Teaching Calculus To 5-Year-Olds 231
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?"
Re:Mischaracterization of problem (Score:5, Interesting)
Up to a point, yes.
I'd point out: I can't do single-digit arithmetic without errors. I never have been able to. I can do math in my head pretty decently; one time on a road trip I got bored, and came up with a km/miles conversion ratio starting from a vague recollection of the number of inches in a meter. I came up with 1.61. Google says 1.60934.
But give me a hundred single-digit addition problems, and I will get a couple of them wrong.
Re:Mischaracterization of problem (Score:4, Interesting)
I agree. But I disagree with TFA's comment about "simple but hard".
Repetitive != Hard
Once you understand the concepts then doing 100 problems is no more difficult than doing 10. It just takes 10x longer to finish them all.
And that is the purpose of assigning a large number of tasks. Someone who does NOT understand the concept can work through 10 problems in an hour. Someone who DOES understand the concepts can do 10 problems in a minute.
So the 100 problem task is used to find those who did not finish because they did not have time because they did not understand the concepts.
Yes. Once they've completed the 100 problem task the first time they've shown that they've mastered the concepts so they can move on.
But we've become so focused on getting a grade (A, B, C ...) for doing the work that we've lost sight of WHY we were doing the work in the first place.
Clickbait Title (Score:4, Interesting)
This article does not contain any description of calculus-like activities that five-year-olds are participating in. There's a lot of 'this is cool' commentary without any description of what 'this' actually is.
Re:I had something similar as a kid (Score:5, Interesting)
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.
Or not teach them maths at all (Score:3, Interesting)
Here it is [slashdot.org]
Basically, during the depression Boston needed to make cuts to the public schools, so they cut maths from all of the schools in the poor neighborhoods until 6th grade. By 7th grade all of the students who only had 1 year of maths were at the level of the students who had 6 years.
It makes some sense to me, math is really just logic, and a child's brain is not wired for logic. Though, part of me also thinks that "math is a young man's game" and you need a way to identify the geniuses before it's too late.
Re:Introducing "advanced" concepts made it easier (Score:4, Interesting)
Exactly. One of the best things my parents did for me while I was growing up was provide "out-of-band" education of that variety. They'd introduce a concept without any of the trappings that typically surround a math lesson, giving me nudges and having me intuit how the concept worked, without putting any pressure on me to learn it right then. If I did, great, but if I didn't, no worries. It made the in-class lessons that came later on significantly easier, since they were just a formalized restatement of concepts that I already understood.
Aside from basic arithmetic, the stuff I pull out of my math toolbox the most often would have to be the way that geometry and calculus taught me to view the world. There aren't many opportunities to FOIL binomials in everyday life*, but if I have some scrap wood and need to figure out how to get the most out of it for a project, geometry has taught me a load of different ways to dissect that shape. If I have a problem that needs to be broken down for an algorithm, the basic idea behind integration (that you can take infinitely small cross sections and sum them together) has numerous applications. If I need a rough approximation of a volume, that same concept can be applied in my head in a few seconds, without any need for busting out a pen and paper or for remembering all of the dx/dy specifics.
And, really, much of that can be taught to kids at a young age. They don't need the "math" of it, so much as they need that way of viewing the world, and you can teach people at a young age how to break down things in those sorts of ways so that they can have an intuition for how things add up, without having to explain sigma notation or whatnot. When they learn integration by parts later on, they should have an "well of course it works that way" attitude, rather than the "wait, you can do that?!" attitude most people learning it seem to have.
* Funny story. I was at a Thanksgiving get-together a few months back, and a high schooler I know came by to ask me for help with her algebra II homework, since her parents hadn't been able to help and I was one of the people there with the most math lessons under my belt. I was able to help her to a point, but a lot of that stuff was just beyond my recollection since the last time I had used it was 15 years prior when I learned it, and without a textbook or other reference guide there, I wasn't able to help. In swoop about a dozen college students to the rescue...or so I thought. In talking it over with them, however, all of them either got stuck at or before the place that I got stuck, so I found myself working with them to try and reformulate the problem using calculus. Finally, a college freshman saved the day, since she had taken algebra II just a year or two prior and still remembered the thing we were all missing. Point is, it was pretty obvious that none of us had used that part of algebra II in the years since we were taught it, whereas calculus was something we all felt much more comfortable applying, despite the fact that it's supposed to be harder.