Jeremy Kun

I have taught enough lectures to high school students (presenting "advanced proofs") and talked to enough geometry teachers who abandoned the two-column geometry proof to know that 1) is not true and 3) is not worth it. The problem is that people build up proofs like they're something to freak out about, or that the proofs that are presented are inherently mechanical because that's what students are taught. You can't expect someone learning to write proofs to be perfect any more than you can expect a first-time drawer to color within the lines. It's okay and should be embraced as an opportunity to reflect and improve. A proof is not complete just because you "got to the answer." It's complete when it's simple, elegant, and easy to explain to others.

I can and have explained beautiful proofs without the need for mechanical proficiency to ten year olds and mathphobes alike. Here are a few examples:

[1]: http://jeremykun.com/2011/06/26/teaching-mathematics-graph-theory/
[2]: http://j2kun.svbtle.com/things-mathematicians-know-proofs-are-beautiful
[3]: http://j2kun.svbtle.com/things-mathematicians-know-more-than-one-infinity
[4]: http://jeremykun.com/2011/06/26/tiling-a-chessboard/

The world is full of these cool problems and proofs. I could literally teach an entire course and do nothing but puzzles involving chessboards. That many teachers ignore these great topics is a problem, but it's certainly for a good reason (the myriad of other problems with high school education).

Jeremy Kun (who is worth reading beyond this article) has a post with very useful insights:

https://j2kun.svbtle.com/mathematicians-are-chronically-lost-and-confused

It begins (and there is far more of value in the post):

Many people who are in this position, trying to learn mathematics on their own, have roughly two approaches. The first is to learn only the things that you need for the applications you’re interested in. There’s nothing wrong with it, but it’s akin to learning just enough vocabulary to fill out your tax forms. It’s often too specialized to give you a good understanding of how to apply the key ideas elsewhere. A common example is learning very specific linear algebra facts in order to understand a particular machine learning algorithm. It’s a commendable effort and undoubtedly useful, but in my experience this route makes you start from scratch in every new application.

The second approach is to try to understand everything so thoroughly as to become a part of it. In technical terms, they try to grok mathematics. For example, I often hear of people going through some foundational (and truly good) mathematics textbook forcing themselves to solve every exercise and prove every claim “left for the reader” before moving on.

This is again commendable, but it often results in insurmountable frustrations and quitting before the best part of the subject. And for all one’s desire to grok mathematics, mathematicians don’t work like this! The truth is that…

To expand a bit further: this is an example of the "rabbit-hole" nature of mathematics. All but the most trivial theorems depend on previous results, and in most cases you cannot realistically follow all the dependencies until you get to the first principles, also known as axioms (and even then, there's the question of which axioms you are willing to accept!)

In order to be able to understand and appreciate mathematical proofs, you have to develop the ability to accept the truthness of a result - and to realize the consequences of it being true - even though you do not yet understand why it is true. You have to learn to accept the ensuing confusion as a natural state of mind (read [0] if this idea intrigues you).

[0] http://j2kun.svbtle.com/mathematicians-are-chronically-lost-and-confused

Ok. First I'm not taking lessons from anyone who plagiarizes. This is a re-write of:

http://j2kun.svbtle.com/you-never-did-math-in-high-school

Which may also be someone's work, but it was one I read recently.

Second, I learned some Calculus in school, how about you? That is not a subject a 1000 years old. So we haven't been teaching it that way for 1000 years.

Third, did this guy not have story problems? Applied Math is basically what this author is claiming we don't have.

Fourth, Graphing is pretty new. We didn't use to visualize plots because we didn't have the resources, I can't tell you when that became part of algebra, but it was in the past 100 years.

I can't believe LA Times ran such a poorly informed piece.