One of the thoughts I've been developing over the past 10 years or so is that you can't teach someone the solution to a problem they don't have. This encompasses the "why" question but even goes beyond it, because even the answer to "why" is often just another level of "why" and/or "who cares", quite reasonably. To learn something, you need a problem, you need to grapple with the problem for a bit, and then you can be presented with a solution. Then, the answer to why is clear: Because it will solve this problem.

Education would still have to artificially give students problems; waiting for students to naturally have a problem for which taking the derivative of a tan function and then leaping in to discuss that just doesn't scale in all sorts of ways. But if we gave them problems first, and let them chew on them for a bit, I think it would work so much better.

But that's an anathema to the current system. It would require not moving students through in cohorts because you need to give students enough time to chew on things and there's no way that will be standardized. And of course it requires admitting that Very Smart People Are Totally Wrong About Education, and I might as well ask for a pony while I'm at it. The Curriculum Must Not Be Changed. The Curriculum Must Not Be Questioned.

I wish someone would pay me for a few years to try to develop a computer-based math curriculum based on this concept. I'm still waiting for education to move to the phase where computers are used as something other than "The Old Curriculum, But On A Computer!" I thought we'd be farther along on that path by now. I seem to have underestimated the inertia of the Holy Curriculum, Hallowed be its Name and Hallowed be The Heroes Who Practice It in its infinite glory, yea verily.

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.

Lol, I've been kicked out of class for asking the teacher to explain why something works in math here in the US

I think it was long division and lattice multiplication in elementary school

Doing math by drawing numbers in predefined shapes so that it magically worked out was the most ludicrous thing I'd ever seen

"Because that's how it works." Wasn't really a satisfactory answer lol.

It's hard to explain to young kids what "out of scope for this class" means. And that the number of people who need these skills vastly the number of people who need to understand the derivation.

Lattice multiplication will probably take algebra to explain. Long division definitely will require algebra. In my school, there's a gap of 5 years between teaching the two. A lot more people in the world need arithmetic than they need algebra (easily over a factor of 10). We can't put off teaching arithmetic till they learn algebra.

A lot of people don't realize that this problem goes all the way to undergrad and grad education in engineering or science. Laplace transforms are very useful, but they require complex analysis to begin to understand. If you blindly apply the integration that is normally taught, the Fourier transforms of several simple functions have integrals that simply, clearly do not converge. Yet we're taught tricks to indirectly calculate them. How is that possible? How do we get a result from something that clearly diverges?

And don't even get me started on the Dirac Delta function.

Recently I picked up an introductory analysis book - it starts from Peano axioms and builds up natural numbers, sets, integers, rationals, and then finally reals. It requires a fair amount of mathematical maturity to explain simple concepts, like how multiplying a positive with a negative could result in a negative, or how multiplying two positive numbers can result in an even smaller number (something that I did get upset about in my school days).

While yes, it is convenient to cherry pick examples where it was taught poorly without intuition, the reality is that if you want to prepare someone to go into, say, engineering, there is a lot of math one needs to cover, and teachers just can't afford to spend time explaining things that are way out of scope.

This is a really important point on this topic.

I don't believe (nor do I think you were claiming) that this means math cannot be taught more intuitively. An even broader example to your point is that Calculus itself was invented without foundation, at least without foundation modern mathematicians would find satisfactory; intuition patched some holes around 'infinity.' But Calculus could none the less be developed by blackboxing its inner workings and justifying its existence by just how spectacularly useful and predictive it was.

Likewise with arithmetic and algebra itself: ancient civilizations who factored numbers or solved equations did not require Peano's work to justify inventing some math. Calculus and all the math before it would indeed rest on foundations absent any Set Theory. The wider perspective is that discovery starts in the middle of a concept, and works out towards its implications and inwards towards its axioms.

The justification taught to those learning math (or anything) need not start from its axioms, but it should start from its history in context to how humans found use for the concept - that is justifying enough and likely more satisfying than learning axioms developed after the fact when the question is 'why?'

Much of the math part was taught in math classes. Engineering classes worked with the math classes to ensure the students were ready for the math there.

Chemistry, physics, thermo, dynamics, electronics, fluid mechanics, electronics, etc., were all math classes, in addition to a solid slate of required math classes.

If one didn't care for math, Caltech was a very very wrong place to attend :-) You either got good at it, or you left. I definitely felt that 4 years of that rewired my brain.

One of my friends in HS was against calculators, and also very stubborn, and got in trouble for asking how to do sin, cos, tan functions without a calculator.

Good old taylor series, of course!

That's certainly the hard way!

There are shortcut formulas, and before calculators a lot of calculations were done using drafting equipment.

Sine Tables Charts is easier if you can bring them to exam.

Just read off the axis on a unit circle. If you need more precision, draw a bigger circle.

Hi sirsinalot, I wrote a book that might be of interest to you: https://nobsmath.com/ = math for adult learners

Here is a PDF preview to get an idea of the contents: https://minireference.com/static/excerpts/noBSmath_v5_previe... and a standalone printable concept map: https://minireference.com/static/conceptmaps/math_concepts.p... There is A LOT to learn, but I assure you that as an adult you can understand the material much more easily, especially with our background in computing now.

I highly recommend you try rekindling your relation with math one day. It has lots of knowledge buzz moments, when you see things fit together. Feel free to email me if you have any questions, or you can post them on this forum for readers of the book https://gitter.im/noBSmath/community

Ditto. I've got more understanding of Math under mathisfun.com over months than in 4 years of edu in Spain akin to up to 10th grade in the US.

Contextless Algebra is not intuitive.