There is plenty of pure mathematics that is beautiful and worth learning on its own without any practical application. A great deal of joy I get from mathematics is the delight in seeing a novel structure I hadn’t before where the proofs fall effortlessly out of the definitions.

Most of pure math does have practical applications - to pure maths! My favorite professors never launched into a subject without a motivating example, even if that motivation was often "Look at x, y, and z. Aren't they awfully similar?". My first exposure to Abstract Algebra started with a little number theory, moved on to rings, then to ideals, and only then to groups. Many people I've talked to about it are surprised we took axioms away rather than adding them, but the way we learned motivated each step. Indeed, groups themselves were introduced with permutations. Similarly, I found measure theory was best introduced by showing how handy cardinality was for finite sets. A "practical" application would have been probability, so perhaps this wasn't exactly application focused, but we certainly didn't start from the definition and work our way out.

I feel the same joy when seeing a derivation of a novel algorithm where the effective procedure falls effortlessly out of the definitions. A good example is Dijkstra's derivation of Smoothsort[1]. It's worth noting that he was educated as a professional mathematician, not as a computing scientist.

[1] https://www.cs.utexas.edu/~EWD/transcriptions/EWD07xx/EWD796...

To be honest, even in pure math this approach ought to be taken. I love math too. But it's not really a very good pedagogical approach even in pure math to start out with a full week of unmotivated definitions.

I have no issue with the problems being posed being very abstract at a suitable level for the student. By the time you hit college, I have no issues with a professor introducing group theory with "Hey, look at this aspect of graph theory, and this aspect of topology, and this aspect of algebra... what commonalities do you think we could abstract from them?" But that's a way better introduction even at that level than "Let's spend 90 minutes giving unmotivated definitions and hoping you pick up the pieces later."

In a conventional school setting I expect the problems to be more concrete, by their nature. I can give another example myself: Taylor polynomials. In my opinion, they're one of the more important things to learn at that level. You can give the students a simple problem: "Having learned sin, cos, and tan, and by this point memorized some of the common values, please develop a procedure for taking an arbitrary sin/cos/tan of an angle." Give them some time to chew on it. They may even come up with some modestly clever things, maybe cover some more special cases or something. But then you can go into how we only "really" know how to add, subtract, multiply, and divide, and here's a tool that allows you to take a wide variety of functions that up to this point only existed in calculus and as magic buttons on your calculator, and turns them into problems we can do with real pencils on real paper using real human brains that do not come with a "sin" button. (And then, heh, be grateful you live in the 21st century and you don't actually have to.)

That's now how I learned them. I learned them as just "Here's some Taylor polynomials. Do these homework problems." And I did. I learned them, and could do the math. It wasn't until years later in my computer hardware class that I realized this is what was motivating them. (Not the literal hardware, because of course Taylor polynomials greatly predate that, but the need to be able to calculate these things prior to computers.) And I'm not saying "oh, that's what they are"; math very often has the characteristic that something is discovered for reason X but then has both mathematical and practical applications well beyond it. My point here is that my understanding of Taylor polynomials is now much richer than what I got in the class I learned them in... but there was no reason for that insight to be delayed and almost coincidentally obtained. It could easily have been conveyed via a different teaching method.

I think you have the right of it: it's hard to teach something like maths to someone who isn't curious or interested. And it is definitely difficult to hook someone's attention.

When my children were still babies and quite young I was reading Zvonkin's book, Math from Three to Seven. And when they reached that age I started playing games with them myself to try and introduce these ideas to them. Like Zvonkin I found that one of my kids was more keen than the other... but the only way to keep them hooked was to avoid the "M" word: maths.

What I think helped was to remind ourselves that we were playing games. Any time I went into an area that required calculation: determining some value -- they would catch on to that and shut down. However if we stuck to exploration and fitting things together and exploring games together I could keep them interested for an hour some days.

And as an adult that's what has kept me interested: Martin Gardners' articles in Scientific American and books; John Conway's playfulness (ONAG, the bloody game of life, etc) -- the stuff that wasn't simply rote calculation which I find many attempts at practical applications seem to focus on.

I can appreciate definitions and proofs now because I've learned the language well enough to piece things together. However it was the fun, the absurd, and the playfulness of the completely impractical that kept me going. Games, thought experiments, what-ifs. That sort of stuff.

Agreed with your overall point, and your specific example. Myself and one of my good friends I met in my physics classes in college both felt the importance of Taylor series had been massively undersold in our calculus courses, because it just kept coming up in our various physics courses. I learned it just as a thing that existed, but we kept relying on them when deriving things in courses like thermal dynamics or mechanics. We would joke that calculus professors should stop the class and just emphasize, "This is really important!" But of course, that wouldn't make the material land any better, for the reasons you've explained.

Re: computer based math curriculum, this is what Jason Roberts is doing with Math Academy here: https://www.mathacademy.us It's mainly targeted to kids, but has adult users as well. It is by far the best self-paced math program I've come across with the widest breadth (from the basics up to graduate level).

He's been working on it for years and talks about it a lot on his podcast: https://techzinglive.com

Edit: The link to use the beta software is here with more details on how the system works: https://www.mathacademy.us/beta-test-information

I’d subscribe to your math app. Khan Academy Kids has greatly accelerated my kids’ language and reading skills to the point where they were reading chapter books to us before Kindergarten.

Brilliant and KA seem to be the leaders in self-directed learning, but I’m still waiting for the device described in Neil Stephenson’s Diamond Age.