## Are the calculus MOOCs any good: After week 1 Saturday, Jan 12 2013

This is a continuation of a previous post.

I’ve been following the two Coursera calculus MOOCs: the elementary introductory to calculus being taught by Dr. Fowler of Ohio State University, and a course designed around Taylor expansions taught by Dr. Ghrist of UPenn, meant to be taken after an introductory calculus course. I’ve completed the ‘first week’ of Dr. Fowler’s course (there are 15 total), and the ‘first unit’ of Dr. Ghrist’s course (there are 5 total), and I have a few things to say – after the fold.

## An elementary proof of when 2 is a quadratic residue Thursday, Aug 23 2012

This has been a week of asking and answering questions from emails, as far as I can see. I want to respond to two questions publicly, though, as they’ve some interesting value. So this is the first of a pair of blog posts. One is a short and sweet elementary proof of when $2$ is a quadratic residue of a prime, responding to Moschops’s comments on an earlier blog post. But to continue my theme of some good and some bad, I’d also like to consider the latest “proof” of the Goldbach conjecture (which I’ll talk about in the next post tomorrow). More after the fold:

## A MSE Collection: Topology Book References Thursday, Jun 21 2012

Today, yet another question was posted on Math.Stackexchange (this time by the new user avatar) asking for topology references. This has been asked a few times before, but somehow the answers are a little bit different. So, as if I were responding to Ramana Venkata’s post on the meta about a consolidated topology resource, based upon the answers at MSE, and to facilitate topology references in the future, I am writing this blog post.

To be clear, this is a compilation of much (not all) of the discussion in the following questions (and their answers): best book for topology? by jgg, Can anybody recommend me a topology textbook? by henryforever14, choosing a topology text by A B, Introductory book on Topology by someguy, Reference for general-topology by newbie, Learning Homology and Cohomology by Refik Marul, What is a good Algebraic topology reference text? by babgen, Learning Roadmap for Algebraic Topology by msnaber, What algebraic topology book to read after Hatcher’s? by weylishere, Best Algebraic Topology book/Alternative to Allen Hatcher free book? by simplicity, and Good book on homology by yaa09d. And I insert my own thoughts and resources, when applicable. Ultimately, this is a post aimed at people beginning to learn topology, perhaps going towards homology and cohomology (rather than towards a manifolds-type, at least for now).

## A MSE Collection: A list of basic integrals Monday, Jun 11 2012

The Math.Stackexchange (MSE) is an extraordinary source of great quality responses on almost any non-research level math question. There was a recent question by the user belgi, called A list of basic integrals, that got me thinking a bit. It is not in the general habit of MSE to allow such big-list or soft questions. But it is an unfortunate habit that many very good tidbits get lost in the sea of questions (over 55000 questions now).

So I decided to begin a post containing some of the gems on integration techniques that I come across. I don’t mean this to be a catchall reference (For a generic integration reference, I again recommend Paul’s Online Math Notes and his Calculus Cheat Sheet). And I hope not to cross anyone, nor do I claim that mixedmath is to be the blog of MSE. But there are some really clever things done to which I, for one, would like a quick reference.

Please note that this is one of those posts-in-progress. If you know of another really slick bit that I missed, please let me know. And as I come across more, I’ll update this page accordingly.

## From the Exchange: Is it unheard of to like math but hate proofs? Monday, Apr 16 2012

A flurry of activity at Math.Stackexchange was just enough to rouse me from my blogging slumber. Last week, the following question was posted.

I have enjoyed math throughout my years of education (now a first year math student in a post-secondary institute) and have done well–relative to the amount of work I put in–and concepts learned were applicable and straight-to-the-point. I understand that to major in any subject past high school means to dive deeper into the unknown void of knowledge and learn the “in’s and out’s” of said major, but I really hate proofs–to say the least.

I can do and understand Calculus, for one reason is because I find it interesting how you can take a physics problem and solve it mathematically. It’s applications to real life are unlimited and the simplicity of that idea strikes a burning curiosity inside, so I have come to realize that I will take my Calculus knowledge to it’s extent. Additionally, I find Linear Algebra to be a little more tedious and “Alien-like”, contrary to popular belief, but still do-able nonetheless. Computer Programming and Statistics are also interesting enough to enjoy the work and succeed to my own desire. Finally, Problems, Proofs and Conjectures–that class is absolutely dreadful.

Before I touch upon my struggle in this course, let me briefly establish my understanding of life thus far in my journey and my future plans: not everything in life is sought after, sometimes you come across small sections in your current chapter in which you must conquer in order to accomplish the greater goal. I intend to complete my undergraduate degree and become a math teacher at a high school. This career path is a smart choice, I think, seeing as how math teachers are in demand, and all the elder math teachers just put the students to sleep (might as well bring warm milk and cookies too). Now on that notion and humour aside, let us return to Problems, Proofs and Conjectures class.

Believe me, I am not trying to butcher pure math in any way, because it definitely requires a skill to be successful without ripping your hair out. Maybe my brain is wired to see things differently (most likely the case), but I just do not understand the importance of learning these tools and techniques for proving theorems, and propositions or lemmas, or whatever they are formally labelled as, and how they will be beneficial to us in real life. For example, when will I ever need to break out a white board and formally write the proof to show the N x N is countable? I mean, let’s face it, I doubt the job market is in dire need for pure mathematicians to sit down and prove more theorems (I’m sure most of them have already been proven anyways). The only real aspiring career path of a pure mathematician, in my opinion, is to obtain a PHd and earn title of Professor (which would be mighty cool), but you really have to want it to get it–not for me.

Before I get caught up in this rant, to sum everything up, I find it very difficult to study and understand proofs because I do not understand it’s importance. It would really bring peace and definitely decrease my stress levels if one much more wise than myself would elaborate on the importance of proofs in mathematics as a post-secondary education. More simply, do we really need to learn it? Should my decision to pursue math be revised? Perhaps the answer will motivate me to embrace this struggle.

I happened to be the first to respond (the original question and answer can be found here, and I’m a bit fond of my answer. So I reproduce it below.

## Reading Math Saturday, Oct 22 2011

First, a recent gem from MathStackExchange:

Task: Calculate $\displaystyle \sum_{i = 1}^{69} \sqrt{ \left( 1 + \frac{1}{i^2} + \frac{1}{(i+1)^2} \right) }$ as quickly as you can with pencil and paper only.

Yes, this is just another cute problem that turns out to have a very pleasant solution. Here’s how this one goes. (If you’re interested – try it out. There’s really only a few ways to proceed at first – so give it a whirl and any idea that has any promise will probably be the only idea with promise).

## From the Exchange Thursday, Jul 28 2011

I speak of Math Stackexchange frequently for two reasons: because it is fantastically interesting and because I waste inordinate amounts of time on it. But I would like to again share some of the more interesting things from the exchange here.