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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.
When I was a kid Mrs. Dunn (one of the parents of a kid at the school) taught an optional "math club" a half hour before school on Wednesdays. I don't remember exactly what we learned (it's since merged with all the "real" math classes I took), but I do remember learning sumnation and some other fairly advanced concepts.
Kids are smart, and they are totally capable of learning a lot of advanced math.
I plan to make sure my children understand what they're taught, and are taught new things based on what they already know. If that means teaching them complex ideas earlier than they would normally learn them then that's fine, but to make that a goal in itself is nonsensical.
Rocky's Boots.
'Nuff said.
I remember being in grade school and being irritated that for the 3rd year in the row I was learning how to do basic math. Then when I got to high school I was pissed off that I was rushed though from algebra to trig in 4 years. I don't think they understood that basic math is easy and higher math is hard and your math level has nothing to do with your grade level.
My school had a one afternoon per week gifted students program. Among other things we did programmed/self paced instruction and classroom work on boolean algebra and basic number theory. This was in the late 1960s in a middle class school district in suburban Pittsburgh (Avonworth.)
The other thing worth noting is how most mathematicians make their breakthrough discoveries before age 30. (Sorry don't have the reference for this, but I've seen it widely discussed.) So that means the earlier we expose kids "with the math gene" to more complex topics, the greater the possibility that stuff will 'stick'.
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?
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.
Dr. Maria Montessori, who before becoming a doctor and then an educator, was an engineering major and loved the math portion of it. Thus in her method that she devised 100 years ago, five-year-olds learn the 3D-geometric equivalent of binomials and trinomials from high school algebra.
u have to bury a treasure for him...
My God can beat up your God. Just kidding...don't take offense. I know there's no God.
For me, having been introduced to the basic idea of a "hard" concept made it a lot easier when the subject was taught in school ten years later. For example, basic cooking introduced me to a lot of math and a little chemistry. At age five, making lemonade was age-appropriate. It made sense that to make half as much lemonade, we'd use half as many lemons. (Ratios). Gee, we used one cup of sugar to make a big jug of lemonade, how much sugar should we use to make half as much? In school, fractions were easy for me - as easy as making lemonade, which I'd been doing for years.
I've said it before, but kids already do simple algebra in elementary school.
3 + [] = 5
and you fill in the box.
Yes and no. In one sense, that's an algebra problem, but not all elementary students are taught to solve it *as* an algebra problem.
I've often seen that problem given to a child like this: "three plus what is five? Come on, three, plus something, is five. What's the something? I have three, and if I add this many more, I get five..." That's not algebra, that's guessing. The child is often thinking "is it one? No. Is it two? Three plus two is five. Yes, its two."
Its only a real algebra problem if its taught this way: "Three plus what is Five? In this bucket I have five apples, and in this bucket I have three apples and some more apples, and they both have the same apples. If I take three apples out of this bucket and three apples out of that bucket, I should still have the same apples in both buckets right? Well this one has two apples left, which means that bucket must have...? How many?"
That's teaching rudimentary algebra to elementary students. The other version is twenty questions.
..Alan Kay's educationally oriented programming language
They said...
Most kids who take math don't learn math
Most kids who take French don't learn French
But, kids who grow up in France, have no problem learning French
We want to create "Mathland" where learning math is natural
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)
As the homeschooling parent of a 5 year old we have learned this first hand. We stumbled upon a set of books called Life of Fred that are "story books" that incorporate math. They were written by a math professor tired getting students that didn't know math and thought it was "hard". He incorporates basic algebra using x from almost the very beginning. They cover many topics that most think of as "advanced math" in simple, natural ways. As the story unfolds Fred has to use math in a variety of situations. It shows that math is practical and teaches it in an accessible way. Even better, the stories are silly and ridiculous and fun for all ages.
I recalled an /. article from 4 years ago with a completely different view of maths for children.
Here it is
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.
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.