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I'm going to fail my first proofs class. How do people even learn this?

by MissSilvy
Tags: class, fail, learn, people, proofs
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MissSilvy
#1
Feb4-10, 10:18 PM
P: 299
A bit of cheese with my whine perhaps but I'm more frustrated because I don't know what to do. I am going to fail this class unless I figure it out before our first exam. I read the chapters, take notes, try to understand the theorems and definitions they present and then fall flat on my face when I try the homework.

The course description is as follows:
"Fundamental ideas used in many areas of mathematics. Topics will include: techniques of proof, mathematical induction, binomial coefficients, rational and irrational numbers, the least upper bound axiom for real numbers, and a rigorous treatment of convergence of sequences and series. This will be supplemented by the instructor from topics available in the various texts. Students will regularly write proofs emphasizing precise reasoning and clear exposition."

Our book is: "Mathematical Thinking: Problem-Solving and Proofs"
http://www.amazon.com/Mathematical-T...6&sr=8-1-fkmr2

The textbook is garbage, with not a single solution and very few worked out problems. I know I can't rely on having solutions for forever, but this is like asking someone to build a house whose never even seen a picture of one. The professor is nice but he only has two office hours a week and his English is a bit poor, so sometimes he can't explain things very well. I'm not looking for an easy way out but I can't sit in office hours and expect him to hold my hand through everything. What else can I do? I really want to pursue math as a minor but if this is the sort of thing they're going to force on us without proper opportunity to actually learn it, frankly I don't need the loss of sleep.
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lisab
#2
Feb4-10, 10:28 PM
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P: 2,979
Sorry to hear it! Maybe you can find students in the class who are willing to form a study group?

I suppose a tutor is also an option. Perhaps you wouldn't need hours and hours of tutoring, just time enough to see what a proof is supposed to look like...just someone to point you in the right direction.
kuahji
#3
Feb4-10, 10:30 PM
P: 395
My first proof class was Discrete Mathematics, we used this textbook http://www.amazon.com/Discrete-Mathe...5343758&sr=8-1. It's a very good textbook. Many people fail their first time through, I did, even though I'm physics major not a mathematics major. I simply went about trying to study for it in the wrong way. In algebra, trig, calculus, etc. you can get away without knowing definitions. With a proofs class you can't do this & you shouldn't try to mimic the authors proof exactly, use some of your own words (it's like trying to memorize a speech verbatim when rather it should just freely flow as long as you hit the main points). The important thing is, get the ideas in your proof right, start your proofs the same way, define problem, solve the problem, state your conclusion.

Proofs fooled me once, now I tutor the subject. Perhaps you aren't making the same mistakes as I did, but again, it's all about definitions, tricks, and relations.

Klockan3
#4
Feb5-10, 12:55 AM
P: 614
I'm going to fail my first proofs class. How do people even learn this?

Isn't it all about understanding? If you understand a subject then you should understand how to prove the things within it. If you don't know how things are connected then you didn't learn to understand but instead just learned to pass tests. Of course if you work really hard you can learn how to pass these classes also but it usually comes together with a ton of understanding.
chiro
#5
Feb5-10, 01:52 AM
P: 4,572
Quote Quote by MissSilvy View Post
A bit of cheese with my whine perhaps but I'm more frustrated because I don't know what to do. I am going to fail this class unless I figure it out before our first exam. I read the chapters, take notes, try to understand the theorems and definitions they present and then fall flat on my face when I try the homework.

The course description is as follows:
"Fundamental ideas used in many areas of mathematics. Topics will include: techniques of proof, mathematical induction, binomial coefficients, rational and irrational numbers, the least upper bound axiom for real numbers, and a rigorous treatment of convergence of sequences and series. This will be supplemented by the instructor from topics available in the various texts. Students will regularly write proofs emphasizing precise reasoning and clear exposition."

Our book is: "Mathematical Thinking: Problem-Solving and Proofs"
http://www.amazon.com/Mathematical-T...6&sr=8-1-fkmr2

The textbook is garbage, with not a single solution and very few worked out problems. I know I can't rely on having solutions for forever, but this is like asking someone to build a house whose never even seen a picture of one. The professor is nice but he only has two office hours a week and his English is a bit poor, so sometimes he can't explain things very well. I'm not looking for an easy way out but I can't sit in office hours and expect him to hold my hand through everything. What else can I do? I really want to pursue math as a minor but if this is the sort of thing they're going to force on us without proper opportunity to actually learn it, frankly I don't need the loss of sleep.
Some things I learned when I did discrete mathematics (involved proofs);

Proofs could be seen as a kind of directed graph where each node represents a transformation of a collection of theorems axioms or other general statements.

In saying this if you are asked to prove something you need to have an intuitive idea about how certain statements are not only linked together but "transform" into one another. Typically one will sketch out the proof at a coarse level and then fill in the gaps. For example if you are asked to prove that all primes are infinite you will typically take a statement about a prime and then transform that statement to identify some characteristic about primes in general.

By "connecting the dots" once you identify the relationship between statements, the rest is searching for those transformations that allow the proof to "flow" from the initial idea to the final statement.

Typically the problem is that proofs make use of quite a lot of trivial statements so that sometimes the proof will take say 10 statements and transform them in various ways to achieve some other statement. In saying this I recommend that you look at how various proofs transform each statement. These are basically you tools in your toolbox.

Some "transformation templates" exist for proving various things too. One of these is mathematical induction. Properties of sequences and series are used to prove bounds that are useful in analysis.

I find myself that proofs are hard but get easier if you can visualize the transformations and think about how each statement is linked and the "flow" of the proof.

Hopefully I've mentioned something that you can make use of.
qspeechc
#6
Feb5-10, 03:51 AM
P: 792
Many students find these books helpful:

http://www.amazon.com/How-Prove-Stru...5354134&sr=1-1
http://www.amazon.com/How-Read-Proof...ref=pd_sim_b_5

Judging by the contents of your course, you may also find the following helpful:
http://www.amazon.com/Analysis-Intro...5354269&sr=1-1

Like others have said, it's all about practice. Practice, practice, practice. Most of us felt hopelessly lost when we first encountered rigorous proofs. I know I did. Proofs are very different from anything you’ve done before— it’s a new way of thinking. So don't stress too much. Know that if you put in enough effort you WILL understand it eventually. The hard work comes first.

One must first understand the mechanics of logic and set theory. For me the two are inseparable, and you cannot artificially treat them separately, as I was taught, and as I see done in many books. It is not enough to know the methods of proof and dis-proof, you need to understand why they are so (and note the particular definitions, e.g. of how the connective ‘or’ is defined— not how we usually use the word…). I found Venn diagrams to cast a light on the dark landscape of proofs. For example, in understanding direct proofs: If you want to prove A implies B, then this is quite the same thing as proving ‘A is a subset of B’, in a sketchy manner of speaking. Draw the Venn diagram with A a subset of B. If you are ‘in A’ then you are ‘in B’. That is proving A implies B. Well now the other proof techniques follow easily from this idea. Contra-positive? Easy! If ‘A is a subset of B’ (you are required to prove A implies B) then if you are ‘outside B’ then you must be ‘outside A’—that’s a contra-positive proof! (I’m sorry if this is all blindingly obvious to you, and I come across as patronising; I apologise. When I was taught proofs we were never given any visualisations of this sort, even though it makes complete sense. I discovered this for myself a year or two later.). This Venn diagram stuff can help you understand proof by contradiction, and dis-proof by counter-example. Try it yourself. Intuitively can see how to prove A implies B or C? How would you set up the Venn diagram? Well, you want to show ‘A is a subset of B union C’. Drawing the diagram shows you this is equivalent to showing that if you are ‘in A’ but (and) not ‘in B’ then you must be ‘in C’. Geddit?

So diagrams and intuition are very important in mathematics. But a diagram is not a proof! Nota Bene! Diagrams only help us to see what’s going on; they are only an aid to formulating our ideas and proofs precisely and rigorously; seeing why things should be a certain way. In fact, diagrams can sometimes be misleading, although they are often helpful-- so beware!

You have to understand the concepts. For example, you should have an intuitive idea of what a convergent sequence is. Perhaps draw a picture. Now how would you define a convergent sequence? Really, try defining it yourself from scratch. Then look at the definition in the book and try and see how it relates to convergence of a series. Compare and contrast with your own definition to see where it falls short. The point is to understand the definition, not to memorise it. Understand it thoroughly and you won't have to memorise the definition, you can reconstruct it at will. But in the beginning it is often helpful to memorise definitions and theorems, word for word.

Theorems are similar stuff. First understand intuitively what is going on. Diagrams are always useful; always draw a picture or visualise it. For instance, you may have seen the theorem that every convergent sequence has one limit. Obvious, but the proof? What if a convergent sequence had two limits? Then what? Well then, the definition of convergence tells us intuitively that all the terms of the sequence should eventually get arbitrarily close to both the limits. Aha! Contradiction. From there you can work on formalising the proof; making it rigorous, mathematical, and clear.

You should understand the limits of definitions and theorems. Why is this definition/theorem like this, and not like that? What happens if I change it slightly here? Why do we need this condition, and what happens if I drop it? Etc.

When you come to a new theorem, cover up the proof and first try and prove it yourself. Give yourself some time, and if you make no headway, look at the first few lines of the proof and try again. You understand a proof far more if you’ve tried working on it yourself first, than if you simply read what is given to you. When you come to a definition/theorem, try and give your own examples. What happens if you drop one of the conditions in the hypothesis? What goes wrong, why doesn’t the conclusion follow?

If you are having trouble with a proof, maybe looking back at other problems will help. Can you use some of the techniques here? How did the author go about proving a similar problem, and will it work here? Also see how the author uses the various types of proof and dis-proof.

Always keep clearly in mind what you can assume and what you can not. Do not make implicit assumptions that are not stated in any definitions or in the hypothesis of the theorem. Do not assume too much. Keep in mind the relevant definitions and previous theorems. When you have some theorem about convergent sequences, recall the definition of convergent series and use it exactly. You may also use theorems about convergent series you have already used. But do not– never!-- use ‘obvious’ statements that you have not proved. No matter how trivial it seems to you, it still needs a proof. Also, many innocuous and trivial-looking statements have very non-trivial proofs.

Quantifiers. The order of quantifiers is very important. Contrast the following:
a) For every integer, there exists another integer larger than it.
b) There exists an integer which is larger than every integer.
If you write that out explicitly with quantifiers (do so), you’ll see the only difference is the change in the position of the quantifiers, yet the two statements say completely different things: the first is correct, the second is patently false.
Also understand how negation works with quantifies and statements. Very important.

Also note the different terminology, and different ways of expressing the same ideas. For instance, ‘A is sufficient for B’ simply means ‘A implies B’, and so on.

These are just some of the pointers I can think of off the top of my head. But the most important thing is to put in a lot of work. Your understanding and mathematical power is a monotonically increasing function of the effort you put in. Hard work is most important, but working intelligently is also important. Constantly think about what you're doing and why. Can this be improved, done better, faster? Etc. Don't do the same thing over and over if it doesn't work. Challenge yourself with hard problems just out of your reach, so when you solve them, you mature a little, mathematically speaking. If you always do easy problems you will never learn anything. But easy problems are good to work on as well. Work on all types of problems, of different levels of difficulty. Don't be discouraged by very difficult problems; keep plugging away, come back to them later.

I hope I have been of some help, though this may be a bit late for the course you are taking now.
snipez90
#7
Feb5-10, 08:11 AM
P: 1,105
Quote Quote by Klockan3 View Post
Isn't it all about understanding? If you understand a subject then you should understand how to prove the things within it. If you don't know how things are connected then you didn't learn to understand but instead just learned to pass tests. Of course if you work really hard you can learn how to pass these classes also but it usually comes together with a ton of understanding.
Yeah I think the OP is actually trying to figure this out, so this seems irrelevant.

My advice is to actually post the exact problems and the precise issues you're having with them on this forum. It's hard to assess your difficulties without knowing precisely what you've tried or at least thought about with regards to the specific problem at hand. I think that understanding the basic idea behind the general techniques of proof: why induction works, basic logic, quantifiers, contradiction, etc. is not difficult.

However, there is a real difference between not knowing how to construct a proof, a difficulty that is relatively easy to overcome, versus not knowing how to actually solve a particular problem, which is more difficult. For instance, if you're given some statement regarding the natural numbers and you need to use induction, people will find it hard to believe that you have no idea where to begin. If you've ever seen an induction proof before, you should automatically have the base case and the inductive hypothesis written down. Figuring out the inductive step however, may require you to solve a sub-problem, and this has nothing to do with proof, and everything to do with problem-solving abilities. The only way to hone the latter ability is to work on more problems.

Given that the course seems to be tailored to a subsequent analysis course (many of the concepts are basic analysis techniques), I would take qspeechc's suggestion and find an analysis text. Most intro analysis texts cover the same concepts, especially in the first few chapters. Perhaps Lay's book is better in this regard than anything else. This way, you'll be able to develop the necessary problem solving abilities in the area of math that you're actually working with.
Klockan3
#8
Feb5-10, 10:22 AM
P: 614
Quote Quote by snipez90 View Post
Yeah I think the OP is actually trying to figure this out, so this seems irrelevant.
It is just that I hate the "think like a computer" approach. It is a very powerful tool when you want to teach people but not when you want to learn. If she had learned the earlier courses more properly this one shouldn't have been a problem in my opinion.

What to do then? Well, I would say that it is a lengthy process where you must reconstruct your view of maths, from the "I get this input so I give that output" to actually thinking.

A formula is a short-cut only to be taken after you have gone the long route at least once. Otherwise you have no idea of what you are doing, this might be fine for the applied sciences but not for the more pure ones.
Edit: And "by the long route" I mean when you basically derive it yourself when you need it instead of memorizing it. When you are at that level it is fine to memorize it afterwards. Get to that level with all the maths you have learned and the proofs based courses are afterwards quite easy.
daveyinaz
#9
Feb5-10, 01:52 PM
P: 225
Quote Quote by Klockan3 View Post
It is just that I hate the "think like a computer" approach. It is a very powerful tool when you want to teach people but not when you want to learn. If she had learned the earlier courses more properly this one shouldn't have been a problem in my opinion.

What to do then? Well, I would say that it is a lengthy process where you must reconstruct your view of maths, from the "I get this input so I give that output" to actually thinking.

A formula is a short-cut only to be taken after you have gone the long route at least once. Otherwise you have no idea of what you are doing, this might be fine for the applied sciences but not for the more pure ones.
Edit: And "by the long route" I mean when you basically derive it yourself when you need it instead of memorizing it. When you are at that level it is fine to memorize it afterwards. Get to that level with all the maths you have learned and the proofs based courses are afterwards quite easy.
What do you mean if she had learned earlier courses more properly? Do you know all the courses the OP has taken prior to taking her proof class? Perhaps, you should avoid commenting on things where you have nothing good to add.
koab1mjr
#10
Feb5-10, 02:14 PM
P: 103
Klocklan3

You can only learn what you are taught and I am not sure where you went to school but i grew up in the US and i understood what I was taught very well but rarely would you find a school that would teach the material to this depth on the first pass.

I did not get a sniff or a "true" proof till college. You sound insulting and condescending, you should look to other avenues to feel better about yourself.


I think a majority of people run into this chasm. Up until this point math was a cake walk no prob then somewhere after calc you get hit like mack truck. Texts and classwork all the sudden demand "mathmatical maturity" which is to be able to think in a generalized abstract fasion and create logical arugments. When you had zero practice its a daunting and abrupt request. I encoutered this with a PDE class which I had to drop due to the heavy proof content. I was not ready. It requires mastery that I was not accostume too and I had to go back and review material with a new mind set. But until you come to this point you do not know about this aspect.

When I first encountered the epsilon delta defintion of a limit of function I struggled, I can't say there exists any tricks to breaking through. Just hard work. I will say paraphrase everyline you see. When you can do that you know the proof line. Don;t memorize though...
It takes time but everything snaps into place and suddenly it all seems so simple and you wonder why you struggled in the first place

The difficulty about math is there isn't a kind of get it state, there is a fence you are either on the know it side or the clueless. You just found out you weren't on the side of the fence you thought you were but that is the first step to climbing over.
MissSilvy
#11
Feb5-10, 05:20 PM
P: 299
Wow, a big than you to Lisab, Kuahji, Chiro, Qspeechc, Snipez90, and Koab1mjr. Your responses are overwhelmingly helpful (I printed this thread for future reference) and I appreciate you guys taking the time to write such long and thoughtful responses. It really does mean a lot to me, at least it gives me hope that this IS possible and I'm not the only person to have struggled with this subject, though it certainly seems like that at times. I'm in the process of checking out several of the books you guys recommended and they seem to be pretty helpful, by and large.

Logically, I knew that at some point I'd encounter the first 'wall' in my university education that everyone pretty much has but I didn't know when. I got good grades all up to this point so the idea of being confronted with REAL math and REAL physics is a bit intimidating. I probably freaked out a wee much but the only way seems to be to push on. Thanks again everyone! You guys made my day :)
atyy
#12
Feb6-10, 12:31 AM
Sci Advisor
P: 8,395
I've never done maths, but the place I learnt why epsilon-delta statements were meaningful were from Schaum's series, where they didn't just state, "for any epsilon", instead they also added "no matter how small", which is not formally needed, but conveys the intent of the formalism.
Werg22
#13
Feb6-10, 12:53 AM
P: 1,520
I've heard good things about this book, and it seems to fall within the ballpark of your class:

http://www.amazon.com/Introductory-M...5438889&sr=1-1

I have another book from the author, and I like his style.
Klockan3
#14
Feb6-10, 02:17 AM
P: 614
Quote Quote by koab1mjr View Post
Klocklan3

You can only learn what you are taught and I am not sure where you went to school but i grew up in the US and i understood what I was taught very well but rarely would you find a school that would teach the material to this depth on the first pass.

I did not get a sniff or a "true" proof till college. You sound insulting and condescending, you should look to other avenues to feel better about yourself.
I am sorry about that, I think my responses got a different tone due to earlier posts by MissSilvy in different threads. Not that I don't like her as a person she just have different viewpoints from mine, that shouldn't get through but I guess it does anyway.

But as far as I know most schools teach rudimentary "proofs". The epsilon delta proofs taught in the first college calculus course aren't that much stranger than what you do when you derive the derivatives in high-school for example, proofs by induction should be the first thing you learn in college and linear algebra should teach you the art of "How to prove the trivial".
Don't you derive the trigonometric identities in high-school, or teaches that an equality is preserved if you do equal operations on both sides no matter the operation? All of these things should be ample of training for formal maths.

To me it seems like the steps are quite small if you make sure to take all of them.
MissSilvy
#15
Feb6-10, 06:36 AM
P: 299
I am sorry about that, I think my responses got a different tone due to earlier posts by MissSilvy in different threads.
Hm? What other posts, if I may ask?
Klockan3
#16
Feb6-10, 09:13 AM
P: 614
I think that it is mostly this thread:
http://www.physicsforums.com/showthread.php?t=299070
and this one:
http://www.physicsforums.com/showthread.php?t=312073

I think that it is a choice in early maths courses to either learn to understand or memorize. The first topic shows that you rely a lot on memorizing which leads to this thread. The second topic do not really have much to do with this one, it is just that I have a different opinion and you made very bold statements there.

Also if you wonder, I am not stalking you I just got a good memory.
MissSilvy
#17
Feb6-10, 11:33 AM
P: 299
Quote Quote by Klockan3 View Post
Also if you wonder, I am not stalking you I just got a good memory.
I didn't think that, but your post made it sound like I offended you in another thread and I was seeking to clarify. But you are correct, thank you :)
daveyinaz
#18
Feb11-10, 12:41 PM
P: 225
Quote Quote by Klockan3 View Post
Also if you wonder, I am not stalking you I just got a good memory.
Could it even be regarded as stalking? I mean, you could easily use the "show other posts from user" button or whatever.


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