Help Gain Mathematical Maturity
March 29, 2011 6:12 AM Subscribe
What are the best ways to develop a solid mathematical maturity?
I'm attending University for Computer Science and although I love the major, I'm having some problem with the math classes. I don't mean Calc I-III, but rather classes like Discrete Mathematics and the ones that require a lot of mathematical maturity to follow the proofs and decipher the notation.
I always did well in the other math classes so I know I'm capable of doing it, I just have been having problems with these ones. I have been reading about mathematical maturity and it seems like I'm lacking it quite a bit. In lecture, I often can't follow along very well because I get stuck on a lot of the silly things like notation and lacking the insight to "see" the best solution rather than the actual material.
I read that one of the best ways to develop this is just to pick up a book on math and work through it. I've been looking for some books but I hope that some people out there have experienced this and can give some pointers and/or books.
Thanks a lot, I really appreciate it.
I'm attending University for Computer Science and although I love the major, I'm having some problem with the math classes. I don't mean Calc I-III, but rather classes like Discrete Mathematics and the ones that require a lot of mathematical maturity to follow the proofs and decipher the notation.
I always did well in the other math classes so I know I'm capable of doing it, I just have been having problems with these ones. I have been reading about mathematical maturity and it seems like I'm lacking it quite a bit. In lecture, I often can't follow along very well because I get stuck on a lot of the silly things like notation and lacking the insight to "see" the best solution rather than the actual material.
I read that one of the best ways to develop this is just to pick up a book on math and work through it. I've been looking for some books but I hope that some people out there have experienced this and can give some pointers and/or books.
Thanks a lot, I really appreciate it.
First off, you are not alone. Developing mathematical maturity is often difficult for many students of mathematics. The curriculum of many math programs has been changing to try to address this issue. One way that math departments try to foster mathematical maturity is to teach classes using Inquiry Based Learning. The IBL approach puts the responsibility on the students to generate proofs, but also supplies the scaffolding for the direction to go. It's a movement that's gaining momentum in the math world, though there are departments and individual professors who argue against it (for very valid reasons).
You might want to check out an IBL textbook to study on your own. Here's one by Marshall, Starbird, and Odell for Number Theory. I am certain that several IBL texts for Discrete Mathematics exist, but I can't find the link right now. BTW, Mike Starbird is a big proponent of IBL and an extremely nice guy. If you couldn't find what you are looking for, it may not be a bad idea to email him and ask for his advice.
posted by El_Marto at 6:46 AM on March 29, 2011 [1 favorite]
You might want to check out an IBL textbook to study on your own. Here's one by Marshall, Starbird, and Odell for Number Theory. I am certain that several IBL texts for Discrete Mathematics exist, but I can't find the link right now. BTW, Mike Starbird is a big proponent of IBL and an extremely nice guy. If you couldn't find what you are looking for, it may not be a bad idea to email him and ask for his advice.
posted by El_Marto at 6:46 AM on March 29, 2011 [1 favorite]
Generally Real Analysis/Advanced Calc is taught as the course where students begin to develop mathematical maturity, and for good reason. Its content is a mix of familiar concepts from Calculus that have been rigorized, which lets you get a sense for the 'parts' that were missing from your earlier grasp of mathematics, as opposed to, say, a course in Algebra, where you'd be encountering all sorts of new concepts for the very fist time, and struggling with how to put them together in the proper way. That isn't to say that Real Analysis has nothing new: in fact, it has a lot. But the "space" you are working with--functions, numbers, graphs, sets--is just a little more familiar.
You don't have to do the entire course to get its benefits. If you have the time, try working though the beginning of a real analysis book like Rudin or Royden (tougher). The library will have both of these. The beginning is where fundamental concepts like open sets, closed sets, functions, etc are defined, and the exercises in those sections will strain your mathematical instinct until you really do get a better sense for how to work in a more mathematically mature environmental.
A final recommendation is A Radical Approach to Real Analysis, which can be helpful for the student who is new to proof. It gives the historical narrative of the development and acceptance of the current proof-based approach. It also has exercises, but will be more valuable for learning about why mathematics made the transition you are currently struggling to make, and why it is a necessary system. I read it after I had already taken Real Analysis, as more of an enjoyment/historical read, but I sense that it would be a very 'natural' introduction to proof-based thinking in general. It earns high marks from me. Good luck!
posted by milestogo at 6:47 AM on March 29, 2011 [1 favorite]
You don't have to do the entire course to get its benefits. If you have the time, try working though the beginning of a real analysis book like Rudin or Royden (tougher). The library will have both of these. The beginning is where fundamental concepts like open sets, closed sets, functions, etc are defined, and the exercises in those sections will strain your mathematical instinct until you really do get a better sense for how to work in a more mathematically mature environmental.
A final recommendation is A Radical Approach to Real Analysis, which can be helpful for the student who is new to proof. It gives the historical narrative of the development and acceptance of the current proof-based approach. It also has exercises, but will be more valuable for learning about why mathematics made the transition you are currently struggling to make, and why it is a necessary system. I read it after I had already taken Real Analysis, as more of an enjoyment/historical read, but I sense that it would be a very 'natural' introduction to proof-based thinking in general. It earns high marks from me. Good luck!
posted by milestogo at 6:47 AM on March 29, 2011 [1 favorite]
Do more proofs. And then more.
posted by advil at 6:56 AM on March 29, 2011 [2 favorites]
posted by advil at 6:56 AM on March 29, 2011 [2 favorites]
I read that one of the best ways to develop this is just to pick up a book on math and work through it.
I'm not so sure this is a great strategy. By all means pick up a math book and try to work through it if you are really interested in the topic, but otherwise you are likely to end up discouraged/feeling-like-a-failure with little to show for it.
"Mathematical maturity" is not a well-defined concept, despite the wikipedia entry. The biggest question you have to ask yourself is: why proofs? I mean, why are proofs useful or necessary at all. You don't actually have to believe they are useful but you have to be able to ask and give an answer to the question. I think one of the bigger problems in transitioning from algebra based algorthmics i.e. calculus, to formal mathematics is a kind of intellectual passivity. You have to be able to see mathematical problems in the first person, as if you were the person asking the question. And it can be harder to ask a question than to answer one given to you from on high.
Personally, I don't think there is any royal road to maturity. You just have to keep on trying and whatever maturity you are allowed will come to you. Everyone has problems with notation, horrible problems with notation, and insight is often illusory.
posted by ennui.bz at 7:10 AM on March 29, 2011
I'm not so sure this is a great strategy. By all means pick up a math book and try to work through it if you are really interested in the topic, but otherwise you are likely to end up discouraged/feeling-like-a-failure with little to show for it.
"Mathematical maturity" is not a well-defined concept, despite the wikipedia entry. The biggest question you have to ask yourself is: why proofs? I mean, why are proofs useful or necessary at all. You don't actually have to believe they are useful but you have to be able to ask and give an answer to the question. I think one of the bigger problems in transitioning from algebra based algorthmics i.e. calculus, to formal mathematics is a kind of intellectual passivity. You have to be able to see mathematical problems in the first person, as if you were the person asking the question. And it can be harder to ask a question than to answer one given to you from on high.
Personally, I don't think there is any royal road to maturity. You just have to keep on trying and whatever maturity you are allowed will come to you. Everyone has problems with notation, horrible problems with notation, and insight is often illusory.
posted by ennui.bz at 7:10 AM on March 29, 2011
When I was young, I had a fair amount of mathematical maturity but was quite immature otherwise. I knew this and it disturbed me and I thought a lot about it. How can I ever achieve maturity? I was looking for an algorithm, though I didn't think of it in those terms. I looked at others, more mature than I was, and tried to copy them--tried to be like them. This, at least in the short run, generally failed, but something must have worked because now, decades later, I'm mature. What was it that worked?
As I teach my math classes, I have to explain to students ways to solve certain types of problems, but when I need to solve such a problem, I rarely turn to such methods, though, in many cases I end up doing the same thing I taught. I think about the problem differently, though--often not in a category at all, and I wonder how I can get my students to think that way. When I try and convey this goal to them, they just become confused. Or angry. They don't want to "think about it differently." They want to know how to get the answer. I want them to understand how someone would come up with such a way to get the answer, if no one was there to tell it to them. Even if they couldn't succeed at at it, I wanted them to know that it was possible to come up with things to try and that it was worth while trying them, even if they failed. That there was something they would gain by the failure, and that, in the end, instead of memorizing a lot of recipes, they would know how to cook.
So, I'm trying and failing to come up with an answer to your question, but I think there's something in the attempt that suggests a sort of answer. Namely, that it takes time, and it takes playing around with things to see what happens, and it takes sufficient interest to keep doing that, even when it doesn't seem to bring one closer to a goal.
posted by Obscure Reference at 9:58 AM on March 29, 2011 [1 favorite]
As I teach my math classes, I have to explain to students ways to solve certain types of problems, but when I need to solve such a problem, I rarely turn to such methods, though, in many cases I end up doing the same thing I taught. I think about the problem differently, though--often not in a category at all, and I wonder how I can get my students to think that way. When I try and convey this goal to them, they just become confused. Or angry. They don't want to "think about it differently." They want to know how to get the answer. I want them to understand how someone would come up with such a way to get the answer, if no one was there to tell it to them. Even if they couldn't succeed at at it, I wanted them to know that it was possible to come up with things to try and that it was worth while trying them, even if they failed. That there was something they would gain by the failure, and that, in the end, instead of memorizing a lot of recipes, they would know how to cook.
So, I'm trying and failing to come up with an answer to your question, but I think there's something in the attempt that suggests a sort of answer. Namely, that it takes time, and it takes playing around with things to see what happens, and it takes sufficient interest to keep doing that, even when it doesn't seem to bring one closer to a goal.
posted by Obscure Reference at 9:58 AM on March 29, 2011 [1 favorite]
I'm with ennui.bz, that picking up a text and trying to work through it is not that great of a plan. Someone upthread mentioned that you should try Royden's Real Anaylsis book; I think that would result in nothing but pure pain for someone who lacks mathematical maturity. If you're going to get any book at all, one of the more friendly ones is Topology by Munkres. That will afford you a much easier transition to proof-based mathematics.
You also may want to consider looking at Proofs from THE BOOK, which is a text containing many extremely elegant proofs. (The title comes from a quote uttered by Paul Erdős when he would see a particularly elegant proof.)
I like Obscure Reference's take on it, that you really need to look at your homework exercises as more of a "first-person" affair. You're the one asking the question, you don't want the "what are they asking for" type of attitude anymore. You need to get away from that somehow...but how?
I think by proving really, really mundane things can get you started. Do you know what a group is? Prove that the identity element in a group must be unique. That's a simple enough thing. Do you know what a homomorphism between groups is? Prove that a homomorphism must map the identity of the domain group to the target group. If you can, do these by yourself without referring to any notes. If you can't, do it with notes. Then try again tomorrow without notes.
Stop being afraid to write words in mathematics. When you're reading a proof, you don't need to necessarily stop every 10 seconds and ask "why are they doing this now?" A proof is often a story that you might need to immerse yourself in a bit, and allow it to unfold. You might not see immediately why the author might start off with defining a certain kind of function, but just let it happen.
Patience is a virtue. It takes a lot of time and a lot of work to develop a mathematical mind.
posted by King Bee at 3:22 PM on March 29, 2011 [1 favorite]
You also may want to consider looking at Proofs from THE BOOK, which is a text containing many extremely elegant proofs. (The title comes from a quote uttered by Paul Erdős when he would see a particularly elegant proof.)
I like Obscure Reference's take on it, that you really need to look at your homework exercises as more of a "first-person" affair. You're the one asking the question, you don't want the "what are they asking for" type of attitude anymore. You need to get away from that somehow...but how?
I think by proving really, really mundane things can get you started. Do you know what a group is? Prove that the identity element in a group must be unique. That's a simple enough thing. Do you know what a homomorphism between groups is? Prove that a homomorphism must map the identity of the domain group to the target group. If you can, do these by yourself without referring to any notes. If you can't, do it with notes. Then try again tomorrow without notes.
Stop being afraid to write words in mathematics. When you're reading a proof, you don't need to necessarily stop every 10 seconds and ask "why are they doing this now?" A proof is often a story that you might need to immerse yourself in a bit, and allow it to unfold. You might not see immediately why the author might start off with defining a certain kind of function, but just let it happen.
Patience is a virtue. It takes a lot of time and a lot of work to develop a mathematical mind.
posted by King Bee at 3:22 PM on March 29, 2011 [1 favorite]
Best answer: I agree that you should try to tackle all of Royden. I suggested the first few chapters as fairly fundamental areas where the definitions can be toyed with and explored profitably. Hopefully you can read them over and ask, "ok, why is this definition chosen? why are all these conditions in the definition? How does this capture everything about the concept?" Trying to figure out the first few exercises should help you start to play with these ideas.
I have a story that I like to tell. The first time I actually tried to solve a problem in Advanced Calc, I sat down in the library and stared at the problem for hours, getting nowhere. I started writing down symbols, words, trying to make headway toward a proof of the proposition. Eventually I built a chain of logic that worked from beginning to end; the proof ended up filling 2 pages. The professor was kind enough to point out that my logic was far too complicated, and I could have gone straight from the first step to the last. I felt a little dumb, and after a few more weeks in the class I laughed at the image of working in the library for hours, unable to see the (what now seemed) simple logical leap from A to B. But there is no way that I could have progressed enough to see that without putting in the time to piece the problem together. Even though it was absurdly simple, the transition between solving a problem and proving a proposition is an immense shift in the thinking style required. When I was sitting in the library, I felt like I was dealing with the most obtuse problem I had ever encountered. Once put in the time to isolate the concepts at play in my mind (even for a simple problem), I was able to see the problems much more clearly. For me, it took a definite amount of banging my head against the table (metaphorically) before I was able to succeed.
posted by milestogo at 4:04 PM on March 29, 2011
I have a story that I like to tell. The first time I actually tried to solve a problem in Advanced Calc, I sat down in the library and stared at the problem for hours, getting nowhere. I started writing down symbols, words, trying to make headway toward a proof of the proposition. Eventually I built a chain of logic that worked from beginning to end; the proof ended up filling 2 pages. The professor was kind enough to point out that my logic was far too complicated, and I could have gone straight from the first step to the last. I felt a little dumb, and after a few more weeks in the class I laughed at the image of working in the library for hours, unable to see the (what now seemed) simple logical leap from A to B. But there is no way that I could have progressed enough to see that without putting in the time to piece the problem together. Even though it was absurdly simple, the transition between solving a problem and proving a proposition is an immense shift in the thinking style required. When I was sitting in the library, I felt like I was dealing with the most obtuse problem I had ever encountered. Once put in the time to isolate the concepts at play in my mind (even for a simple problem), I was able to see the problems much more clearly. For me, it took a definite amount of banging my head against the table (metaphorically) before I was able to succeed.
posted by milestogo at 4:04 PM on March 29, 2011
damnit, of course I mean I agree that you shouldn't try to tackle all of Royden
posted by milestogo at 4:06 PM on March 29, 2011
posted by milestogo at 4:06 PM on March 29, 2011
milestogo - I think I just misunderstood what you meant by "pick up Royden". Indeed, the first chapter or two is basic enough stuff about the real numbers and sets and so forth. It could be a good starting ground. As soon as he starts talking about measure, I'd move on to something else until I was a bit more "seasoned".
and dammit, I want to mapping the identity of the domain group to the identity of the target group above...blargh, I suck.
posted by King Bee at 4:42 PM on March 29, 2011
and dammit, I want to mapping the identity of the domain group to the identity of the target group above...blargh, I suck.
posted by King Bee at 4:42 PM on March 29, 2011
My Advanced Calc class experience was, uh, different... Only my prior knowledge of proof techniques and notation got me through that class. I would not in any way recommend it to someone who is still learning notation for proofs.
This guy has a lot of slides which you may find helpful. I also like the book A Transition to Advanced Mathematics (Smith, Eggen, and Andre). The current edition is 7th edition, but you can find 5th and 6th very cheaply--or the library might have it.
Finally, see if your school has a math tutoring center. I'm always excited when someone walks in with "fun math" questions instead of more damn calculus. This is one of the harder things to teach, but talking through the thought process for different proofs is a good way to get practice.
posted by anaelith at 5:12 PM on March 29, 2011
This guy has a lot of slides which you may find helpful. I also like the book A Transition to Advanced Mathematics (Smith, Eggen, and Andre). The current edition is 7th edition, but you can find 5th and 6th very cheaply--or the library might have it.
Finally, see if your school has a math tutoring center. I'm always excited when someone walks in with "fun math" questions instead of more damn calculus. This is one of the harder things to teach, but talking through the thought process for different proofs is a good way to get practice.
posted by anaelith at 5:12 PM on March 29, 2011
This thread is closed to new comments.
posted by Dr. Eigenvariable at 6:24 AM on March 29, 2011