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Academic Guidance Academic Guidance - Academic guidance for those pursuing a college degree... what college? Grad school? PhD help?


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June 18th, 2014, 10:12 PM   #1
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I require your assistance with math!

I feel like I might be posting in the wrong section but that is fine, if so please move this thread to the appropriate section.

I am planning on God willingly attending university this fall and enrolling in the undergraduate pure maths program at the UoC here in Canada, and I had several queries I wanted to make pertaining to my mathematical education.

First off, I am sure that my mathematical knowledge is very limited and lacking and this is holding me back from being able to study with ease and comfort. For instance, I have a very weak mathematical vocabulary, my problem solving unfortunately consists of the rudimentary standard here is how to solve this problem, but I never (although I do admit I was not ready at the time) learned the whys and hows or the theory of such problems (from elementary to high-school), so this is a problem. I also canno solve problems simply because I lack the background, for instance I was a terrible student in high school for certain reasons and later after a dramatic change in my life I was transformed into a different person and through some kind of desire began my quest to become a more mathematically fluent person. Going back to the main topic, I was a terrible student, (the curriculum also did not go very deep so it was a problem on two sides), and I cannot therefore solve geometry problems from simple to more challenging problems because I lack the knowledge.

My second problem is all too common and can be remedied with enough time and effort, and that is the creativity or brainstorming to solve problems and secondly the problem solving. I am working on a maths proofs textbook (Chatrand, Polemeni, and Zhang, Mathematical Proofs 2nd Ed.) and I got stuck on chapter 4 problem 4.9 because I lack the imagination or the insight to be able to get to the right idea and solve the problem. I believe this problem is also coupled with my lack of knowledge and background, because with more knowledge the steps should become clear. I therefore come to you mathematical students of all levels for your advice, for references to any resources that would faciliate growth in their varying forms (websites, books, pdfs, i am hesitant on videos) for problem solving and such. I have at the moment how to solve it by polya, but I am at a loss at applications for bad knowledge banks.

I appreciate all feedback, thanks and blessings.

Last edited by Algeria; June 18th, 2014 at 10:17 PM.
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June 18th, 2014, 10:48 PM   #2
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Better to share the problem here. Make a thread with question.
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June 19th, 2014, 06:16 AM   #3
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Sharing individual problems is a good way to get help on individual questions. You could try asking for hints only, not complete solutions. That would perhaps help your problem solving.

On the knowledge side the first thing to say is: don't worry. The University should not admit you if you don't have sufficient knowledge. You might have to work hard to fill some holes, but that shouldn't stop you from understanding the course.

I too have big holes in my geometry knowledge. I use wikipedia, wikiproof and other web resources as references especially for geometric theorems. The key is to never look only at the result, but to read the proof and be suure that you understand it. My approach to learning is first to learn what tools are available (in terms of theorems and methods) and where I can find the detail. Learrning the detail myself comes later, but I never use a theorem that I can't understand. If I think a theorem is sufficiently useful, I try to learn the proof. This helps to fix the result in my mind and also allows me to derive it if I can't remember it clearly.

Finally, at university and online, always work a problem as far as you can before asking for help, but ALWAYS ask for help if you need it. Sitting quietly staring at a problem won't solve it. Often leavinng it to turn over in the back of your mind while you do something else is good for getting solutions.
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June 19th, 2014, 09:09 AM   #4
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Okay sure, here are two problems.

4. Let there be a number x such that x is an integer and x is > or = to 8. Prove that x may be expressed in the form 3a + 5b for some integers a and b.

So I know that x can be either odd or even and that is how far I got, and then I set forth some cases to show that they can be written in the form 3a + 5b, but I could not get to the general case as I would be assuming the very thing I am trying to prove.

4.9) Let b = a mod (n), prove that b^2 = a^2 (n). I got to the part that n divides a-b so a-b = nx for some integer x, now I am at a loss at how to manipulate the equation to yield a^-b^2.

If anyone has good encyclopedia like websites for knowledge please let me know, I will try to work on learning the ideas, thanks!
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June 19th, 2014, 09:32 AM   #5
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4.8$$) This is a question about modular arithmetic. If you work in $\mod{3}$ you need only to think about three cases: $n = 3k$, $n=3k+1$ and $n=3k+2$. $b$ need only take the values $0, 1, 2$ to be able to make any $n$.
4.9$$) What about $a+b$?
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