|
On September 28 2013 21:03 Toadesstern wrote:
oh another thing, though probably not what you're asking for:
Just like with us men, the percentage of chicks looking decent and being normal is a lot better if you're looking at those who're doing maths to become a teacher. At least here in Germany they have to do the same Ana 1-3 and LA 1-3 and not some kind of "math for biologists" (or whatever it'd be for teachers) so you'll be sitting in the same lectures.
The ones doing math to get a bachelor and plan on making a masters afterwards are a lot more nerdy, which I probably don't have to explain to anyone here. So if you're looking for people who are a bit more down to earth you should be looking over there. People starting out in maths, when comming from highschool tend to be huge dicks the first 1 or 2 semesters until they realize they aren't geniuses.
This post makes me sad.
Superprotip: Even if you study maths, you are allowed to socialize with everyone else. In fact, it may be refreshing to not limit your social circle to other maths students, whether they intend to become teachers or not.
|
I wish you a lot of luck in math. I already did my first year in linear algebra and analysis 1, and it is hard at times (to me analysis was really challenging). But its also the most exciting and rewarding thing you can learn. I still fondly remember writing a paper on analytic functions and learning about taylor approximations for the first time, and now recognizing how much power they have and how commonly they are used in physics.
I think the only thing you need to do is just stick through it, talk to your TA's, maybe find recommended books on linear algebra (if you find yours hard to understand), and go through those. Eventually I think you'll find it becomes easier and easier.
I would be a bit frightened if you are already getting confused by vector spaces, linear independence/dependence, and the basis for a set. Maybe you are being taught very poorly, but at least in my course I remember that things tend to get complicated fairly quickly after you cover those basics. I mean the replacement theorem can be a bit confusing, but that's okay because its a big theorem  .
I'm guessing you will probably learn the replacement theorem, learn all of the corollaries, then start using a basis to compose a matrix with which you can describe a linear transformation (a type of function) in terms of the basis vectors of the domain and codomain...then you'll learn about the range space and null space of this transformation, after which all of these concepts blend together in very intricate ways, leading to theorems that help solve systems of linear equations.
Although I don't think you really need to understand it at a deep level to actually do it; reduced row echelon form is really basic stuff to do for solving a system of linear equations; so is finding an inverse of a matrix. So is finding the eigenvalues of a matrix. So you can still probably do well in the course if you just do it mechanically, but of course if it comes to proving a theorem then you really need to understand. I wish I could just send you a knowledge bomb of everything I remember or mail you my algebra textbook, but I love my algebra textbook and have to work on physics problems myself!
|
learn the theorems and know everything about them.
learn the proofs. Except maybe if it is a really annoying one that does a hundred uninteresting things. But probably learn that too because you will never come up with it yourselv when a test asks you about it.
DO ALL THE ASSIGNMENTS WITHOUT LOOKING AT ANYTHING OTHER THAN THE ASSIGNMENT.
You HAVE to do this. If you look stuff up, you cheat yourself. watching something and understanding it is different from doing it yourself without help. This will take a lot of your time, but there is a reason math students dont have many courses.
If you dont do this, you WILL fail your test and you WILL be one of those idiots complaining "waaah but we never did a task like this before this test".
Having said that, you ALSO will need to learn how to solve all the tasks you did in your assignments instantly without thinking about ot for more than a minute. That is purely for the final course test. The test will assume you know how to do that stuff and if you dont, you will run out of time really fast. I had this happen to me before on my first math test since i assumed i would be asked about proofs and shit, but they asked for a lot of actual calculating stuff that we did in assignments. I barely passed despite working hard all semester.
Learning groups are good, but dont waste time with inefficient ones that arent getting done anything. Also you have to solve almost all the assignments yourself or you are cheating yourself.
Its gonna be a lot of hard work, especially in the first 2 years. YOu will have to sacrifice a lot.
My courses had a failure rate of 80%.
Also, dont be the guy that sticks around for years even though it did not work out. If you feel like pure math is not for you, you should switch.
|
This confuses me. Our math assignments were really hard and it took regularly multiple days to solve them. The exams were "rather" easy in comparison, mostly due to time limits. so you should take that with a grain of salt.
also: why hasn't nobody pointed out that plexa's example is wrong? the union of subspaces is, in general, not closed under vector addition.
|
On September 29 2013 04:59 Hryul wrote:
This confuses me. Our math assignments were really hard and it took regularly multiple days to solve them. The exams were "rather" easy in comparison, mostly due to time limits. so you should take that with a grain of salt.
also: why hasn't nobody pointed out that plexa's example is wrong? the union of subspaces is, in general, not closed under vector addition.
I think he meant the sum instead of the union 
i also think he is wrong in how you learn math. If you try to learn all your definitions by heart before actually knowing what you're talking about, you're just setting yourself up to be a monkey savant, and you'll never be any good.
Proper math should be learned through trial and error.
1)Read your lessons quickly
2)Try to do some exercises.
3)Go read specific chapters in the book to help you when you are stuck.
4)Do more exercices.
5)Repeat steps 3) and 4)
5)Once you have a good feeling (intuition) for what the objects means and how they work, go back and learn your lessons by heart.
If you try to learn your lesson by heart without first knowing what it means, you'll spend hours not having fun, and chances are, you'll get decent grades when it comes to solving direct applications of theroems but still have no clue what you are doing.
|
On September 29 2013 05:23 Geiko wrote:Show nested quote +On September 29 2013 04:59 Hryul wrote:
This confuses me. Our math assignments were really hard and it took regularly multiple days to solve them. The exams were "rather" easy in comparison, mostly due to time limits. so you should take that with a grain of salt.
also: why hasn't nobody pointed out that plexa's example is wrong? the union of subspaces is, in general, not closed under vector addition. I think he meant the sum instead of the union i also think he is wrong in how you learn math. If you try to learn all your definitions by heart before actually knowing what you're talking about, you're just setting yourself up to be a monkey savant, and you'll never be any good. Proper math should be learned through trial and error. 1)Read your lessons quickly 2)Try to do some exercises. 3)Go read specific chapters in the book to help you when you are stuck. 4)Do more exercices. 5)Repeat steps 3) and 4) 5)Once you have a good feeling (intuition) for what the objects means and how they work, go back and learn your lessons by heart. If you try to learn your lesson by heart without first knowing what it means, you'll spend hours not having fun, and chances are, you'll get decent grades when it comes to solving direct applications of theroems but still have no clue what you are doing.
You learn definitions, you don't learn them by heart, this is very very different.
Your system seems extremely unefficient to me.
|
Definitions are made to be as concise and precise as possible. Most of todays axioms, definitions and theorems were written way after they were actually invented and used.
For instance the definition of a line (AB) in a metric space (E,d) is
{P€E / d(A,B)=|d(A,P)±d(P,B)|}.
If you don't have a intuitive knowledge of what a line truly is, you're never going to able to use this definition other than in simple problems.
You have to have a good feeling of what mathematical objects are before learning your definitions.
|
My thing wasnt a priority guide.
Of course you first look at things and figure them out. But after that my suggestion is to learn it by heart first and then do the assignments with it, in order to control yourself if you are capable of doing what is asked of you. With the internet, it is extremely easy to cheat yourself out of exercise and self-controlling the weekly assignments provide.
I dont know how it is in the netherlands, but in germany you are required to get around an average of like 75% in the tests in your bachelor to let the university start you in the master. So considering a lot of early test have a failing quota of 70% or more, i always made sure i did the things i listed here.
How the exams look...if they are easy deductions of learned definitions or hardcore time constrained stress tests of previous assignments or focused on construction new proofs with a lot of spare time, but a lot of needed exercise in doing this....you cant know unless you get some hints from somewhere.
As i said, it happened to me that i focused on the wrong thing once, so i started to learn specifically for the time stresstests. Maybe your university has better professors and ou dont need to do it, i just gave advice on how i handled it.
And yes, my weekly assignments took multiple days to finish, but i only had 2 real courses per semester.
Other prople that study economics or something have their whole week filled with courses and shit.
|
On September 29 2013 06:05 Geiko wrote:
Definitions are made to be as concise and precise as possible. Most of todays axioms, definitions and theorems were written way after they were actually invented and used.
For instance the definition of a line (AB) in a metric space (E,d) is
{P€E / d(A,B)=|d(A,P)±d(P,B)|}.
If you don't have a intuitive knowledge of what a line truly is, you're never going to able to use this definition other than in simple problems.
You have to have a good feeling of what mathematical objects are before learning your definitions.
You get intuitive knowledge of most mathematical objects by doing those simple problems, which you solve by applying those definitions, hence you need to know them before doing any exercise. Learning you lessons also makes you manipulate those object in a very interesting way, if the lesson is well-made, because there will be examples and basic properties just after those definitions.
For me people who focus on exercises are the monkey savant, because they tend to know how to solve many problems because they have already seen them before, but they have simply no understanding of what's really going on.
Edit : typically, people who only do exercises end up doing stupid stuff like applying d'Alembert criterium on a geometric serie, and that's the best case...
|
On September 29 2013 06:19 corumjhaelen wrote:
You get intuitive knowledge of most mathematical objects by doing those simple problems, which you solve by applying those definitions, hence you need to know them before doing any exercise.
That's what I'm saying You get intuitive knowledge of mathematical objects by applying definitions to simple problems. But you never truly understand their purpose until you've figured out by yourself how they work and why they are useful.
Telling people to work on their lessons very hard before doing any problems is like giving them a book full of problems with the solutions written right next to the problems. Most people will understand the solution, and those same people will immediately forget how to do it next time they see it. If you spend time on problems you can't solve with your current knowledge, you understand why you need the elements in your lessons.
In hindsight it's easy to think that the best way to learn math is to learn definitions, then apply them intelligently, because that's what you do in math once you've understood. But the people who advocate such a way of teaching forget that this is not at all how they themselves learned math. They forget that they were deriving and integrating before knowing what it meant, that they were working in finite commutative rings before knowing what those were etc...
|
On September 29 2013 07:44 Geiko wrote:Show nested quote +On September 29 2013 06:19 corumjhaelen wrote:
You get intuitive knowledge of most mathematical objects by doing those simple problems, which you solve by applying those definitions, hence you need to know them before doing any exercise. That's what I'm saying You get intuitive knowledge of mathematical objects by applying definitions to simple problems. But you never truly understand their purpose until you've figured out by yourself how they work and why they are useful. Telling people to work on their lessons very hard before doing any problems is like giving them a book full of problems with the solutions written right next to the problems. Most people will understand the solution, and those same people will immediately forget how to do it next time they see it. If you spend time on problems you can't solve with your current knowledge, you understand why you need the elements in your lessons. In hindsight it's easy to think that the best way to learn math is to learn definitions, then apply them intelligently, because that's what you do in math once you've understood. But the people who advocate such a way of teaching forget that this is not at all how they themselves learned math. They forget that they were deriving and integrating before knowing what it meant, that they were working in finite commutative rings before knowing what those were etc...
What I mean by learning a lesson seems to be different than what you mean. The idea is not to read the course over and over, the idea is to take a blank paper, copy a few headlines, and being able to rewrite your course from A to Z, in your own way, not by heart...
Of course that's impractical in a master's degree, but I think there's nothing more important to do in undergraduate classes.
Edit : and that's how I worked in practice you know...
Edit 2 : what is a finite commutative ring I worked in ?
Every maths concept is not as intuitive as you make it sound, plus intuition is such a good recipe for disaster in so many cases, I'm a bit dubious at what you say.
|
On September 29 2013 07:51 corumjhaelen wrote:Show nested quote +On September 29 2013 07:44 Geiko wrote:On September 29 2013 06:19 corumjhaelen wrote:
You get intuitive knowledge of most mathematical objects by doing those simple problems, which you solve by applying those definitions, hence you need to know them before doing any exercise. That's what I'm saying You get intuitive knowledge of mathematical objects by applying definitions to simple problems. But you never truly understand their purpose until you've figured out by yourself how they work and why they are useful. Telling people to work on their lessons very hard before doing any problems is like giving them a book full of problems with the solutions written right next to the problems. Most people will understand the solution, and those same people will immediately forget how to do it next time they see it. If you spend time on problems you can't solve with your current knowledge, you understand why you need the elements in your lessons. In hindsight it's easy to think that the best way to learn math is to learn definitions, then apply them intelligently, because that's what you do in math once you've understood. But the people who advocate such a way of teaching forget that this is not at all how they themselves learned math. They forget that they were deriving and integrating before knowing what it meant, that they were working in finite commutative rings before knowing what those were etc... What I mean by learning a lesson seems to be different than what you mean. The idea is not to read the course over and over, the idea is to take a blank paper, copy a few headlines, and being able to rewrite your course from A to Z, in your own way, not by heart... Of course that's impractical in a master's degree, but I think there's nothing more important to do in undergraduate classes. Edit : and that's how I worked in practice you know... Edit 2 : what is a finite commutative ring I worked in ? Every maths concept is not as intuitive as you make it sound, plus intuition is such a good recipe for disaster in so many cases, I'm a bit dubious at what you say.
I'm sure you worked with modulus (11≡3[8]) before knowing what Z/8Z (finite commutative ring) was.
Also if you have no intuition of a math concept, then you haven't understood it. It's that simple. You might know how to solve a couple of problems with it. You might know how to use it in some circumstances, but you don't know what it is. Intuitionis what you need to be good in math and by that I don't mean get Master degrees or PHDs or even good grades in high school. Then once you have intuition, to be truly brilliant you need to be able to write flawless proofs. Not the other way around.
|
On September 29 2013 08:06 Geiko wrote:Show nested quote +On September 29 2013 07:51 corumjhaelen wrote:On September 29 2013 07:44 Geiko wrote:On September 29 2013 06:19 corumjhaelen wrote:
You get intuitive knowledge of most mathematical objects by doing those simple problems, which you solve by applying those definitions, hence you need to know them before doing any exercise. That's what I'm saying You get intuitive knowledge of mathematical objects by applying definitions to simple problems. But you never truly understand their purpose until you've figured out by yourself how they work and why they are useful. Telling people to work on their lessons very hard before doing any problems is like giving them a book full of problems with the solutions written right next to the problems. Most people will understand the solution, and those same people will immediately forget how to do it next time they see it. If you spend time on problems you can't solve with your current knowledge, you understand why you need the elements in your lessons. In hindsight it's easy to think that the best way to learn math is to learn definitions, then apply them intelligently, because that's what you do in math once you've understood. But the people who advocate such a way of teaching forget that this is not at all how they themselves learned math. They forget that they were deriving and integrating before knowing what it meant, that they were working in finite commutative rings before knowing what those were etc... What I mean by learning a lesson seems to be different than what you mean. The idea is not to read the course over and over, the idea is to take a blank paper, copy a few headlines, and being able to rewrite your course from A to Z, in your own way, not by heart... Of course that's impractical in a master's degree, but I think there's nothing more important to do in undergraduate classes. Edit : and that's how I worked in practice you know... Edit 2 : what is a finite commutative ring I worked in ? Every maths concept is not as intuitive as you make it sound, plus intuition is such a good recipe for disaster in so many cases, I'm a bit dubious at what you say. I'm sure you worked with modulus (11≡3[8]) before knowing what Z/8Z (finite commutative ring) was. Also if you have no intuition of a math concept, then you haven't understood it. It's that simple. You might know how to solve a couple of problems with it. You might know how to use it in some circumstances, but you don't know what it is. Intuitionis what you need to be good in math and by that I don't mean get Master degrees or PHDs or even good grades in high school. Then once you have intuition, to be truly brilliant you need to be able to write flawless proofs. Not the other way around.
Yes but it wasn't an object to me, which is very different, I had no intuition of Z/nZ, I was reasonning in Z. And for instance, thinking that every commutative ring is a bit like Z won't help you much doing anything on commutative ring.
I might have expressed myself wrongly about intuition, but what I don't like with your approach is the idea that intuition must come first. I encountered differentiable function way before I knew what it was. Now if I still had the same intuitive idea of what differentiable functions are as I had in high school, I'd still be like Cauchy, persuaded that non derivable functions are a rare exception, and more or less acting as if every function I encounter were differentiable.
You get good intuition of an object by looking at and manipulating definitions, not the other way around. Unless you want to act as if you were rediscovering every mathematic concept ever invented, but in that case I fear you'll get stuck in 1820 at best.
|
|
|
Very nice article, describes my line of thought exactly, especially
The “post-rigorous” stage, in which one has grown comfortable with all the rigorous foundations of one’s chosen field, and is now ready to revisit and refine one’s pre-rigorous intuition on the subject, but this time with the intuition solidly buttressed by rigorous theory. (For instance, in this stage one would be able to quickly and accurately perform computations in vector calculus by using analogies with scalar calculus, or informal and semi-rigorous use of infinitesimals, big-O notation, and so forth, and be able to convert all such calculations into a rigorous argument whenever required.) The emphasis is now on applications, intuition, and the “big picture”. This stage usually occupies the late graduate years and beyond.
I guess you're also right about the fact that there are many ways to learn
|
Switch to CS. You know you want to.
|
The only general advice I could extract from this thread is: Work really hard. Everyone seems to have different ideas about how to learn Math. I will however make a list of all the definitions I had so far for every class today and see if I understand them. I haven´t been doing much at all in the weekend because I want to see my girlfriend and friends who are 2 hours away. Ahhh, reading this was a little bit discouraging. But I find it fun, I'll make it work somehow.
|
Aotearoa39261 Posts
On September 29 2013 05:23 Geiko wrote:Show nested quote +On September 29 2013 04:59 Hryul wrote:
This confuses me. Our math assignments were really hard and it took regularly multiple days to solve them. The exams were "rather" easy in comparison, mostly due to time limits. so you should take that with a grain of salt.
also: why hasn't nobody pointed out that plexa's example is wrong? the union of subspaces is, in general, not closed under vector addition. I think he meant the sum instead of the union Haha yes  fixed.
|
Aotearoa39261 Posts
On September 29 2013 08:06 Geiko wrote:Show nested quote +On September 29 2013 07:51 corumjhaelen wrote:On September 29 2013 07:44 Geiko wrote:On September 29 2013 06:19 corumjhaelen wrote:
You get intuitive knowledge of most mathematical objects by doing those simple problems, which you solve by applying those definitions, hence you need to know them before doing any exercise. That's what I'm saying You get intuitive knowledge of mathematical objects by applying definitions to simple problems. But you never truly understand their purpose until you've figured out by yourself how they work and why they are useful. Telling people to work on their lessons very hard before doing any problems is like giving them a book full of problems with the solutions written right next to the problems. Most people will understand the solution, and those same people will immediately forget how to do it next time they see it. If you spend time on problems you can't solve with your current knowledge, you understand why you need the elements in your lessons. In hindsight it's easy to think that the best way to learn math is to learn definitions, then apply them intelligently, because that's what you do in math once you've understood. But the people who advocate such a way of teaching forget that this is not at all how they themselves learned math. They forget that they were deriving and integrating before knowing what it meant, that they were working in finite commutative rings before knowing what those were etc... What I mean by learning a lesson seems to be different than what you mean. The idea is not to read the course over and over, the idea is to take a blank paper, copy a few headlines, and being able to rewrite your course from A to Z, in your own way, not by heart... Of course that's impractical in a master's degree, but I think there's nothing more important to do in undergraduate classes. Edit : and that's how I worked in practice you know... Edit 2 : what is a finite commutative ring I worked in ? Every maths concept is not as intuitive as you make it sound, plus intuition is such a good recipe for disaster in so many cases, I'm a bit dubious at what you say. I'm sure you worked with modulus (11≡3[8]) before knowing what Z/8Z (finite commutative ring) was. Also if you have no intuition of a math concept, then you haven't understood it. It's that simple. You might know how to solve a couple of problems with it. You might know how to use it in some circumstances, but you don't know what it is. Intuitionis what you need to be good in math and by that I don't mean get Master degrees or PHDs or even good grades in high school. Then once you have intuition, to be truly brilliant you need to be able to write flawless proofs. Not the other way around.
Intuition is what separates B students from A students, in my experience anyway. That intuition with definitions means they can construct novel approaches to problems because they can "see" what is going on. Even with amazing intuition, you're pretty stuck if you don't know what your working with. For most papers that I've taught being able to recite a definition accounts for at least 50% of the paper (higher in some of the lower level ones). After learning definitions building up some level of flexibility with them (via examples -- most definitions are motivated by one particular class of examples fortunately!) then you can start throwing in the intuition.
tldr; I don't see a way for a math student to excel in a paper without first being able to understand the formal definitions of what they are working with before moving to solve problems via intuition.
|
Guys, if you had read the op before the edit, maybe you'd understand that he's not Ramanujan. He admitted doing lots exercises, and getting a linear algebra question wrong at an exam because he didn't know what linear independance meant.
You're not helping him here.
|
|
|
|
|
|
EPT
