MariaDroujkova

It's a deep question, and I'd like to warn that the answer is somewhat disturbing in its implications. Yes, some of it is WHAT is taught - the number crunching without patterns, the primitive yet tedious topics instead of beautiful adventures of the mind, medieval content not linked to current trends. The "what" part is relatively easy to address: there are wonderful materials out there! Innovative books, cool computer simulations, hands-on construction sets, etc.

And a part of the problem HOW math is taught: we do need to mind what we know about human learning, such as spaced repetition for memory, the power of multiple examples that come from your peer group, the motivation of making mathematics your own.

Yet the most difficult part that tends to stay off-screen is WHY math is taught. Advanced math is taught as a gatekeeper, as a means not to starve. It trickles all the way down - I hear parents of children as young as five or six say that if they don't push math now, the child will fail forever. To quote a presentation: "Why do we need to know multiplication? One reason is that multiplication is on many tests kids take. The story goes like this: if kids don’t know multiplication facts, they will fail tests, which means they won’t get into college, which means no career, which means epic fail of the whole life. For want of a nail, the kingdom is lost."

So people hate math because they learn it out of fear. How can we help kids learn math for meaningful, joyful, loving reasons? That's what it's all about...

Thank you for making this connection. Computing is one of the awesome ways for children to learn gentle, powerful math. The machine does child's bidding, it takes care of the tedious stuff, and the child has the power to work at the level of concepts, to fly! There are some very neat computer tools I've used with children as young as three - like Scratch from MIT.

First of all, let's be clear that a 5yo can understand the concept and idea of limits or integrals, but probably not computations of them from something like AP Calculus test. It's such a FUN task to ask yourself - how can I help a little kid to play with derivatives? And to figure it out! A lot of it is in storytelling and hands-on play.

Zeno's paradoxes or Hotel Infinity stories are perennial favorites for finite and infinite limits: https://en.wikipedia.org/wiki/Zeno%27s_paradoxes

Here are our hands-on activities for integrals and derivatives, from the Inspired by Calculus course. They have particular examples.

Integration as 3D printing, layers on an onion, LEGO building - "What Would 3D Printer Do?": https://drive.google.com/file/d/0B6enMfoYXJb3RjRzUks0bVpQaHc/view?usp=sharing

Integration as mosaic, beads, pixel art: https://drive.google.com/file/d/0B6enMfoYXJb3N0Z5SXFFdEdBeXM/view?usp=sharing

Integrals and derivatives in "X-ray vision" metaphor - making flipbooks! https://drive.google.com/file/d/0B6enMfoYXJb3cWxhUTB3SGlIM3M/view?usp=sharing

I hope these examples help to get started. There are more, like this interactive toy for beautiful rotations - kids love it: http://www.zefrank.com/string_spinv2/

I am sorry to hear you've been hurt by meaningless math. One of good ways to "get" math, as an adult, is to watch short, recreational videos about topics that weren't spoiled for you as a child. Like this, Infinity Elephants: https://www.youtube.com/watch?v=DK5Z709J2eo

You are probably doing TONS of applied math while fixing cars, by the way. I once took our math circle on a scavenger hunt to a car mechanic, and even little children found many different math ideas.

Imagine an interesting shape, like the Millennium Falcon. How much space does it occupy? Or maybe, how much plastic would you need to 3D print it? Now, imagine building that spaceship out of LEGO blocks. You can then count the blocks to estimate the volume. This, in a nutshell, is integration: building a shape out of easy, simple little shapes.