall picture is given to you. If you have difficulty doing this, or if you don't understand the solutions given, please ask for help in the tutorials.Model solutions
I will not provide model solutions to past examination papers, because they encourage poor quality learning. If you wish to use past exam papers as an aid to your revision, you may bring your solutions to one of my office hours, and I will give you feedback on them as far as time allows.Revision Revision is an essential part of the learning process. Here are a few suggestions which you may find useful. First, read your notes in suitably small chunks. Make sure you understand the main points: if not, consult books or the www as necessary, and try some more exercises. Second, write new notes summarising the old ones. Third, summarise the summary. When you think you understand, try explaining it to someone else (in writing, if you can't find a willing victim to listen to you). Only when you can do this can you really say you have understood.
Other course material
Web ResourcesSome positive comments: Prof Cameron's notes are useful, as is having the question booklet early. Also the `topic a week' structure.
Some misconceptions:
"solutions to only the feedback questions are given" - this is not true,
there are solutions to at least 25 other questions in the printed lecture notes. You just have
to look for them.
"why are we given 20-30 questions per week" - I have set between 4 and 8
questions each week. I have put extra questions on the web-site which you can
attempt if you wish, or use for revision later, or ignore if you prefer.
"I have to spend a lot of hours going back over the notes..." - this was
presented as a criticism, but this is exactly what you are supposed
to do.
"there is little or no help on the actual set question for the week" -
this is in order to make the marking fairer to everybody. The practice questions
are there to help you get to grips with the necessary material, after which
the feedback question is (usually) fairly straightforward.
Robert A. Wilson
Created 10 August 2010
Updated 13 April 2011