I am writing this post because one of the best things happened to me while I was blogging my thought process. I wrote a blog about some mathematical questions from Polya when someone actually responded that I had been mistaken. Some of you might think this is not a big deal, but it actually is for me. The reason why it is important is because it attributes to my philosophy in learning mathematics. I believe in learning by trial and error, and a major emphasis on the error. I truly believe if you fail enough times at an objective, you will eventually fail to 'fail'. Hence, you will then achieve your objective.

The contribution of a reader (its hard to believe I get any) pointing out a new view on a thought process is a very important because it makes me a better mathematician. I put a lot of effort into writing these post so that if anyone needed help, then they can look up what I have written and receive help. One of these reason I try to put a lot of detail into the post is because there is a lot work that can go unseen to a reader.

The problem occurred when I was explaining what triangle numbers are, till this point I had not found a way to properly explain the significance of the pattern. The reader named Brian pointed out that the pattern was clear, the difference in triangle numbers is the natural counting numbers in order. At first, I was thinking he had to be wrong. Why was the pattern so clear? Then I realized afterwards, he was right. At the time, my mind was focused on the relevance of the pattern, while he was pointing out to something I said. He was right, I had miss typed my conclusion.

It is these little observations that I find are so valuable, I had just literally found another point of view. I think the more perspective you can obtain on a problem, the easier it is to solve. The only way I ended up with this valuable observation was to simply make the mistake in the first place. It can be difficult to write a blog post and calculate an equation at the same time. I worry about grammar, sentence structure, and density of a paragraph. Is what I am talking about making any sense? Typically, I work out a problem completely before I write up a conclusion. I felt this time 'I could get away with it', luckily I was wrong. Had I not typed up the wrong words, and he had not pointed out my mistake, then I would not have gained such valuable knowledge. This is what I like to call the 'essence of mathematical error'. This process is what I dedicate this website too, I want to show contrast. I want to show you the wrong way and the right way to solve a problem. Remember, there can be many correct ways to solve a problem, and even infinitely more ways to solve a problem incorrectly.

To me, these small opportunities are the fundamental building blocks of learning. I can now talk about triangle numbers and show people how I made such mistakes. With hope, my readers will in turn not make the same mistakes I made. I hope they make new ones, that way they can perfect their own skill even better than I had. My hope is to keep making useful mistakes, that way I can pave the road to understanding even more clearly.

These types of situations can be find in classrooms all over the world. You probably have noticed a similar situation, where the teacher was moving from 'step 5' to 'step 6' and some students got lost. That small disconnect is hard to pin point sometimes, it takes a long time to realize where the mistake was made. Think about how much time can be wasted just on trying to find where the confusion is, but the longer you spend the more reward you receive in understanding. Of course, when you try to pin point the confusion a lot of students who already understand can become really frustrated.

There is no shame in not knowing the answer, your skills is not lessened by lack of knowledge. Your skills decreases when you decide not to do something about your lack of knowledge. Half of the questions I received in the tutoring center began with the same statement, 'I don't know, but we will make it up as we go along!'. So I sat there with the student and pieced together the question slowly. Step by step, we found a clear path to understanding. While me and the student worked at the problem, we made at least a dozen mistakes. Some of the mistakes would seem absurd to other advanced mathematical students. This did not detour us from accomplishing our goal of clarity, sometimes we had to even give up.

I remember this one occasion where me and this student worked on a problem for a whole hour, by the end of that hour we had exhausted our brains. Finally, I was ready to stop trying and seek help from other resources. I look up on the Wolfram Alpha website and obtain the answer. Turns out me and the student were on the right track, we just had to literally take it two steps further. Had we not given up, then we would have came to the same conclusion. All those absurd techniques we applied to the problem helped us understand how to really remember the solution. Plus, it was really pleasing to know that we would have ended up with the same answer as the 'powerful' Wolfram Alpha engine. After spending a whole hour on one question, and you try all sorts of different techniques, there is a high chance you will remember it.

All these observations that I write about come from my personal experience of working on mathematics. I was an engineer drop out before I took up mathematics, so naturally I had a bit of a disadvantage. I never had a mathematical mind set like students that started in a mathematics degree. I was stuck with an 'engineering mind' set, not that I am complaining. The way I was trained, I learned to break down a system into individual pieces. I then observe each of these individual pieces separately, so I could better understand them as a whole. This mental thinking is how I work mathematical problems. I grew up with an 'engineering' mind set, its why I think everyone is an engineer in their own respect. If you practice something long enough, you begin to understand it. Eventually you will begin to know how it works and become an engineer of it. Kind of like how you have 'toaster' engineers, they may not know anything else but they do know how a toaster works.

Reference:

- Polya, George.
*Mathematics And Plausible Reasoning: Induction And Analogy In Mathematics*. 6th ed. Vol. 1. Princeton, New Jersey: Princeton University Press, 1954. 1-9. 2 vols. Print. *Wikipedia, the free encyclopedia*. N.p., n.d. Web. 04 Jun 2013. <http://en.wikipedia.org>. Path: http://en.wikipedia.org/wiki/Triangular_number.