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Teaching Calculus To 5-Year-Olds
Doofus writes "The Atlantic has an interesting story about opening up what we routinely consider 'advanced' areas of mathematics to younger learners. The goals here are to use complex but easy tasks as introductions to more advanced topics in math, rather than the standard, sequential process of counting, arithmetic, sets, geometry, then eventually algebra and finally calculus. Quoting: 'Examples of activities that fall into the "simple but hard" quadrant: Building a trench with a spoon (a military punishment that involves many small, repetitive tasks, akin to doing 100 two-digit addition problems on a typical worksheet, as Droujkova points out), or memorizing multiplication tables as individual facts rather than patterns. Far better, she says, to start by creating rich and social mathematical experiences that are complex (allowing them to be taken in many different directions) yet easy (making them conducive to immediate play). Activities that fall into this quadrant: building a house with LEGO blocks, doing origami or snowflake cut-outs, or using a pretend "function box" that transforms objects (and can also be used in combination with a second machine to compose functions, or backwards to invert a function, and so on).' I plan to get my children learning the 'advanced' topics as soon as possible. How about you?"
Doing the same thing 100x is only "simple but hard" if you can actually do it accurately. The point of that sort of practice is to make it easy.
Any teacher handing that out to someone who can already do it isn't doing their job properly. However, handing it out to someone who can't do it and needs to practice is perfectly reasonable.
In my experience, with young children your best chance at teaching them these things is to relate it to their current interest. My 4 year old is really into maps right now, he draws me one every day at his preschool. I've been showing him different maps and trying to relate the concept of directions etc. With his interest in drawing hopefully I can work in the alphabet at some point too. It's a tricky task to put things in terms a 4 year old mind and attention span can digest without overwhelming them.
I came to the datacenter drunk with a fake ID, don't you want to be just like me?
Ahh. I have the answer to that one! The answer is the same as "why does e^(pi * i) = -1", in a very non-obvious way, but it's very simple.
Why is the derivative of e^x = e^x? Because that's what makes e special - we picked 'e' to make that true. if you look at exponential curves for various bases, it becomes clear that somewhere between 2 and 3 this neat thing happens, and it turns out to be quite handy. If you play around with a graphing calculator it becomes obvious that it must be true for some number, and you can observe/discover "oh, that's e - so that's why its called the natural log".
Why is the derivative of sin(x) equal to cos(x)? Because we use radians. If you measure angles in degrees or grads or whatever, it doesn't work out this way. But if you study simple harmonic motion (which back in the days if record players everyone did), or just think about a point moving around a circle as viewed edge-on, it you will observe/discover that there's this neat property something moving that way: it's velocity as seen edge on is the same as it's position as seen edge on, rotated 90 degrees.. This is really visually obvious with a toothpick stuck to the outside of the spinning platter of a record player!
Once you grok that visually, then clearly there must be some way of measuring angles such that the derivative of sin(x) is cos(x), because that's what those functions mean: the position as viewed from the side, and the position as viewed from the side after rotating 90 degrees! It just so happens that choosing the range [0 2pi) for angles makes the math work out properly. Proving why it's 2pi and not some other value, like proving why it's e and not some other value, is a mess, but you can just observe that some such value must exist for both cases.
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The article didn't make this terribly clear, but people seem to be missing the point.
If you teach the concepts through hands-on interactive play, kids as young as five can understand the concepts underlying Calculus without too much difficulty. This also happens to be one of the best times in your life for learning, when the brain is rapidly forming new connections.
Her point is teach the concepts, teach the patterns, teach kids how to find patterns, and how to internalize mathematical knowledge.
The mechanical drudgery of formal language, writing out and solving equations, etc comes later on but builds on the fundamental understanding developed much earlier in life.
Natural != (nontoxic || beneficial)