Discrete math/combinatorial course

[emyt]

Hi, I'm currently taking a seminar course , it's like a "small group" course (like a lecture as a tutorial) in mathematics. It's open for anybody, most people there actually aren't math students. It's basically a discrete math/combinatorial/"problem solving" driven course but the thing is, I don't like it at all!

I've done some problems in combinatorics before, but it was actually enjoyable.. this is something else. My professor basically leads the class into just guessing for patterns ( for example, in a geometric question, he might create a chart of results and tell people to see patterns (without explanations even) ). It's really annoying because I'd rather see how it works deductively, and not empirically. I'm thinking of dropping out of that course, but before I do, I have to ask: does that seem like a fair way to treat the material? if so, then I probably just have to suck it up and adjust to it.. thanks

 

[hearts_hacker]

most of the fun combinatorics problems ask to find out such patterns... you cannot do any worthwhile combinatorics without enjoying finding these patterns...

 

[rasmhop]

It's really annoying because I'd rather see how it works deductively, and not empirically.

I'm sorry, but while math is usually presented deductively most of it isn't deductive in the first place. Consider for instance how when you're reading a graduate textbook a good student may very well be able to do all or at least most proofs by himself, but many of the theorems were once hard research problems so why can a student do it today? This is because they're led to ask the right questions by the author who makes the correct definitions, present the correct theorems, and in the right level of generality. Real mathematics consists of making guesses, conjectures, definitions and exploring the extents to which we can go. Even the proofs often consists of making guesses as to what directions to take, what objects to introduce, what cases to split into, what lemmas to prove. The deductive part is just used to confirm or disproof our guesses.

However that is not to say that the type of guessing presented in your course will help you with this. If it's always about guessing simple patterns like the type presented on IQ tests, then it may not be that beneficial. This may help non-math majors to get adjusted to guessing in math, but you may be prepared for more advanced problems. Just remember that it may not be all useless and maybe you should give it a chance. Sometimes it's beneficial to stretch your mind and try different types of thinking. It may be hard, but perhaps you'll come out on the other end of the course with improved problem solving skills. As this is a small course, perhaps you should ask your professor for his advice on how this applies outside this course.

 

[emyt]

I'm sorry, but while math is usually presented deductively most of it isn't deductive in the first place. Consider for instance how when you're reading a graduate textbook a good student may very well be able to do all or at least most proofs by himself, but many of the theorems were once hard research problems so why can a student do it today? This is because they're led to ask the right questions by the author who makes the correct definitions, present the correct theorems, and in the right level of generality. Real mathematics consists of making guesses, conjectures, definitions and exploring the extents to which we can go. Even the proofs often consists of making guesses as to what directions to take, what objects to introduce, what cases to split into, what lemmas to prove. The deductive part is just used to confirm or disproof our guesses.

However that is not to say that the type of guessing presented in your course will help you with this. If it's always about guessing simple patterns like the type presented on IQ tests, then it may not be that beneficial. This may help non-math majors to get adjusted to guessing in math, but you may be prepared for more advanced problems. Just remember that it may not be all useless and maybe you should give it a chance. Sometimes it's beneficial to stretch your mind and try different types of thinking. It may be hard, but perhaps you'll come out on the other end of the course with improved problem solving skills. As this is a small course, perhaps you should ask your professor for his advice on how this applies outside this course.

you have a point, and I will talk to my professor. Anyway, I know what you mean by how there is a lot of "guessing" in math - I have had experience with this ( guessing which theorems may be relevant for a proof.. et c). But these are guided guesses, there is more "sense" in guessing these things - whereas adding tables and results and then saying that x is like y in pascal's triangle is weird to me.. I guess I'm just not used to empiricy in math. For example, if you asked someone in that class why they might think that their formula is true, they might just reply "because they give me the same values as the values found in our chart"

 

[rasmhop]

you have a point, and I will talk to my professor. Anyway, I know what you mean by how there is a lot of "guessing" in math - I have had experience with this ( guessing which theorems may be relevant for a proof.. et c). But these are guided guesses, there is more "sense" in guessing these things - whereas adding tables and results and then saying that x is like y in pascal's triangle is weird to me.. I guess I'm just not used to empiricy in math. For example, if you asked someone in that class why they might think that their formula is true, they might just reply "because they give me the same values as the values found in our chart"

If you're advanced enough to have some exposure to independent formulation of theorem and proofs, then I don't think you'll learn much from this type of problem solving. For people unfamiliar with math at this level, these types of problems are approachable and may help them adjust. I remember using this type of reasoning way back, but as I advanced mathematically I advanced to more abstract heuristics and problem solving methods. I think they have limited application at most. I remember a problem I had a while back where I managed to reduce it to determining a function f. I managed to find the values of f at 0, 1 and 2 and deduce that it was a polynomial of degree 3. Then I just guessed a polynomial of degree 3 that fitted the values at 0, 1 and 2 and was able to deduce that this was the unique solution. In other cases I have sometimes spotted sequences like 0,3,8,15,24 in working with certain objects and recognizing that this is the start of n^2 -1 can help (though I would never take it on faith just because the first 2000 values matched). For instance I once forget the closed form for 1^3+2^3+3^3+...+n^3 when I needed it on an exam, but thought it was something simple like n^2(n+1)^2/4, but since I could easily reason that the result would be a fourth degree polynomial I just needed to check the first 4 values. This kind of reasoning is widely applicable and useful, but it doesn't sound exactly like what you do. It does sound like this class isn't really targeted at you to be honest though.

 

[ice109]

If you're advanced enough to have some exposure to independent formulation of theorem and proofs, then I don't think you'll learn much from this type of problem solving. For people unfamiliar with math at this level, these types of problems are approachable and may help them adjust. I remember using this type of reasoning way back, but as I advanced mathematically I advanced to more abstract heuristics and problem solving methods. I think they have limited application at most. I remember a problem I had a while back where I managed to reduce it to determining a function f. I managed to find the values of f at 0, 1 and 2 and deduce that it was a polynomial of degree 3. Then I just guessed a polynomial of degree 3 that fitted the values at 0, 1 and 2 and was able to deduce that this was the unique solution. In other cases I have sometimes spotted sequences like 0,3,8,15,24 in working with certain objects and recognizing that this is the start of n^2 -1 can help (though I would never take it on faith just because the first 2000 values matched). For instance I once forget the closed form for 1^3+2^3+3^3+...+n^3 when I needed it on an exam, but thought it was something simple like n^2(n+1)^2/4, but since I could easily reason that the result would be a fourth degree polynomial I just needed to check the first 4 values. This kind of reasoning is widely applicable and useful, but it doesn't sound exactly like what you do. It does sound like this class isn't really targeted at you to be honest though.

how did you reason it had to be a fourth degree polynomial?

 

[rasmhop]

how did you reason it had to be a fourth degree polynomial?

$$f(n) = \sum_{i=1}^n i^3$$

It's pretty clear* that an expression of this form is a polynomial so we just need to show it has degree 4.

We have,
$$f(2n) = \sum_{i=1}^{2n} i^3 > \sum_{i=n+1}^{2n} i^3 \geq \sum_{i=n+1}^{2n} n^3 = n^4$$
so f grows at least as fast as a fourth degree polynomial and therefore can't be of third degree or lower.
$$f(n) = \sum_{i=1}^n i^3 < \sum_{i=1}^n n^3 = n^4$$
so f grows at most as fast a a fourth degree polynomial and therefore can't be of fifth degree or higher.

*
EDIT: When I say clear I mean that it's fairly well-known and most importantly that I knew it was true and therefore could use it. It's not that it's completely trivial.

 

[emyt]

interesting. Thanks for your input rasmhop, I ended up dropping the course. I will wait for until my upper years before I take a serious discrete mathematics course (and until then, I'll be doing some reading on my own)