Aphalleon
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I'm planning to study Maths with Economics at university. I'm already doing A-level Maths and Further Maths, both of which i really enjoy. I have heard however, that university maths is completely different will more emphasis on logic and proof and is nothing like A-level Further Maths "carried on"


Are there any online lectures, books etc. which can give me a "feel" for uni maths?
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Bobifier
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If you look on NRich you will find a lot of Maths problems in a similar style to those which you will encounter at university. If you want a book, I have always found that Excursions into Mathematics bridges the gap very well, though it is quite expensive.
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Raiden10
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(Original post by Aphalleon)
I'm planning to study Maths with Economics at university. I'm already doing A-level Maths and Further Maths, both of which i really enjoy. I have heard however, that university maths is completely different will more emphasis on logic and proof and is nothing like A-level Further Maths "carried on"


Are there any online lectures, books etc. which can give me a "feel" for uni maths?
One great book for this sort of purpose is Liebeck's "A Concise Introduction to Pure Mathematics". You could also try "Concepts of Modern Mathematics" by Ian Stewart, although I haven't read it it has some very good reviews and doesn't hold back.

It's true that uni maths is not A-level Further Maths "carried on". It's more like "starting again". You take a few steps backward, and upon stepping forward again you can see many avenues to which you were oblivious before, some of which will be entirely separate from what previously appeared to be the entirety of mathematics.

The closest thing to A-level Maths or Further Maths "carried on" is something called mathematical methods, an area of applied mathematics. That is just one small part of university mathematics. The purpose of A-level Maths/Further Maths is (1) to prepare engineers and scientists, people who aren't doing maths at uni, for the mathematical parts of their university courses, and (2) to raise the level of abstraction slightly from GCSE and have students be aware of the properties of the sets N, Z, Q, R and C which will provide essential examples to help make sense of the abstract point of view of university mathematics.

So yes, it's very different! Unless you have prodigious qualities, not until you are about 18 will be you ready to tolerate the abstraction of university maths. There is no guarantee but if you really like maths at the moment, there is a *good* chance you will like "serious" university mathematics. If it turns out you don't (or if you do) you can go into finance and make loads of money.
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Jodin
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Well you could read some lecture notes, Warwick seems to make some of theirs available (I'm unsure exactly what the rules are regarding it). Still if you search "foundations lecture notes" on Google for example you ought to get Warwicks lecture notes for a 1st year module pretty much with the intention of getting you into uni maths, as such there's nothing difficult but I guess it gives an idea of the kind of rigour we're talking (though Analysis would likely do a better job of that), and some parts are pretty interesting too, I find them to be very well written so learning from them should be quite simple too .
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pomme de terre
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Most of Manchester's lecture notes are online http://www.maths.manchester.ac.uk/un...its/index.html

just click on the unit you want to look at the lectures for, then click "view online course materials". (not all of them are up there though)
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alibee
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Don't freak yourself out reading them though. I look at online notes before my lectures sometimes and they often make very little sense! There's something about having it actually explained to you that makes it sit in your head a bit better.
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Spungo
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Looking at lecture notes now is not a good idea. I looked at a few after I had done all of my exams of one of the courses I'm actually doing now, and it was completely incomprehensible and filled me with dread.
As mentioned before, "a concise introduction to pure mathematics" is a very nice book to get, it follows on from A levels smoothly and explains things in easy to understand ways. It has a few exercises at the end of each section just to reinforce some ideas, really good for doing a bit of over the summer.
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Raiden10
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(Original post by Spungo)
Looking at lecture notes now is not a good idea. I looked at a few after I had done all of my exams of one of the courses I'm actually doing now, and it was completely incomprehensible and filled me with dread.
As mentioned before, "a concise introduction to pure mathematics" is a very nice book to get, it follows on from A levels smoothly and explains things in easy to understand ways. It has a few exercises at the end of each section just to reinforce some ideas, really good for doing a bit of over the summer.
What's maths like at Durham? I heard they have a very good (research) reputation for the area of geometry/topology.

Do they cover Cantor's diagonal argument in the first year? Curiously I heard that they didn't.
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rowzee
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I will probably not add anything worthwhile to the discussion, but you can anticipate the majority of pure maths course to follow the structure Definition, Theorem, Proof in that order with the occasional example thrown in.
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Spungo
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(Original post by Raiden10)
What's maths like at Durham? I heard they have a very good (research) reputation for the area of geometry/topology.

Do they cover Cantor's diagonal argument in the first year? Curiously I heard that they didn't.
I don't really know. I've only been to 2 lectures on fourier series there.
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Narev
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You can look at this link for an introduction to Analysis: http://www.ueltschi.org/teaching/201...otes-MA131.pdf. Not structured in form of lecture notes / textbook, but more of exercises to help you think.
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.ACS.
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The worst thing you can do is look at first year lecture notes and example exam papers. I did this and decided to opt for Economics as opposed to Maths and Statistics, and it's the biggest regret I've ever made. The example exam papers scared me into thinking I wouldn't cope with university level mathematics, but in my economics degree I've opted to specialise in theoretical econometrics, and make heavy use of matrix algebra and calculus constantly. (The lecture notes for one of my second year modules.)

In all honesty, if you think you want to study Maths at university, and found you have really enjoyed A-Level Maths and A-Level Further Maths and didn't particularly struggle with either, then go for it.
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mastermind_107
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(Original post by Aphalleon)
I'm planning to study Maths with Economics at university. I'm already doing A-level Maths and Further Maths, both of which i really enjoy. I have heard however, that university maths is completely different will more emphasis on logic and proof and is nothing like A-level Further Maths "carried on"


Are there any online lectures, books etc. which can give me a "feel" for uni maths?
The STEP examinations which Cambridge uses as its de facto entrance exam for the Mathematics Tripos give a good insight into degree-level mathematics. Their main website is here (where you can find past papers) and a booklet which was prepared to help candidates with STEP preparation is here.
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monkyvirus
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To be honest it depends which kind of math modules you intend on studying as you will usually need to take a combination of pure, applied and statistics. The pure is what will be really different from school most of the other stuff is way more complicated but your way of thinking stays the same.

On the whole it won't actually be that different just more proof and while this may sound bizarre a pretty useful thing to do would be to get a second hand copy of Professor Layton and crack on through the puzzles which aren't hard but they'll help you find a new way of thinking with particular emphasis on logic skills. Before you scoff my flat-mate and I have had a couple of questions in pure tutorials that I knew the answer to because I was asked the same kind of thing in prof L

Looking at uni level stuff will just confuse you in the same a GCSE student would be put of by an A-Level textbook if they just started to browse one.

as mastermind_107 says STEP is a good place to look as it is pretty much a representation of 1st year maths at uni, many an undergraduate has proclaimed they would've aced STEP if they'd taken it after 1st year :P
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Raiden10
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(Original post by mastermind_107)
The STEP examinations which Cambridge uses as its de facto entrance exam for the Mathematics Tripos give a good insight into degree-level mathematics. Their main website is here (where you can find past papers) and a booklet which was prepared to help candidates with STEP preparation is here.
I don't think STEP will provide much insight into degree mathematics. It gives you insight into the skills required to excel at exams at Warwick and Oxbridge maths courses. It's essentially A-level mathematics with the difficulty set to "fiendish".

A-level and STEP are still set in the same year - 1700. On the other hand uni maths is set in the 1800s and 1900s, a different era.

[OTOH when I think of "uni mathematics" I think of equivalence relations, vector spaces, metric spaces, graphs, automorphism groups, partial orders, monomorphisms. Perhaps if you are more likely to think about physics, differential equations, stats those do link more to Newton's era.]
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rr0
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Hi,

You can see some course notes on the Open University web site (for maths and many other subjects).

You can see a list of maths modules here -
http://www3.open.ac.uk/study/undergr...roduct-courses

And you can see some of the course notes here -
http://openlearn.open.ac.uk/course/

I have taken the following courses ...
MST121 - Using mathematics
MS221 - Exploring mathematics
M208 - Pure mathematics (Introduction to Linear Algebra, Analysis and Group Theory)
MST209 - Mathematical methods and models (mechanics)

And I hope to do the following ...
M338 - Topology (recently started)
M336 - Groups and geometry
M337 - Complex Analysis
M381 - Number Theory and mathematical logic
MS324 - Waves, diffusion and variational principles
MST326 - Mathematical methods and fluid mechanics
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mastermind_107
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(Original post by Raiden10)
I don't think STEP will provide much insight into degree mathematics. It gives you insight into the skills required to excel at exams at Warwick and Oxbridge maths courses. It's essentially A-level mathematics with the difficulty set to "fiendish".

A-level and STEP are still set in the same year - 1700. On the other hand uni maths is set in the 1800s and 1900s, a different era.

[OTOH when I think of "uni mathematics" I think of equivalence relations, vector spaces, metric spaces, graphs, automorphism groups, partial orders, monomorphisms. Perhaps if you are more likely to think about physics, differential equations, stats those do link more to Newton's era.]
I think the STEP examinations act as a good transition from A Level work to 1st year undergraduate work. If you learn all your undergraduate stuff now, what's the point of going to university?! Also, you don't want to be learning things "wrongly" so that when you do take a lecture course on so and so, you have to unlearn what you know first.

On the other hand, becoming more fluent with the stuff you already know by practising questions is better preparation.

Also, it's the style of STEP questions which make them such a nice transition from A Level to undergraduate studies. You get choice in the questions you answer and the way you have to write up solutions to these is quite different to the standard rote learning of A Level answers.
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Narev
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(Original post by mastermind_107)
I think the STEP examinations act as a good transition from A Level work to 1st year undergraduate work. If you learn all your undergraduate stuff now, what's the point of going to university?! Also, you don't want to be learning things "wrongly" so that when you do take a lecture course on so and so, you have to unlearn what you know first.

On the other hand, becoming more fluent with the stuff you already know by practising questions is better preparation.

Also, it's the style of STEP questions which make them such a nice transition from A Level to undergraduate studies. You get choice in the questions you answer and the way you have to write up solutions to these is quite different to the standard rote learning of A Level answers.
I don't think anyone here is advocating learning all the course material before starting University. But it's good to know the basic outlines, idea of proofs, etc before starting University. At the very worst - you'll end up with a headstart over several people.

It's also hard to learn things wrongly for mathematics - it's either true, or false, or open problem. If you look at the lecture notes some people posted, I doubt you can "learn the wrong stuff" from them.

*Barring any occasional typos.

I also doubt practicing many A level mathematics questions will help in University mathematics, but that's just my opinion.
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rowzee
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(Original post by Narev)
It's also hard to learn things wrongly for mathematics - it's either true, or false, or open problem. If you look at the lecture notes some people posted, I doubt you can "learn the wrong stuff" from them.
Not entirely right - especially for someone with little or no experience with the subject. Maths is very scrupulous and the tiniest details make all the difference; with no guidance people tend to leave out important parts, claim things are true without any justification (and mostly they are wrong anyway). I am supervising a bunch of first-years this year and they were experiencing exactly this (some still are, but overall they've gotten a lot better). It is hard to avoid unless your always right on the ball.

That said, I think just reading the notes can't do much harm, however without practicing its hard to learn anything at all.
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Narev
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(Original post by rowzee)
Not entirely right - especially for someone with little or no experience with the subject. Maths is very scrupulous and the tiniest details make all the difference; with no guidance people tend to leave out important parts, claim things are true without any justification (and mostly they are wrong anyway). I am supervising a bunch of first-years this year and they were experiencing exactly this (some still are, but overall they've gotten a lot better). It is hard to avoid unless your always right on the ball.

That said, I think just reading the notes can't do much harm, however without practicing its hard to learn anything at all.
Possibly. I would have thought someone reading the notes consistently (or do the Analysis workbooks) probably won't make such errors.

I am also supervising a bunch of first years this year, and even with guidance, I find that some still don't read the notes / recommended textbooks / handouts I give -_- (and then ask me about it in the next supervision), making these errors.
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