I feel like I dont know you anymore, math.
October 22, 2007 2:52 PM Subscribe
How do I make the transition, in my brain, to studying graduate/advanced mathematics?
Throughout High School and the first two years of college, I have excelled at pretty much your standard math courses: Calc I, II & III, Differential, Linear Algebra, etc, even (thus far into the semester) in Advanced Calc.
But I am in a class called Random Walks, and I just cannot function (heh) on the level that they are manipulating equations. They are pulling out summations signs from everywhere, doing god-knows-what with them, etc. I really just don’t know where to begin. This class leaves your typical math classes in the dust. So I'm wondering: how does one make the transition from friendly "This is how we solve things, now go solve these things" classes to the kind of classes with textbooks that pretty much walk you through an incredibly complex derivation with wild symbols you've never seen before, throwing in some problems along the way for good measure. Note: I am confident that if the writer of the book were next to me, and i could ask him all the questions i wanted, I could understand the material. In that respect, it isn't above my head.
I want to learn how to manage successfully in this new type of math class world, because right now, I just give up every time I try, unless I have someone explaining it to me. I get the feeling math grad students have tackled this transition. Any tips on how i get the right mindset to compete in this class would be appreciated. Thanks!
Throughout High School and the first two years of college, I have excelled at pretty much your standard math courses: Calc I, II & III, Differential, Linear Algebra, etc, even (thus far into the semester) in Advanced Calc.
But I am in a class called Random Walks, and I just cannot function (heh) on the level that they are manipulating equations. They are pulling out summations signs from everywhere, doing god-knows-what with them, etc. I really just don’t know where to begin. This class leaves your typical math classes in the dust. So I'm wondering: how does one make the transition from friendly "This is how we solve things, now go solve these things" classes to the kind of classes with textbooks that pretty much walk you through an incredibly complex derivation with wild symbols you've never seen before, throwing in some problems along the way for good measure. Note: I am confident that if the writer of the book were next to me, and i could ask him all the questions i wanted, I could understand the material. In that respect, it isn't above my head.
I want to learn how to manage successfully in this new type of math class world, because right now, I just give up every time I try, unless I have someone explaining it to me. I get the feeling math grad students have tackled this transition. Any tips on how i get the right mindset to compete in this class would be appreciated. Thanks!
You need a study group. I could not have completed my third and fourth year mathematics assignments without a group. This is not a "copy your homework" group. Instead, you can brainstorm and talk each other through a solution that works, which you then go home and write up independently. Your brainstorming session should take place *before* office hours, so you can go talk to the prof if you are still lost.
posted by crazycanuck at 3:08 PM on October 22, 2007 [1 favorite]
posted by crazycanuck at 3:08 PM on October 22, 2007 [1 favorite]
Seconding talking to your professors. I'm having a hard time finding profs that know me well for recommendations because I never asked anyone for help.
posted by pravit at 3:10 PM on October 22, 2007 [2 favorites]
posted by pravit at 3:10 PM on October 22, 2007 [2 favorites]
I'm going to repeat this, because, really, it bears repeating.
Go talk to your professor.
My father was a professor. He loved it when students showed up to his office hours, and they usually kept coming back.
I'm a sophomore in college and finally got around to trying it (I kept talking myself out of it before). It's not nearly as scary as you'd think, and any half-decent professor will enjoy helping you figure things out.
This is they are paid for.
posted by phrontist at 3:14 PM on October 22, 2007
Go talk to your professor.
My father was a professor. He loved it when students showed up to his office hours, and they usually kept coming back.
I'm a sophomore in college and finally got around to trying it (I kept talking myself out of it before). It's not nearly as scary as you'd think, and any half-decent professor will enjoy helping you figure things out.
This is they are paid for.
posted by phrontist at 3:14 PM on October 22, 2007
Learning to work with summations can be pretty tough. Is it possible that you just don't have the background that others do? Maybe you need to find a text that will help you understand these type of manipulations. I know that when I first encountered summations (and double and triple summations) in equations, I got pretty confused. Things became clearer over time, but, in retrospect, I should have spent a few weeks practicing to wrap my head around them when I first started to get into difficulty.
posted by ssg at 3:27 PM on October 22, 2007
posted by ssg at 3:27 PM on October 22, 2007
Given your previous classes it doesn't really sound like you have the prerequisites for what I imagine a class in Stochastic Processes i.e. Random Walks would require (i.e. real analysis/statistics/probability)... which is to say I'm not sure exactly how you got where you are but I don't see any reason why you shouldn't be struggling with this material.
Random Walks is really a special topic class in mathematics, it will use special methods and ideas that may not come up in your standard courses. Of course what happens in the courses you mentioned is different from university to university but it sounds like you are not so happy with series which you might get more comfortable with in either a real or complex analysis class. They are touched on only briefly in perhaps Calc III and DiffEq. Reading between the lines, there are alot of little tricks with series that you might not see in *any* math class (though you might in physics classes!)
My suggestion would be to go to the professor with the specific things you don't understand. DO NOT go to him or her and say "I'm really not getting the material." It sort of sounds like you might be getting hung up on special tricks which you could get past possibly with some further demonstration.
By the way, unless you are at a 4 year college the professors are most certainly not paid to do anything other than perfunctory service for undergraduates. That's how you get to breathe the rarified air with Prof. Hot Shot. Repeat, don't go with the assumption that the Prof. cares about how you a feeling/doing. However most mathematicians can't resist explaining something they understand to an appreciative audience so definitely give it a try.
posted by geos at 3:48 PM on October 22, 2007
Random Walks is really a special topic class in mathematics, it will use special methods and ideas that may not come up in your standard courses. Of course what happens in the courses you mentioned is different from university to university but it sounds like you are not so happy with series which you might get more comfortable with in either a real or complex analysis class. They are touched on only briefly in perhaps Calc III and DiffEq. Reading between the lines, there are alot of little tricks with series that you might not see in *any* math class (though you might in physics classes!)
My suggestion would be to go to the professor with the specific things you don't understand. DO NOT go to him or her and say "I'm really not getting the material." It sort of sounds like you might be getting hung up on special tricks which you could get past possibly with some further demonstration.
By the way, unless you are at a 4 year college the professors are most certainly not paid to do anything other than perfunctory service for undergraduates. That's how you get to breathe the rarified air with Prof. Hot Shot. Repeat, don't go with the assumption that the Prof. cares about how you a feeling/doing. However most mathematicians can't resist explaining something they understand to an appreciative audience so definitely give it a try.
posted by geos at 3:48 PM on October 22, 2007
First off, don't feel bad. A class in random walks is pretty sophisticated compared to the more mechanistic classes that you've taken. Probability concepts are often not advanced in a technical sense, but draw from a wide background of techniques and use a lot of slick manipulation. Most people start getting into upper level math via subjects like analysis, geometry, and algebra which work the advanced concepts without requiring quite as much breadth and practice with manipulation. That said, I'm sure you have the background to understand each individual step (one of my favorite things about random walks), from the courses you've mentioned.
Study groups and office hours, in that order, are the two best things to do as a math student. Just having someone to bounce ideas off of and mutually catch dumb errors is great, but the biggest advantage to a group is that you wind up explaining what you are doing to other people. All of upper level math is explanatory writing in some sense, so getting practice at this sort of thought is very useful. It takes a little while to get up to speed, and the best way to figure it out is just to keep going and then look back and realize how far you've come.
Another helpful option in this specific case is to get online and look for courses in random walks (there is definitely one on MIT opencourseware ) or stochastic processes ( a lot of overlap, but often a different emphasis ). Find class web sites that have posted homeworks and solutions and use them to test yourself and learn from.
posted by Schismatic at 3:57 PM on October 22, 2007
Study groups and office hours, in that order, are the two best things to do as a math student. Just having someone to bounce ideas off of and mutually catch dumb errors is great, but the biggest advantage to a group is that you wind up explaining what you are doing to other people. All of upper level math is explanatory writing in some sense, so getting practice at this sort of thought is very useful. It takes a little while to get up to speed, and the best way to figure it out is just to keep going and then look back and realize how far you've come.
Another helpful option in this specific case is to get online and look for courses in random walks (there is definitely one on MIT opencourseware ) or stochastic processes ( a lot of overlap, but often a different emphasis ). Find class web sites that have posted homeworks and solutions and use them to test yourself and learn from.
posted by Schismatic at 3:57 PM on October 22, 2007
Also, go back over your notes. Rewrite them, trying to fill in and understand the details along the way. Sit down with the material you've seen in class and go over it well enough that you can do every step. If there's a manipulation you don't understand, play with it for a while and then talk to the prof if you still can't get it. In fact, this is probably the first thing to do before everything else.
posted by Schismatic at 4:08 PM on October 22, 2007
posted by Schismatic at 4:08 PM on October 22, 2007
What geos says (profs don't care) may or may not be true. My experience has been that most profs certainly do care about their students, but have lots of other commitments (research, grad students, other prof stuff) which limits their available time. The 'office hour' is meant to formalize and bound the amount of time the prof gives to students outside of lecture. As long as you go in with the attitude that the professor's time is valuable, i.e. demand outstrips supply, the prof will probably be friendly. This simply means preparing and going as far as you can before getting stuck, rather than simply walking in and saying "I don't get it".
About your question in particular. I have faced some impenetrable courses in the past and I know how intimidating it can be to stare at pages full of equations that just look like little black squiggles on the page. It's like a foreign language. However, and you know this already of course, the book is attempting to communicate with you so it should be possible to figure out what it is saying. Equations are just shorthand for plain english, after all.
I would recommend picking something that stumps you and working through it methodically. Write out every step and figure out the derivation to the next step. Do this all the way through until you get to the end or get stuck. Then you can tell the prof, "I don't see how they went from here to there". But you will find that working your way through will internalize the concept and there is really no other way to do it.
Personally when I am in lectures for courses like this, I am tempted to turn off my brain, let the equations whiz over my head, and work it all out later. I learned towards the end of my undergrad that by writing out everything that happened, even the stuff that was pre-printed on the notes, I was much more able to figure out what it meant and thus actually get something out of the lecture. (It also meant I had to learn to write really fast.)
posted by PercussivePaul at 4:30 PM on October 22, 2007
About your question in particular. I have faced some impenetrable courses in the past and I know how intimidating it can be to stare at pages full of equations that just look like little black squiggles on the page. It's like a foreign language. However, and you know this already of course, the book is attempting to communicate with you so it should be possible to figure out what it is saying. Equations are just shorthand for plain english, after all.
I would recommend picking something that stumps you and working through it methodically. Write out every step and figure out the derivation to the next step. Do this all the way through until you get to the end or get stuck. Then you can tell the prof, "I don't see how they went from here to there". But you will find that working your way through will internalize the concept and there is really no other way to do it.
Personally when I am in lectures for courses like this, I am tempted to turn off my brain, let the equations whiz over my head, and work it all out later. I learned towards the end of my undergrad that by writing out everything that happened, even the stuff that was pre-printed on the notes, I was much more able to figure out what it meant and thus actually get something out of the lecture. (It also meant I had to learn to write really fast.)
posted by PercussivePaul at 4:30 PM on October 22, 2007
Well, the methods that you need are almost certainly coming from two graduate/advanced undergrad sets of classes.
First is real analysis. This is where I learned topics like when/how you can rearrange summations/integrals, put limits through summations/integrals, etc.
Second is math stat. Here you learn the theory behind most of the things that you want to do with probability objects. You might also want a class in measure-theory based probability.
You get better by doing these, and doing them right. The math grad students have tackled many of these, and you'd be surprised how well techniques synergize between different math classes. Study groups will definitely help you get better grades, and see how people approach these problems "live" which can be very different from the nice clean worked up version that they show you later (aside: this is why you should never integrate in public). In your average mathematician's toolbox of random-ass methods/techniques I expect there to be about 10^3 elements.
Hehe, move the hyphen
posted by a robot made out of meat at 5:15 PM on October 22, 2007
First is real analysis. This is where I learned topics like when/how you can rearrange summations/integrals, put limits through summations/integrals, etc.
Second is math stat. Here you learn the theory behind most of the things that you want to do with probability objects. You might also want a class in measure-theory based probability.
You get better by doing these, and doing them right. The math grad students have tackled many of these, and you'd be surprised how well techniques synergize between different math classes. Study groups will definitely help you get better grades, and see how people approach these problems "live" which can be very different from the nice clean worked up version that they show you later (aside: this is why you should never integrate in public). In your average mathematician's toolbox of random-ass methods/techniques I expect there to be about 10^3 elements.
Hehe, move the hyphen
posted by a robot made out of meat at 5:15 PM on October 22, 2007
Nthing that the classes you've taken thus far are mechanics classes rather than theory classes. If you have time for the extra credits, try an intro theory class like Discrete Math. It won't help you with your specific problem (summations) but it definitely gets you into a problem solving mode and gives you a good framework for discussing problem solving, attacking proofs, and so on. (As a bonus, Discrete Math is probably the most fun that you can have in a math class, ever.)
posted by anaelith at 5:16 PM on October 22, 2007
posted by anaelith at 5:16 PM on October 22, 2007
What geos says (profs don't care) may or may not be true. My experience has been that most profs certainly do care about their students, but have lots of other commitments (research, grad students, other prof stuff) which limits their available time.
I wasn't saying that prof's don't care, some do and some honestly don't; but that you shouldn't be working on the assumption that they do.
As an grad student and instructor, when a student comes into my office I would really love if they had figured out what they didn't get, had tried to get it, and then gotten stuck. Then, hopefully I can figure out where they went wrong and show them how to do it.
basically it's the details, figuring out the piddling steps in the calculation even if you don't really understand where it is going. milestogo is definitely above his weight class and is going to have to deal with feeling sort of lost (and this feels really bad and no one is going to make it feel any better). but if he can grab onto to the details and pull himself along it won't be a class wasted.
posted by geos at 6:56 PM on October 22, 2007
I wasn't saying that prof's don't care, some do and some honestly don't; but that you shouldn't be working on the assumption that they do.
As an grad student and instructor, when a student comes into my office I would really love if they had figured out what they didn't get, had tried to get it, and then gotten stuck. Then, hopefully I can figure out where they went wrong and show them how to do it.
basically it's the details, figuring out the piddling steps in the calculation even if you don't really understand where it is going. milestogo is definitely above his weight class and is going to have to deal with feeling sort of lost (and this feels really bad and no one is going to make it feel any better). but if he can grab onto to the details and pull himself along it won't be a class wasted.
posted by geos at 6:56 PM on October 22, 2007
I am a math professor. I have office hours every week. I have them because I want students to come and talk to me. Math classes are supposed to be hard, and transitioning to higher level math classes is the hardest time of all. We're here to help out with that.
The advice in the thread is very good -- talk to other students, work on problems until you get stuck and then come to office hours to explain the sticking point, go over the notes. I don't have anything practical to add -- I just want to say that you're going through what every single math student goes through. If you go to graduate school, you'll go through it again at the beginning of grad courses, and then again when you start writing your thesis. It's hard, but the rewards are great, and the fact that it hurts doesn't mean you're doing it wrong!
posted by escabeche at 7:33 PM on October 22, 2007 [1 favorite]
The advice in the thread is very good -- talk to other students, work on problems until you get stuck and then come to office hours to explain the sticking point, go over the notes. I don't have anything practical to add -- I just want to say that you're going through what every single math student goes through. If you go to graduate school, you'll go through it again at the beginning of grad courses, and then again when you start writing your thesis. It's hard, but the rewards are great, and the fact that it hurts doesn't mean you're doing it wrong!
posted by escabeche at 7:33 PM on October 22, 2007 [1 favorite]
Math grad student here.
One thing I've learned in the last couple of years is that the best way for me to learn something is to write it out myself, step by step. Even if that just means copying it word for word, somehow that just makes it all stick in my head a lot better.
It sounds like the formalism and technical details are getting the best of you. This may be different for some people, but for me, I can't really understand a theorem or its proof until I've gotten to the point where I'm familiar with the notation and definitions that I'm working with.
More succinctly, I can't understand what a theorem is saying if I have to flip back a few pages to check the definitions while I'm reading through the statement of it. My advice is first to make sure that you have committed all the definitions to memory (by writing them down as many times as it takes) while you're working through the proofs.
There is no shortcut to learning difficult mathematics. You have to just do it over and over again until you get it.
posted by number9dream at 9:52 PM on October 22, 2007 [1 favorite]
One thing I've learned in the last couple of years is that the best way for me to learn something is to write it out myself, step by step. Even if that just means copying it word for word, somehow that just makes it all stick in my head a lot better.
It sounds like the formalism and technical details are getting the best of you. This may be different for some people, but for me, I can't really understand a theorem or its proof until I've gotten to the point where I'm familiar with the notation and definitions that I'm working with.
More succinctly, I can't understand what a theorem is saying if I have to flip back a few pages to check the definitions while I'm reading through the statement of it. My advice is first to make sure that you have committed all the definitions to memory (by writing them down as many times as it takes) while you're working through the proofs.
There is no shortcut to learning difficult mathematics. You have to just do it over and over again until you get it.
posted by number9dream at 9:52 PM on October 22, 2007 [1 favorite]
Oh, you can also take in OLD problem sets/examples as questions. Since here you obviously have tried to work things out, and there's no element of free-homework. Look over the solution sets, step by step and ask yourself which steps you couldn't come up with, then go ask how the TA/prof came up with them. There's two elements to understanding the construction of an argument
1) that it logically holds (the easy part).
2) why you would think to construct this particular argument.
Getting that second motivation step is the key, and a good professor can guide you through that (TAs frequently can not).
posted by a robot made out of meat at 4:36 AM on October 23, 2007
1) that it logically holds (the easy part).
2) why you would think to construct this particular argument.
Getting that second motivation step is the key, and a good professor can guide you through that (TAs frequently can not).
posted by a robot made out of meat at 4:36 AM on October 23, 2007
number9dream gives good advice. One of the hardest things to get students who are taking a more theoretical math course for the first time to do is to actually read the textbook in the manner number9 describes. It's really important to understand the definitions before moving on to the theorems before moving on to the proofs, before moving on to more definitions... You need to read with a pencil in hand copying down the arguments and filling in the details that are left out. For example, if there is some symbol or word that I am using for the first time, I'll often make a note of what the new thing means in the margin of whatever I'm writing/reading; I'll do this EVERY time I use/read it for a while, until I can instantly look at it and know the definition. I do this with the assumptions and consequences of a theorem that I'm using as well. Once you get better at parsing the book/symbols, you might not have to go so slowly. But, I must say 4 years in to grad school, I still copy pretty much everything down while reading.
Then, when you attempt the problems, you'll have a complete understanding of what's going on (also when you ask the professor questions, you'll have a better idea of what you don't understand, and why you don't understand it).
It may seem overwhelming now, but this careful reading/thinking process is one of the things I now love best about math. It's a really good feeling when you figure something out, and it's that feeling that made me go to grad school in the first place.
posted by bluefly at 5:27 AM on October 23, 2007
Then, when you attempt the problems, you'll have a complete understanding of what's going on (also when you ask the professor questions, you'll have a better idea of what you don't understand, and why you don't understand it).
It may seem overwhelming now, but this careful reading/thinking process is one of the things I now love best about math. It's a really good feeling when you figure something out, and it's that feeling that made me go to grad school in the first place.
posted by bluefly at 5:27 AM on October 23, 2007
Do you have a "math lounge"? There were some tables near our math classes where students did homework & it saved me. It didn't require a formal study group but because of schedules I often ran into the same people.
posted by ejaned8 at 8:14 AM on October 23, 2007
posted by ejaned8 at 8:14 AM on October 23, 2007
Get a different textbook on the same subject that takes a different approach to the material. The library is a likely spot to find this.
On rereading your question, it sounds like what you really want to know is how to deal with doing work that constantly makes you feel that you are in over your head. I wish that someone had told me that this happens. It probably feels more overwhelming since you have understood everything fairly well up until now. Think of this as exercise for your brain, resulting in greater fitness levels than you had before.
I've taken a lot of classes where students are not necessarily expected to accomplish anything more than a tiny portion of what is thrown at them, and at some point you will probably encounter one of those. This isn't the sort of thing that will be on the syllabus. If you do all you can and flunk, you know it wasn't that sort of class -- but if it is that sort of class, you can pull down a decent grade even if you feel constantly confused.
posted by yohko at 8:57 PM on October 23, 2007
On rereading your question, it sounds like what you really want to know is how to deal with doing work that constantly makes you feel that you are in over your head. I wish that someone had told me that this happens. It probably feels more overwhelming since you have understood everything fairly well up until now. Think of this as exercise for your brain, resulting in greater fitness levels than you had before.
I've taken a lot of classes where students are not necessarily expected to accomplish anything more than a tiny portion of what is thrown at them, and at some point you will probably encounter one of those. This isn't the sort of thing that will be on the syllabus. If you do all you can and flunk, you know it wasn't that sort of class -- but if it is that sort of class, you can pull down a decent grade even if you feel constantly confused.
posted by yohko at 8:57 PM on October 23, 2007
I'm a little late, but perhaps I can help. Here's the deal: I was terrible at mechanistic mathematics, but random walks, probability distribution and "higher math," comes easy to me. But I am (or was) good at philosophy and logic, and I have a feeling that those carried over in helping me understand stochastic processes, martingales, etc. Here's my general synthesis:
It is more important to understand the meta-mathematics at work than the technical details. I was sloppy ... as was Louis Bachelier (that's where our similarities end). I luckily had a professor who saw that I understood the process and could explain what was going on despite my code being less than ideal. If that makes any sense. Textbooks are generally no better in that they throw equations at you and expect you to read between the line. Most likely you were able to score well without understanding what was really going on, which is the fault of your instructors for taking a memorization and complete the equation approach.
I recommend picking up Louis Bachelier's Theory of Speculation, the new copy, from Amazon or the library. Also read Einstein's paper on Brownian motion. I've learned more from mathematics by reading journal articles and original papers than textbooks. A thorough lit review should have you understanding what exactly is going on, beyond the equations on a page. I find that papers or commentary pre-WWII are generally more forgiving and try to explain thing without resorting to equations at first.
In other words, mathematics is a tool to explain often complex concepts. Such concepts can be explained different ways, but most accurately in the context of mathematical logic. With the concepts and commentary, you should understand why the summations are coming out of nowhere. It is not as if some genius simply wrote equations on a page, there's a lot of conceptual thought.
I have a lot on random walks, and I think you'll find a lot on the Internet. It is related to a important concept in finance and there's a lot of texts out there that try to explain it differently than what a mathematics textbook would try to do. Message me if you want to see what I might be able to pull up from PDFs, etc.
posted by geoff. at 5:11 PM on October 25, 2007
It is more important to understand the meta-mathematics at work than the technical details. I was sloppy ... as was Louis Bachelier (that's where our similarities end). I luckily had a professor who saw that I understood the process and could explain what was going on despite my code being less than ideal. If that makes any sense. Textbooks are generally no better in that they throw equations at you and expect you to read between the line. Most likely you were able to score well without understanding what was really going on, which is the fault of your instructors for taking a memorization and complete the equation approach.
I recommend picking up Louis Bachelier's Theory of Speculation, the new copy, from Amazon or the library. Also read Einstein's paper on Brownian motion. I've learned more from mathematics by reading journal articles and original papers than textbooks. A thorough lit review should have you understanding what exactly is going on, beyond the equations on a page. I find that papers or commentary pre-WWII are generally more forgiving and try to explain thing without resorting to equations at first.
In other words, mathematics is a tool to explain often complex concepts. Such concepts can be explained different ways, but most accurately in the context of mathematical logic. With the concepts and commentary, you should understand why the summations are coming out of nowhere. It is not as if some genius simply wrote equations on a page, there's a lot of conceptual thought.
I have a lot on random walks, and I think you'll find a lot on the Internet. It is related to a important concept in finance and there's a lot of texts out there that try to explain it differently than what a mathematics textbook would try to do. Message me if you want to see what I might be able to pull up from PDFs, etc.
posted by geoff. at 5:11 PM on October 25, 2007
This thread is closed to new comments.
posted by LobsterMitten at 3:01 PM on October 22, 2007