Maths beyond A Level without doing a degree
I wasn't sure quite where to put this, since it didn't really belong in the A Level Maths forum.
I love Maths. I can sit around and work my way through problems all day long and just genuinely enjoy it - I've done all the A Level Maths I possibly can (Regular, Further and Additional, although additional isn't aggregated past AS since I missed an exam).
I enjoyed Pure the most, followed closely by Mechanics. Statistics could be interesting at S3/S4 level, but Decision was pretty nonsensical outside of topics like Game Theory.
The problem is, I didn't like the idea of the end result of having done a degree in Maths (usually the result is either go into Banking, Economics etc or Teaching - there are very few actual 'academic' positions AFAIK), so I chose a different field to do at undergraduate (along with other reasons).
I was wondering if there was anything I could do beyond A Level that would keep my interest in Maths alive; any books or websites etc to use and problems to do that sort of build on A Level maths and go from there - i.e, they introduce a topic and explain it and give you some problems to solve (I know the dynamic of Maths changes once you get to university in that it isn't particularly about "Here's X - integrate it") - I've done STEP papers in my spare time, but the reality is that they are firmly grounded in A Level Maths - the questions are just a lot more difficult.
I'm not looking to get any qualification in it, or to not focus on my degree and I'm not trying to pretend I would be doing it at / understanding it at the level of someone actually doing a Maths degree, but I would just like to make sure that I have the same passion for Maths in 10 years time, rather than just slowly forgetting it all because it isn't being used.
Thanks for reading.
I love Maths. I can sit around and work my way through problems all day long and just genuinely enjoy it - I've done all the A Level Maths I possibly can (Regular, Further and Additional, although additional isn't aggregated past AS since I missed an exam).
I enjoyed Pure the most, followed closely by Mechanics. Statistics could be interesting at S3/S4 level, but Decision was pretty nonsensical outside of topics like Game Theory.
The problem is, I didn't like the idea of the end result of having done a degree in Maths (usually the result is either go into Banking, Economics etc or Teaching - there are very few actual 'academic' positions AFAIK), so I chose a different field to do at undergraduate (along with other reasons).
I was wondering if there was anything I could do beyond A Level that would keep my interest in Maths alive; any books or websites etc to use and problems to do that sort of build on A Level maths and go from there - i.e, they introduce a topic and explain it and give you some problems to solve (I know the dynamic of Maths changes once you get to university in that it isn't particularly about "Here's X - integrate it") - I've done STEP papers in my spare time, but the reality is that they are firmly grounded in A Level Maths - the questions are just a lot more difficult.
I'm not looking to get any qualification in it, or to not focus on my degree and I'm not trying to pretend I would be doing it at / understanding it at the level of someone actually doing a Maths degree, but I would just like to make sure that I have the same passion for Maths in 10 years time, rather than just slowly forgetting it all because it isn't being used.
Thanks for reading.
(edited 12 years ago)
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What degree are you doing?
Well this is wrong. It's common but there are plenty of other things.
Original post by TwilightKnight
The problem is, I didn't like the idea of the end result of having done a degree in Maths (usually the result is either go into Banking, Economics etc or Teaching - there are very few actual 'academic' positions AFAIK), so I chose a different field to do at undergraduate (along with other reasons).
The problem is, I didn't like the idea of the end result of having done a degree in Maths (usually the result is either go into Banking, Economics etc or Teaching - there are very few actual 'academic' positions AFAIK), so I chose a different field to do at undergraduate (along with other reasons).
Well this is wrong. It's common but there are plenty of other things.
Original post by refref
What degree are you doing?
I don't really know what difference knowing what degree I'm doing makes, but I'm doing Medicine.
Original post by refref
Well this is wrong. It's common but there are plenty of other things.
Well this is wrong. It's common but there are plenty of other things.
I know that there are other options, but what I was trying to say was that the only option I would see myself enjoying having done a degree in Maths would be as an academic, and there are many fewer spots in this country than there are in say, America. It's irrelevant now as I've made my choice, but I just wanted to know if there was anything I could do to keep my Maths going, more as a hobby than anything else.
(edited 12 years ago)
Original post by SimonM
Do a closely related degree: Physics, CompSci, etc. and take as much maths as possible
I'm already (going to be) doing another unrelated degree. It isn't a question of "I like Maths, but not enough to do a Maths degree" - if I was going to do anything else other than Medicine, I would do Maths.
What I'm looking for is a way to keep my Maths interest alive whilst also maybe going a bit beyond A Level. I've done all the A Level Maths you can possibly do and I can't really just re-tread old ground over and over.
Original post by angelmxxx
Open university modules maybe?
I had considered that, but I didn't want to do this 'officially'. Maths is more of a hobby now and I never really wanted it to feel like I had to be doing it, because if it's a choice between my degree and Maths, I'm going to pick the degree. Also, the cost of doing OU modules is prohibitively expensive for a full time student. It's not something I've ruled out altogether, but it would have to wait until long into the future, at which point I would probably consider doing Maths full time at an actual physical university.
I guess I should explain more what I'm looking for: I'm hoping to find some websites or a collection of books etc that teach you the method and then give you unstructured problems to solve (if that's possible). I'm not averse to sitting there for hours until I figure it out if the problems are challenging or if they method isn't explained and I have to research elsewhere. I'm not entirely sure if what I'm asking is possible on the level I'm asking for, but I thought it would be a waste if I didn't ask.
Original post by TwilightKnight
I guess I should explain more what I'm looking for: I'm hoping to find some websites or a collection of books etc that teach you the method and then give you unstructured problems to solve (if that's possible). I'm not averse to sitting there for hours until I figure it out if the problems are challenging or if they method isn't explained and I have to research elsewhere. I'm not entirely sure if what I'm asking is possible on the level I'm asking for, but I thought it would be a waste if I didn't ask.
I guess I should explain more what I'm looking for: I'm hoping to find some websites or a collection of books etc that teach you the method and then give you unstructured problems to solve (if that's possible). I'm not averse to sitting there for hours until I figure it out if the problems are challenging or if they method isn't explained and I have to research elsewhere. I'm not entirely sure if what I'm asking is possible on the level I'm asking for, but I thought it would be a waste if I didn't ask.
http://projecteuler.net/index.php?section=about
Depends on what you're looking for.
General introductory books transitioning to degree level are:
- What is Mathematics?. This covers some basic set and number theory, geometry and analysis. Nothing is covered in a lot of depth but its very well written. The last chapter is now a little out of date (in places) but even so is still a good read
- A Concise Introduction to Pure Mathematics. This covers the foundation part of most first year courses, covering again basic set and number theory but in a very different style to What is Mathematics?
I would spend some time reading through this course guide, which give a brief description of the sort of thing you do at degree level. A lot of it is not about solving complicated equations but about understanding the mathematics you already know from A-Level. It is a lot more rigorous which might take some getting used to.
Once you have a feel for what you want to pursue, there are lecture notes aplenty around the internet. From a simple search:
Differential Equations
Probability
Analysis
I purposefully chose three different institutions as there are loads of lecture notes around - e.g. see Cambridge (I'm sure there are better examples of Cambridge notes) and Oxford examples
The quality of notes is not dependent on the reputation of the university. As I was studying I used lectures notes from all over the world, just keep trying until you find something that works.
American universities have a different style of course but again have some excellent notes.
If you're looking for questions, at university they're called problem sheets or example sheets. Most university courses have these public, even if the notes aren't so there will be lots for you to choose from.
Finally, if you're at university doing Medicine I'm sure you can browse their maths books until you find something that gets your interest.
(Specific examples of books/notes will be easier once you've narrowed your interests a bit)
General introductory books transitioning to degree level are:
- What is Mathematics?. This covers some basic set and number theory, geometry and analysis. Nothing is covered in a lot of depth but its very well written. The last chapter is now a little out of date (in places) but even so is still a good read
- A Concise Introduction to Pure Mathematics. This covers the foundation part of most first year courses, covering again basic set and number theory but in a very different style to What is Mathematics?
I would spend some time reading through this course guide, which give a brief description of the sort of thing you do at degree level. A lot of it is not about solving complicated equations but about understanding the mathematics you already know from A-Level. It is a lot more rigorous which might take some getting used to.
Once you have a feel for what you want to pursue, there are lecture notes aplenty around the internet. From a simple search:
Differential Equations
Probability
Analysis
I purposefully chose three different institutions as there are loads of lecture notes around - e.g. see Cambridge (I'm sure there are better examples of Cambridge notes) and Oxford examples
The quality of notes is not dependent on the reputation of the university. As I was studying I used lectures notes from all over the world, just keep trying until you find something that works.
American universities have a different style of course but again have some excellent notes.
If you're looking for questions, at university they're called problem sheets or example sheets. Most university courses have these public, even if the notes aren't so there will be lots for you to choose from.
Finally, if you're at university doing Medicine I'm sure you can browse their maths books until you find something that gets your interest.
(Specific examples of books/notes will be easier once you've narrowed your interests a bit)
Original post by shamika
Depends on what you're looking for.
General introductory books transitioning to degree level are:
- What is Mathematics?. This covers some basic set and number theory, geometry and analysis. Nothing is covered in a lot of depth but its very well written. The last chapter is now a little out of date (in places) but even so is still a good read
- A Concise Introduction to Pure Mathematics. This covers the foundation part of most first year courses, covering again basic set and number theory but in a very different style to What is Mathematics?
I would spend some time reading through this course guide, which give a brief description of the sort of thing you do at degree level. A lot of it is not about solving complicated equations but about understanding the mathematics you already know from A-Level. It is a lot more rigorous which might take some getting used to.
Once you have a feel for what you want to pursue, there are lecture notes aplenty around the internet. From a simple search:
Differential Equations
Probability
Analysis
I purposefully chose three different institutions as there are loads of lecture notes around - e.g. see Cambridge (I'm sure there are better examples of Cambridge notes) and Oxford examples
The quality of notes is not dependent on the reputation of the university. As I was studying I used lectures notes from all over the world, just keep trying until you find something that works.
American universities have a different style of course but again have some excellent notes.
If you're looking for questions, at university they're called problem sheets or example sheets. Most university courses have these public, even if the notes aren't so there will be lots for you to choose from.
Finally, if you're at university doing Medicine I'm sure you can browse their maths books until you find something that gets your interest.
(Specific examples of books/notes will be easier once you've narrowed your interests a bit)
General introductory books transitioning to degree level are:
- What is Mathematics?. This covers some basic set and number theory, geometry and analysis. Nothing is covered in a lot of depth but its very well written. The last chapter is now a little out of date (in places) but even so is still a good read
- A Concise Introduction to Pure Mathematics. This covers the foundation part of most first year courses, covering again basic set and number theory but in a very different style to What is Mathematics?
I would spend some time reading through this course guide, which give a brief description of the sort of thing you do at degree level. A lot of it is not about solving complicated equations but about understanding the mathematics you already know from A-Level. It is a lot more rigorous which might take some getting used to.
Once you have a feel for what you want to pursue, there are lecture notes aplenty around the internet. From a simple search:
Differential Equations
Probability
Analysis
I purposefully chose three different institutions as there are loads of lecture notes around - e.g. see Cambridge (I'm sure there are better examples of Cambridge notes) and Oxford examples
The quality of notes is not dependent on the reputation of the university. As I was studying I used lectures notes from all over the world, just keep trying until you find something that works.
American universities have a different style of course but again have some excellent notes.
If you're looking for questions, at university they're called problem sheets or example sheets. Most university courses have these public, even if the notes aren't so there will be lots for you to choose from.
Finally, if you're at university doing Medicine I'm sure you can browse their maths books until you find something that gets your interest.
(Specific examples of books/notes will be easier once you've narrowed your interests a bit)
Thank you!
That's all wonderful and has opened my eyes to the fact that what I wanted to do is possible. I've already started turning the Oxford and Cambridge lecture .pdfs into eBooks.
look into the cambridge schedules for maths...first it gives you a better idea of the main theorems for each course and the relevant books. I'll warn you though that enjoying the "problem" solving at A-level might not translate to the rigours required at degree level, though i don't want to dicourage you. most books are of the form of chapters with theory, some problems aimed at exam purposes, no solutions are usually given,
Lecture notes ARE a good idea though courses usually need some bookwork.
Warning; i wouldnt go too far into Number theory, this gets harder and harder requiring other course material, try groups,rings , vector calculus, differential equations,
Lecture notes ARE a good idea though courses usually need some bookwork.
Warning; i wouldnt go too far into Number theory, this gets harder and harder requiring other course material, try groups,rings , vector calculus, differential equations,
Original post by TwilightKnight
Thank you!
That's all wonderful and has opened my eyes to the fact that what I wanted to do is possible. I've already started turning the Oxford and Cambridge lecture .pdfs into eBooks.
That's all wonderful and has opened my eyes to the fact that what I wanted to do is possible. I've already started turning the Oxford and Cambridge lecture .pdfs into eBooks.
No worries
I have other examples of resources if anyone wants them
Original post by shamika
No worries
I have other examples of resources if anyone wants them
I have other examples of resources if anyone wants them
The more the better!
Original post by TwilightKnight
I wasn't sure quite where to put this, since it didn't really belong in the A Level Maths forum.
I love Maths. I can sit around and work my way through problems all day long and just genuinely enjoy it - I've done all the A Level Maths I possibly can (Regular, Further and Additional, although additional isn't aggregated past AS since I missed an exam).
I enjoyed Pure the most, followed closely by Mechanics. Statistics could be interesting at S3/S4 level, but Decision was pretty nonsensical outside of topics like Game Theory.
The problem is, I didn't like the idea of the end result of having done a degree in Maths (usually the result is either go into Banking, Economics etc or Teaching - there are very few actual 'academic' positions AFAIK), so I chose a different field to do at undergraduate (along with other reasons).
I was wondering if there was anything I could do beyond A Level that would keep my interest in Maths alive; any books or websites etc to use and problems to do that sort of build on A Level maths and go from there - i.e, they introduce a topic and explain it and give you some problems to solve (I know the dynamic of Maths changes once you get to university in that it isn't particularly about "Here's X - integrate it") - I've done STEP papers in my spare time, but the reality is that they are firmly grounded in A Level Maths - the questions are just a lot more difficult.
I'm not looking to get any qualification in it, or to not focus on my degree and I'm not trying to pretend I would be doing it at / understanding it at the level of someone actually doing a Maths degree, but I would just like to make sure that I have the same passion for Maths in 10 years time, rather than just slowly forgetting it all because it isn't being used.
Thanks for reading.
I love Maths. I can sit around and work my way through problems all day long and just genuinely enjoy it - I've done all the A Level Maths I possibly can (Regular, Further and Additional, although additional isn't aggregated past AS since I missed an exam).
I enjoyed Pure the most, followed closely by Mechanics. Statistics could be interesting at S3/S4 level, but Decision was pretty nonsensical outside of topics like Game Theory.
The problem is, I didn't like the idea of the end result of having done a degree in Maths (usually the result is either go into Banking, Economics etc or Teaching - there are very few actual 'academic' positions AFAIK), so I chose a different field to do at undergraduate (along with other reasons).
I was wondering if there was anything I could do beyond A Level that would keep my interest in Maths alive; any books or websites etc to use and problems to do that sort of build on A Level maths and go from there - i.e, they introduce a topic and explain it and give you some problems to solve (I know the dynamic of Maths changes once you get to university in that it isn't particularly about "Here's X - integrate it") - I've done STEP papers in my spare time, but the reality is that they are firmly grounded in A Level Maths - the questions are just a lot more difficult.
I'm not looking to get any qualification in it, or to not focus on my degree and I'm not trying to pretend I would be doing it at / understanding it at the level of someone actually doing a Maths degree, but I would just like to make sure that I have the same passion for Maths in 10 years time, rather than just slowly forgetting it all because it isn't being used.
Thanks for reading.
Take a look at Academic Earth and Mathematics lectures from MIT OpenCourseWare. Enjoy!
Shamika's suggestions would seem helpful. In principle there's no reason why you couldn't learn some university maths, in a self-taught way.
On the pure side:
(1) "Transition" from A-level to degree level: The number systems, N, Q, R and C, proof by induction
(2) Sets, functions and relations; the fundamentals of pure mathematics
(3) Sequences and series, followed by an introduction to real analysis
(4) An introduction to abstract algebra; groups, rings and fields (probably slightly focusing on groups), isomorphisms and homomorphisms (of groups rings and fields), isomorphisms and homomorphisms generally
(5) Linear algebra
Those aren't actual names of courses as such, rather a very brief description of courses that might, and do, exist.
But that would be the basics that you would cover in the first two years at a normal uni. Or in Fresher's week at Cambridge.
On the pure side:
(1) "Transition" from A-level to degree level: The number systems, N, Q, R and C, proof by induction
(2) Sets, functions and relations; the fundamentals of pure mathematics
(3) Sequences and series, followed by an introduction to real analysis
(4) An introduction to abstract algebra; groups, rings and fields (probably slightly focusing on groups), isomorphisms and homomorphisms (of groups rings and fields), isomorphisms and homomorphisms generally
(5) Linear algebra
Those aren't actual names of courses as such, rather a very brief description of courses that might, and do, exist.
But that would be the basics that you would cover in the first two years at a normal uni. Or in Fresher's week at Cambridge.
(edited 12 years ago)
Original post by shamika
- What is Mathematics?. This covers some basic set and number theory, geometry and analysis. Nothing is covered in a lot of depth but its very well written. The last chapter is now a little out of date (in places) but even so is still a good read
- A Concise Introduction to Pure Mathematics. This covers the foundation part of most first year courses, covering again basic set and number theory but in a very different style to What is Mathematics?
- What is Mathematics?. This covers some basic set and number theory, geometry and analysis. Nothing is covered in a lot of depth but its very well written. The last chapter is now a little out of date (in places) but even so is still a good read
- A Concise Introduction to Pure Mathematics. This covers the foundation part of most first year courses, covering again basic set and number theory but in a very different style to What is Mathematics?
The Liebeck book is a very good book. It doesn't contain as much maths as the other one. On the other hand, quantity of mathematics is not the only factor. Liebeck is much more "accurate" as a foundation for modern mathematics.
"What is Mathematics?" is very dated (it's endorsed by Einstein which is nice, but it certainly dates the book, must be to about world war two). It doesn't contain the more algebraic, structural bent of modern mathematics. These days it would be better called "Introductory analysis and introductory geometry" or something like that. And in that regard it is a classic.
But it doesn't talk about stuff like injective functions and equivalence relations. Things that are bread-and-butter, ubiquitous, in modern pure mathematics. It's something of an in-between mathematics as Gauss knew it, and mathematics as it is today.
I don't think What is Mathematics? was meant to be used as a proper textbook even when it was written, I think it's merely a taster for more rigorous maths. In that regard it far excels Liebeck's book. Textbooks these days just don't have the same flair. I agree with your list re pure topics to start on, but I would add a little bit of elementary number theory to have lots of examples of proving things and restrict (4) to just groups.
I also second vitamortis' suggestion of the MIT notes, it's a great resource
I also second vitamortis' suggestion of the MIT notes, it's a great resource
Original post by TwilightKnight
I wasn't sure quite where to put this, since it didn't really belong in the A Level Maths forum.
I love Maths. I can sit around and work my way through problems all day long and just genuinely enjoy it - I've done all the A Level Maths I possibly can (Regular, Further and Additional, although additional isn't aggregated past AS since I missed an exam).
I enjoyed Pure the most, followed closely by Mechanics. Statistics could be interesting at S3/S4 level, but Decision was pretty nonsensical outside of topics like Game Theory.
The problem is, I didn't like the idea of the end result of having done a degree in Maths (usually the result is either go into Banking, Economics etc or Teaching - there are very few actual 'academic' positions AFAIK), so I chose a different field to do at undergraduate (along with other reasons).
I was wondering if there was anything I could do beyond A Level that would keep my interest in Maths alive; any books or websites etc to use and problems to do that sort of build on A Level maths and go from there - i.e, they introduce a topic and explain it and give you some problems to solve (I know the dynamic of Maths changes once you get to university in that it isn't particularly about "Here's X - integrate it") - I've done STEP papers in my spare time, but the reality is that they are firmly grounded in A Level Maths - the questions are just a lot more difficult.
I'm not looking to get any qualification in it, or to not focus on my degree and I'm not trying to pretend I would be doing it at / understanding it at the level of someone actually doing a Maths degree, but I would just like to make sure that I have the same passion for Maths in 10 years time, rather than just slowly forgetting it all because it isn't being used.
Thanks for reading.
I love Maths. I can sit around and work my way through problems all day long and just genuinely enjoy it - I've done all the A Level Maths I possibly can (Regular, Further and Additional, although additional isn't aggregated past AS since I missed an exam).
I enjoyed Pure the most, followed closely by Mechanics. Statistics could be interesting at S3/S4 level, but Decision was pretty nonsensical outside of topics like Game Theory.
The problem is, I didn't like the idea of the end result of having done a degree in Maths (usually the result is either go into Banking, Economics etc or Teaching - there are very few actual 'academic' positions AFAIK), so I chose a different field to do at undergraduate (along with other reasons).
I was wondering if there was anything I could do beyond A Level that would keep my interest in Maths alive; any books or websites etc to use and problems to do that sort of build on A Level maths and go from there - i.e, they introduce a topic and explain it and give you some problems to solve (I know the dynamic of Maths changes once you get to university in that it isn't particularly about "Here's X - integrate it") - I've done STEP papers in my spare time, but the reality is that they are firmly grounded in A Level Maths - the questions are just a lot more difficult.
I'm not looking to get any qualification in it, or to not focus on my degree and I'm not trying to pretend I would be doing it at / understanding it at the level of someone actually doing a Maths degree, but I would just like to make sure that I have the same passion for Maths in 10 years time, rather than just slowly forgetting it all because it isn't being used.
Thanks for reading.
Also, send me a PM with your email address and I will send you a PDF version of Stroud's excellent Engineering Mathematics and Advanced Engineering Mathematics textbooks.
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