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.

Can’t wait for lesson one

Excited to read this – great idea, Michael.

Thanks!

I guess this one is turning out like the safe.

Sorry, but I’m not sure I’m understanding this comment…

Like a safe, it’s easier to open from the inside (Houdini)? If you “grok” the beauty of maths (get inside it), the problems become more interesting.

That’s what I think it ought to mean, anyway :)

Well if that’s not what it means, I’m sure I’d like your interpretation better, so let’s go with it! :)

Nice introduction. Last fall semester i had my ass handed to me by trigonometry, but this only made me more curious. I like math, and want to have a better relationship with it, but am very afraid. Thank you for putting this site together.

No problem, my pleasure, glad you like it! And I’m also glad that you didn’t let your bad experience with trig scare you away from math. The problem with this subject is that it’s somewhat of an unstable equilibrium–it’s all built on what comes before and so one bad experience, or bad teacher, can turn away a student forever. It is rare, however, that one bad history teacher or English teacher turns the student away forever. This is just a challenge that math has to live with, and good on you for not letting that happen.

Great Work Done. As a retired teacher of Mathematics for college students I have become enthusiastic in your posts. So clear is the the understanding! Presently I am working to make Mathematics popular and lovable among the children. Your posts are helping me a lot to explain things in a better manner.

Awesome, thanks! I’m glad to hear that you’ve found it useful in your classes. If you don’t mind me asking, could you elaborate on exactly how you’re using it? Don’t worry, I’m not asking for any kind of credit or monetary reward or citation or anything like that (I promise), I’m just curious to know how you’re using these lessons, as I one day hope to implement this sort of stuff in a classroom setting. Thanks again!

good-work-sir

trig was not much of a buddy of mine…am hoping with your work….I can be inspired into being a mathematician…calculus did it for me…#motivation….am looking forward to the lesson man.

Awesome, I hope you enjoy!

Can’t wait to read through this whole site. I lost my love of math a long time ago, but all my mathematician friends keep telling me how amazing it is once you get out of the rut. This site looks awesome for doing just that!

Awesome, I really hope this site can do just that, or even at least play a small role in doing so. I hope you enjoy!

Hello,

I just encountered your site. This ‘part’ of math is exactly the reason why I flunked out of college. My teacher didn’t present the information very well and she didn’t even care to help us out when we asked for help.

I am ready to give it another try. Therefore I need to know if you still develop further lessons and if this site is still ‘active’?

Thank you very much

Yup! I am indeed still coming up with new lessons, etc., (albeit not with the frequency that I would like, although hopefully that will change soon). I hope you enjoy, and I hope this time around is a bit better!

Thank you very much, Michael! I will start my first lesson this afternoon!

Thank you again for your effort and your sharing personality!

Jakomien

My pleasure, I hope you enjoy!

Hi, I’m just finishing high school but unfortunately got behind in math and had to stop after algebra 2. However, I would love to fundamentally reunderstand what math really is. Would elementary level of math knowledge prevent me from understanding the content on your site? If so, what resources would you recommend to get back up to speed? I just finished learning about matrices and geometric series.

Hi John Doe, you should be just fine understanding this site with the background that you have. Just be sure to go slowly through difficult parts and allow yourself to reread confusing passages as many times as necessary to get a deep understanding of them. Best of luck, and do keep me posted on your progress!

Cheers!