There is a world of mathematics out there that most people live and die without ever getting to see. In order to see this world, one usually has to complete an entire high school math curriculum as well as the first year or two of an undergraduate math major’s coursework before getting to start studying it. And by then, many students are already turned off by the subject, usually due to the emphasis in these earlier years on repetition, standardized testing, and rote learning. While these aspects of math are indeed important, they are not the whole story. This site is here to introduce its readers to the rest of the story, while trying to focus on the good parts.

It is generally thought that the type of mathematics that one studies in the second or third year of an undergraduate curriculum is too abstract and/or too rigorous to be introduced earlier. I beg to differ with this conventional wisdom, and believe that this “other world” of math can and should be introduced earlier, as it is this “other world” that often gets students to really fall in love with the subject. It is in this “other world” of math that we see, for example, exactly when it is the case that 1+1=2. In other words, the more you venture into this world, the more you see that this is not necessarily the case! And the reasons for this are a large part of what makes the subject so fun.

There is no doubt that to have a deep and thorough knowledge of the material presented here one must do a lot more than read this site. The aim here is simply to introduce and inspire. The math that is taught in middle and high school, while completely necessary, is only the tip of the iceberg. I claim, and attempt to prove via this site, that this iceberg gets cooler and cooler the more visible it becomes. To start exploring it, head on over to the lessons, where we explore this “other world” of rigorous, abstract mathematics in a completely user friendly way. No experience necessary, I promise!

Mathematical scales

When an aspiring musician takes to the piano for the first time, he or she does not spend all day playing scales. Instead, the budding pianist will play a few really easy and really fun songs that are used to impress his or her friends. After a little while some scales might be incorporated into the musician’s practice routine, but this is done with care because everyone knows that practicing scales sucks.

Imagine living in a society where every aspiring musician had to spend the first 10 years of his or her training doing nothing but scales. In other words, there is nothing but scales on the radio, on her iPod, in her classes, and on her practice schedule. Scales, scales, and scales. All day, every day. Only after about 10 years can the musician move on to more beautiful music if he or she desires. How many musicians do you think that society would produce? How many musicians would be scared away from music within the first year or two?

This clearly sounds absurd, right? Then why do we do this with our aspiring mathematicians? Almost every single potential future mathematician spends his or her first 10 years of schooling doing almost nothing but the equivalent of “mathematical scales”. Calculate this, simplify that. Jeez, shoot me now. Those who dislike or depart from math likely do so for this reason, and the good news is that this can change.

Mathematical “Mary had a little lamb”

Songs like “Mary had a little lamb” are great for young kids to learn on the piano because it teaches them that music can be fun and, well, musical. So why don’t we do the same for our budding mathematicians? Why don’t we teach our younger mathematicians the bits and pieces of real, beautiful, and abstract mathematics that makes the field so supremely cool? In other words, why don’t we teach the mathematical versions of “Mary had a little lamb”? To be honest, I don’t know why we don’t. But for those who want to put the days of “hating math” (if that be the case) behind them, or those who simply want to learn a bit more about what math is “really” all about, look no further than these lessons and this blog, where we shamelessly put “conventional” education behind us and focus on what’s interesting!

I’ll warn you, though, that you might end up really loving math.