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Teaching Calculus To 5-Year-Olds

Soulskill posted about 5 months ago | from the finding-the-limit-as-age-approaches-zero dept.

Math 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?"

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231 comments

Mischaracterization of problem (5, Insightful)

Anonymous Coward | about 5 months ago | (#46399483)

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.

Re:Mischaracterization of problem (5, Interesting)

seebs (15766) | about 5 months ago | (#46399615)

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 (-1, Troll)

Anonymous Coward | about 5 months ago | (#46399649)

Are you retarded? Did you suffer a traumatic brain injury as a very, very small child?

Re:Mischaracterization of problem (0)

Anonymous Coward | about 5 months ago | (#46399713)

The best of the internet, ladies and gentlemen, right here!

Re:Mischaracterization of problem (5, Funny)

Mitchell314 (1576581) | about 5 months ago | (#46399735)

Yes, it's called embryonic development. It affects millions of people around the world and leads to impaired math abilities, where the affected cannot handle hundreds of mental calculations before making an error. The only known cure is to spend years in a basement alone eating cheetos, while insulting others' trivial math and lingual mistakes.

Re:Mischaracterization of problem (3, Funny)

Anonymous Coward | about 5 months ago | (#46400597)

I like you ideas, and would like both to subscribe to your newsletter, and build a train wreck of a web site where a multitude of these embryonically impaired people can co-mingle and share fantasies about Natalie Portman.

Re:Mischaracterization of problem (3, Funny)

gstoddart (321705) | about 5 months ago | (#46399949)

But give me a hundred single-digit addition problems, and I will get a couple of them wrong.

You may find you're aided by taking off your shoes. it's worked for me for years. ;-)

Very inconvenient at the grocery store though.

Re:Mischaracterization of problem (1)

camperdave (969942) | about 5 months ago | (#46400175)

You think taking off your shoes in a grocery store is bad... try settling a restaurant bill.

Re:Mischaracterization of problem (1)

LynnwoodRooster (966895) | about 5 months ago | (#46401535)

Yeah - remind me to never split the bill with you!

Re:Mischaracterization of problem (0)

Anonymous Coward | about 5 months ago | (#46400215)

Especially if you have to count to 21 (or 22 depending on what sex you are)

Re:Mischaracterization of problem (1, Funny)

SQLGuru (980662) | about 5 months ago | (#46400497)

This is slashdot. We all count in binary on our fingers. So -- FOUR.

Re:Mischaracterization of problem (0)

Anonymous Coward | about 5 months ago | (#46401113)

Actually, binary gets you a lot more options. 1 hand can count to 31 dec, as there are 32 dec states for a set of 5 bits. With shoes off, you should be able to count to (2^20 - 1) dec = 1,048,575 dec.

Re:Mischaracterization of problem (1)

Immerman (2627577) | about 5 months ago | (#46401283)

Whoosh. Count in binary on your fingers - represent four.

Re:Mischaracterization of problem (1)

ahem (174666) | about 5 months ago | (#46400521)

Wait! Are you saying that there's a sex with two penises?

Re: Mischaracterization of problem (1)

Nialin (570647) | about 5 months ago | (#46400697)

Dude did a reddit AMA and everything. It was awesome.

Re:Mischaracterization of problem (1)

Hognoxious (631665) | about 5 months ago | (#46401467)

In Thailand some can count to 23.

Umm, so I'm told.

Re:Mischaracterization of problem (3, Interesting)

khasim (1285) | about 5 months ago | (#46399913)

Doing the same thing 100x is only "simple but hard" if you can actually do it accurately.

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.

Any teacher handing that out to someone who can already do it isn't doing their job properly.

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.

Re:Mischaracterization of problem (2)

Ken_g6 (775014) | about 5 months ago | (#46400099)

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.

I disagree completely. Repetition leads to boredom. Boredom leads to difficulty concentrating. Difficulty concentrating makes it hard.

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.

When I started being home-schooled (for health reasons, not religious reasons), my Mom bought Saxon math books. They may still have a large number of problems, e.g. 100, but they mix up old types of math problems with newly learned types. That way I didn't forget old learning and I was less bored, while still learning new material.

Re:Mischaracterization of problem (1)

khasim (1285) | about 5 months ago | (#46400537)

When I started being home-schooled (for health reasons, not religious reasons), my Mom bought Saxon math books. They may still have a large number of problems, e.g. 100, but they mix up old types of math problems with newly learned types. That way I didn't forget old learning and I was less bored, while still learning new material.

Why would you forget addition if you were doing multiplication?

I disagree completely. Repetition leads to boredom. Boredom leads to difficulty concentrating. Difficulty concentrating makes it hard.

So you are saying that you would have trouble completing 100 addition problems right now because it would be "hard" for you? A task that a child could complete in 10 minutes.

Re:Mischaracterization of problem (1)

AK Marc (707885) | about 5 months ago | (#46401293)

So you are saying that you would have trouble completing 100 addition problems right now because it would be "hard" for you?

Yes. I would have trouble completing the 100 assigned problems, as I'd throw the paper in the trash and go do something more interesting. It is "hard" to sit for 10 minutes in a boring mindless repetitive task.

Re:Mischaracterization of problem (1)

khasim (1285) | about 5 months ago | (#46401449)

Yes. I would have trouble completing the 100 assigned problems, as I'd throw the paper in the trash and go do something more interesting.

And I think that that says everything that needs to be said on the subject.

It's not that it is "hard" it is that you do not want to do it.

Re:Mischaracterization of problem (1)

Hognoxious (631665) | about 5 months ago | (#46401547)

You're not a team player.

Re:Mischaracterization of problem (4, Funny)

Hognoxious (631665) | about 5 months ago | (#46401511)

my Mom bought Saxon math books

Yf Hrthringmir haet twee battleaxen, uend Gwindmir haet neu een, hoewveel Waeolces cowd yeach slaythen in an qvartel hooer?

Re:Mischaracterization of problem (1)

slinches (1540051) | about 5 months ago | (#46400309)

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.

You're assuming that the speed at which the problems are solved is positively correlated with fundamental understanding of the concepts. For problems like multiplication, this isn't really the case. Someone who memorizes the "times tables" may have a less complete understanding of the concept but finish quicker.

This is the flaw in timed math assignments with a large number of problems. It penalizes taking time to think about the problems and come up with the correct answer in favor of rote memorization. And worst of all is these tests are given before the students have had the time to fully learn the subject and recognize the patterns that would make memorization easier.

Re:Mischaracterization of problem (1)

suutar (1860506) | about 5 months ago | (#46401165)

When I was a kid, the 100 problem task was assigned as punishment, pure and simple. (It was also actually "these 10 problems, 10 times each".)

Re:Mischaracterization of problem (1)

AK Marc (707885) | about 5 months ago | (#46401273)

Repetitive != Hard

excessive repitition is hard in that it's difficult to stay motivated long enough to finish. Easy to do a problem, hard to do the entire assignment properly.

Re:Mischaracterization of problem (1)

Immerman (2627577) | about 5 months ago | (#46401521)

I would think the context was sufficient to clarify which of the several meaning of hard was being used.

By your own example doing 100 problems requires 10x as much time and mental energy doing 10. What word would you use to describe the increase in labor? Clearly digging a swimming pool with a spoon is qualitatively different than digging a seed-hole.

I had a horrible time with math in grade school, especially multiplication - my brain just doesn't store trivia well: 7*6 = ....? Couldn't tell you offhand. On the other hand I'm quite good at understanding and interrelating the underlying concepts - so I can say okay 6*7 = 6 + 6*6 (one that I do happen to remember) = 42, but that really increases the workload when doing it dozens or hundreds of times. Once I got to algebra, where it was understanding and application of concepts and patterns rather than memorization of trivia I excelled, and now one of my degrees is actually in mathematics.

Re:Mischaracterization of problem (1)

SaXisT4LiF (120908) | about 5 months ago | (#46400251)

Yes and no. There are certainly some benefits to repeated practice in developing the speed and accuracy of computations. The problem is that some people may never master these low-level computations due to undiagnosed cognitive disabilities (i.e. discalculia or problems with working memory) and this content is being used as a gate-keeper to higher-level mathematics which the person could potentially master with appropriate support. Different types of mathematical activities use different areas of the brain. Assigning more arithmetic practice to someone with a cognitive disability isn't going to magically make the problem go away, so why not focus on the math skills they *can* learn instead?

Re:Mischaracterization of problem (2)

avandesande (143899) | about 5 months ago | (#46400755)

Much like music a strong grasp of basic arithmetic helps you learn to visualize problems and develop an intuitive sense for math. I don't think there is any other way to get this other than practice.

Re:Mischaracterization of problem (1)

Hognoxious (631665) | about 5 months ago | (#46401651)

Not so sure. I knew a guy who had an MA (from Oxford, no less) in Maths and he absolutely sucked at mental arithmetic; he could never have worked as a bartender in the days before the tills did the magic for you.

On the other hand he knew many conceptual things that I'd never even heard of before.

That's not to say that learning your tables is useless; it's precaching commonly used calculations and burning them into your ROM.

I had something similar as a kid (2)

machineghost (622031) | about 5 months ago | (#46399531)

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.

Re:I had something similar as a kid (1)

NotDrWho (3543773) | about 5 months ago | (#46399753)

Some kids are smart

FTFY

Re:I had something similar as a kid (3, Insightful)

ackthpt (218170) | about 5 months ago | (#46399767)

The trick is getting to kids before their idiot peers who casually go around saying things like "Math is hard", "I can't do math, it's difficult", "Math is only for really super smart people."

Math is actually pretty easy, but once you've convinced yourself it's hard it becomes twice the battle, first to get past that mental barrier about how impossible it is.

Same applies to many areas of study. I was coding like a coding fool on National Coding Day and my High School counselor wouldn't let me into the programming classes because my math grades needed to be higher. Pfft, like math is more prevailing than logic. Anyway, plenty of misconceptions on what people are really capable of, particularly at a very young age.

I think there's a growing culture of morons who think you should molly coddle kids rather than get those little brains working during the time in their lives when they are capable of learning the fastest.

Re:I had something similar as a kid (4, Insightful)

lgw (121541) | about 5 months ago | (#46399827)

Calculus, taught properly, is incredibly easy and intuitive because it's all geometry - you can teach it visually, with no numbers.

Area under a curve? No harder to understand qualitatively than the area of any other shape. Slope of a curve at a point? Again, quite easy to understand with construction paper cut-outs of curves, and a ruler.

And there are plenty of real physics problems that can be solved with simple geometry! Make a drawing of velocity over time that tells a story of a trip. With constant acceleration, all the shapes will be triangles and rectangles. Find the area to find the distance travelled.

For actual curves, you can make them from wood and weigh them to find the integral. Awesome hands-on fun that completely de-mystifies calculus. Not sure a kid would be ready for it by 5, but 8-10, no problem.

Re:I had something similar as a kid (2)

CastrTroy (595695) | about 5 months ago | (#46400409)

Doing hands on geometrical calculus is easy, and can be understood quite easily. What's I actually found difficult, was not the concept, but the memorization of how go obtain the integral or derivative of a functions. So many rules, that seemingly had no logic to them. The derivative of sin(x) is cos(x). Why? most students probably couldn't tell you that. Looking at a proof I found [math.com] , it actually seems quite non-obvious, and not something most beginner calculus students could figure out on their own.

Re:I had something similar as a kid (5, Interesting)

lgw (121541) | about 5 months ago | (#46400839)

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.

Re:I had something similar as a kid (1)

CastrTroy (595695) | about 5 months ago | (#46401063)

While I probably didn't pick the best example, as looking at the graphs of sin(x) and cos(x) you can see that one is the derivative of the other, there are plenty of more complicated rules out there. Polynomials are quite easy as well. But once you get into more complicated functions applying all the rules can be frustrating. Often the questions are more about reducing a function to something else that's easily derivable than about how to actually find the derivative.

Re:I had something similar as a kid (2)

lgw (121541) | about 5 months ago | (#46401147)

Sure, but by that point you're doing computation, not learning the principle involved. Few people find doing computation to be the fun or interesting part of math, which is why we automate it. Doing enough exercises to be good at it, like memorizing multiplication tables, is worthwhile eventually, but it's a terrible place to start.

Re:I had something similar as a kid (1)

AK Marc (707885) | about 5 months ago | (#46401407)

Calculus, taught properly, is incredibly easy and intuitive because it's all geometry - you can teach it visually, with no numbers.

The problem isn't that the students are dumb, but that the teachers aren't allowed to teach to the children in the class. The materal and methods are set for the state, and there's little negotiation available.

How about me? (4, Insightful)

kruach aum (1934852) | about 5 months ago | (#46399567)

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.

Re:How about me? (2)

metlin (258108) | about 5 months ago | (#46399965)

I have always wondered why puzzles were never included in any educational system. Logical puzzles, spatial manipulation, patterns, and lateral thinking challenges go a long way towards improving general intelligence and learning abilities. Much more so than, say, memorizing multiplication tables. It also helps them with those complex ideas that you spoke of.

Instead, kids are taught to hate math and hate puzzles, and standardized tests are a joke.

My grandfather was a mathematician and he taught me that geometry and algebra were essentially the same when I was about 7. So, as I grew up, I could "visualize" every equation and that improved my problem solving ability. I cannot help but feel that teaching multiple complex ideas earlier will help children's creativity as they learn to combine them (i.e. spatially visualize a problem to look for patterns and use that to solve it as an algebraic equation).

Re:How about me? (1)

ShanghaiBill (739463) | about 5 months ago | (#46400165)

I have always wondered why puzzles were never included in any educational system. Logical puzzles, spatial manipulation, patterns, and lateral thinking challenges ...

My kids attend public school in California. Their math assignments regularly include puzzles of all the types you mention, as well as other recreational math, often adopted directly from the grand master [wikipedia.org] .

Instead, kids are taught to hate math and hate puzzles, and standardized tests are a joke.

My kids like math, enjoy the puzzles, and the standardized tests (at least in math) are quite good (and often include questions requiring insight, that most people would consider "puzzles").

Introducing "advanced" concepts made it easier for (2)

raymorris (2726007) | about 5 months ago | (#46400327)

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.

Re:Introducing "advanced" concepts made it easier (3, Interesting)

Anubis IV (1279820) | about 5 months ago | (#46401573)

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.

DragonBox Algebra App (1)

Anonymous Coward | about 5 months ago | (#46399589)

The DragonBox Algebra App is intended to teach Algebra to 5 year olds using very similar mechanics and incentives as Angry Birds.

Rocky's Boots (3, Funny)

Etherwalk (681268) | about 5 months ago | (#46399631)

Rocky's Boots.

'Nuff said.

Re:Rocky's Boots (1)

ackthpt (218170) | about 5 months ago | (#46399911)

Rocky's Boots.

'Nuff said.

I really loved playing the original Paradroid on the C64. Had to beat computer intelligence at toggling logic gates. Makes you think much faster when looking at logic circuits later. :D

I agree with all of the things. (4, Insightful)

dicobalt (1536225) | about 5 months ago | (#46399701)

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.

Re:I agree with all of the things. (2)

NotDrWho (3543773) | about 5 months ago | (#46399845)

The reason that you were irritated is because you were one of the smart kids. I felt the same way in school, until one day a teacher told me that they weren't constantly reviewing the basics for *me*. They were doing it for the other 90% of the kids in the class who weren't like me.

If my parents had been able to afford a private school, or if I had access to a "gifted" school, it would have been different (and much better). But in a public school, you can't fault teachers for having to teach to the lowest common denominator. They can't leave the dumb kids behind just because we're smart. We can't forget that just because we're an unrepresentative sample on slashdot.

Re:I agree with all of the things. (0)

Anonymous Coward | about 5 months ago | (#46400371)

The reason that you were irritated is because you were one of the smart kids. I felt the same way in school, until one day a teacher told me that they weren't constantly reviewing the basics for *me*. They were doing it for the other 90% of the kids in the class who weren't like me.

If my parents had been able to afford a private school, or if I had access to a "gifted" school, it would have been different (and much better). But in a public school, you can't fault teachers for having to teach to the lowest common denominator. They can't leave the dumb kids behind just because we're smart. We can't forget that just because we're an unrepresentative sample on slashdot.

Yes, that's what the teachers think they're doing.

In practice what they're doing is training children that they don't have to learn they juts need to "cram" before the tests because no one actually expects them to retain nontrivial amounts of their skills from year to year until they get to high school/college.

Re:I agree with all of the things. (1)

camperdave (969942) | about 5 months ago | (#46400639)

Back when I was in grade school, probably grade 4 or 5, there was this reading comprehension system. It had a bunch of colored levels, and on each level there would be ten booklets. Each booklet had a story and a question sheet. You would mark your answers on an answer sheet using the same color pencil crayon as the level you were on.

They should develop the same sort of thing for mathematics.

Re:I agree with all of the things. (1)

pjt33 (739471) | about 5 months ago | (#46400935)

They had something similar for mathematics [nationalst...tre.org.uk] back when I was in primary school, except that rather than 10 booklets there were dozens of cards. The teacher would assign each pupil 10 cards, and then we could do them in the order we wanted (as long as no-one else was using the card we wanted). I loved it.

Gifted classes (0)

Anonymous Coward | about 5 months ago | (#46401049)

When I was a kid, my school offered advanced courses for "gifted" students... But from what I've heard, this is no longer the case today due to "No child left behind." It seems the opposite of the direction we should be going, providing a challenge for 90% rather than boredom for 10%.

Teaching Abstract Algebra to preschoolers (1)

davidwr (791652) | about 5 months ago | (#46399731)

We teach preschoolers some specific examples of abstract algebra:

Today is Friday. Friday is the 6th day of the week. What day will it be 3 days from now? *hold up a calendar*

It is 11 o'clock. What time will it be two hours from now? *hold up an analog clock and point to the hour hand*

You get the idea.

Re:Teaching Abstract Algebra to preschoolers (1)

NatasRevol (731260) | about 5 months ago | (#46400069)

Adding numbers doesn't compute as abstract algebra to me.

Boolean algebra & number theory in 5th grade (2)

david.emery (127135) | about 5 months ago | (#46399737)

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'.

NYC schools already doing it (1)

alen (225700) | about 5 months ago | (#46399773)

in first grade there are pre-algebra and problem solving concepts being taught now. at least in my kid's public school
last night i had a huge argument with him about the proper strategy to use to solve a problem. i had to google the common core lesson plans to help him

Problem Has Been Solved For Generations (1)

LifesABeach (234436) | about 5 months ago | (#46399791)

Teach the children Art, and Music.

"Advanced" topics??? LOGIC? (0)

Anonymous Coward | about 5 months ago | (#46399807)

If you want to teach your kids advanced topics, start with LOGIC. Normally, logic and proofs are only introduced in universities. But there is nothing stopping a 5 year old from learning that stuff. All you need is basic arithmetic, at most.

Fields, vector spaces, topology, etc. The actual logic and thinking behind them are not out of grasp of a 5 year old with a good teacher. You know, definitions, and even theorems with proofs. Like why 5+2 is larger than 3+3? Use field axioms alone to prove that.

If that is beyond what you know, then give up. Teaching kids advanced arithmetic (ie. calculus) when they don't know basics is stupid and will just result in frustration.

father of 4 year old, align with interest is key (5, Insightful)

trybywrench (584843) | about 5 months ago | (#46399809)

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.

Re:father of 4 year old, align with interest is ke (2)

schlachter (862210) | about 5 months ago | (#46400269)

u have to bury a treasure for him...

Cognition (0)

Anonymous Coward | about 5 months ago | (#46401719)

A five year old is not a fully developed adult. They lack certain cognitive abilities in general, such as an understanding of the lingual construct of the passive voice. You are usually doing well if the child at that age understands the concept of whole numbers (0, 1, 2, 3, 4, etc) and limited rational real numbers.

I strongly doubt that children at that age can generally understand a limit problem or the idea that numbers are infinitely divisible. In basic calculus, you are calculating the area under a curve by adding up an infinite number of zero-width trapezoids and coming up with a number that may be negative.

Age Appropriate? (0, Funny)

Anonymous Coward | about 5 months ago | (#46399825)

I'm curious how age-appropriate calculus is to a 5 year old. Perhaps we need a car analogy? Calculus is to a 5 year old as a car is to a dead person. Yes you can give a dead person a car... and with scaffolding give them the support needed to drive... and with autonomous cars they can get around where they need to go... Wouldn't we all be better off if you gave that car to a living person and and a coffin to the dead person? Instead of getting your kid to learn calculus, try teaching them how to pilot drones and blow up women and children with no remorse so they can get jobs with the US Army when they grow up.

Caluclus is not inherently hard (0)

Anonymous Coward | about 5 months ago | (#46399885)

you can make calculus - or any other subject - arbitrarily hard by choice of problem.
But if you get the idea of x and y as variables - a really hard concept - then there is no reason you can't do some integration and differentiation.

high school math teacher chiming in (0)

Anonymous Coward | about 5 months ago | (#46399901)

These are commonly called "rich mathematical tasks." The idea is that the problems have a low point of entry, and a high ceiling for extension. They are also called "open ended" problems.

Some websites that come to mind that may be good resources are http://nrich.maths.org/frontpage and Dan Myers blog (Dan Myers pretty much goes across the country discussing these sorts of problems).

Clickbait Title (3, Interesting)

Capt.Albatross (1301561) | about 5 months ago | (#46399915)

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: Clickbait Title (2, Funny)

Anonymous Coward | about 5 months ago | (#46400065)

"And I'm gonna be a really cool parent." Then reality sets in.

Re: Clickbait Title (1)

avandesande (143899) | about 5 months ago | (#46400705)

That's so true. You start out with all kinds of high goals for you kids and by the time they are teenagers you are just happy if they stay out of trouble and will be able to take care of themselves when they are an adult.

Abstrate Thinking skills may be required (1)

oxnyx (653869) | about 5 months ago | (#46399927)

I took a lot special education due to having dyslexia. However the memory that sticks out the most is sitting in a Normal Grade 10 Math Class in High School with 3/4 of the class very upset about the idea of "x" entering into Math. Really they had a HUGE problem with it. Apparently that normal because you actually need to get quiet a long way before you develop abstract thinking skills. Now I'm sure some of the children of people on slashdot are really good at it at a young age however there are limits. Similar ever try to teach someone in Grade 2 about Irony or Theme in English Lit? I understand one of the hardest things to do is as Magician is impress 2 years with a rabbit coming out of a top hat because they do not see an reason why that would not be normal. Get the kids good at factoring and able to handle BEDMAS. It sort like says my child will be independent at 18 therefore at 2 his/she should know 1/9 of all though skills but can't keep their room clean. Younger children may be the best learn but older people can handle more complex ideas if slower.

Algebra in elementary school (1)

mattack2 (1165421) | about 5 months ago | (#46399957)

I've said it before, but kids already do simple algebra in elementary school.

3 + [] = 5

and you fill in the box.

Re:Algebra in elementary school (2)

dnavid (2842431) | about 5 months ago | (#46400449)

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.

Interesting idea (1)

ErichTheRed (39327) | about 5 months ago | (#46399969)

I think that one of the problems with the way math is taught in schools is the fact that very little is done to explain how calculations students are doing can be applied to actual problems. Now that I'm older, went through a science education in college and work in a technical field, I understand this. However, one of my problems early on was that I never really felt comfortable doing math problems. It sounds really stupid, but I must have some sort of disability -- I can't do basic arithmetic in my head. The numbers just don't stick in my head the way they need to when you're doing multi-column addition or multiplication. My wife, a finance wizard, laughs at and pities me at the same time when I'm manually figuring out a tip. When I was learning math back in the Jurassic period, the students who were "good at math" were the ones who could easily do calculations in their head and just had a feel for numbers. Calculators in the early grades were unheard of back then. And this skill is still what a trader needs -- they need to be able to make a decision in 5 seconds based on a calculation they do in their head. It's also a skill you need to do well on the SATs, since they basically contain two 30-minute timed algebra and arithmetic tests.

What I'm saying is that math is more than basic arithmetic and algebraic manipulation. If you can get a student to understand what you mean when you say exponential growth, and how it relates to something they care about, then students will understand it more. I remember hating grade school math with the endless arithmetic drills, and later, the rote memorization of procedures for fractions, long division, etc. I also remember going through high school algebra just memorizing the exact steps to complete the crazy factoring/simplification problems and not understanding _anything_. It literally took me until about halfway through high school, when science classes actually got somewhat challenging and delivered meatier material, to make any sort of connection.

Calculus and other applied math should be at least touched on earlier on in the school career. I think it would help students who don't necessarily have the skills that would make them "good at math" to at least understand some of it. People I know who understand math well say it's like a foreign language, so maybe we should be teaching useful phrases for travellers more than we teach verb conjugation and sentence structure...

Please answer my question, it's really bugging me (-1)

Anonymous Coward | about 5 months ago | (#46399993)

I have always had this question (sorry it's off topic)... when you read soulskill, is it read like:
Souls kill? or Soul Skill?

Montessori trinomials (2)

michaelmalak (91262) | about 5 months ago | (#46400091)

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 [cabdevmontessori.com] of binomials and trinomials from high school algebra.

Teach Logic instead (1)

pieisgood (841871) | about 5 months ago | (#46400095)

If you want to prepare children for higher level mathematics and all that learning it implies, please start with logic. The idea of teaching young kids calculus is a bit absurd and not nearly as helpful as a foundation in logic. When you have a malleable mind that is still growing and rapidly changing giving an early foundation in how to think critically and how to approach abstract questions would seem to have a larger benefit than having them think about calculus.

Re:Teach Logic instead (0)

Anonymous Coward | about 5 months ago | (#46400375)

You must not have kids.

Teaching kids logic and critical thinking is something that starts at age 0 and goes through at least age 18, if not longer. (Just getting children to think about cause and effect or getting them to learn to ask who, what, why, where, when, and how after hearing, reading, or seeing anything takes years.)

Teaching kids the principles of calculus - i.e., the geometric interpretation of derivatives and integration - is easy, takes about 5 minutes, and can be done with children at least as young as 9 years of age. Furthermore, when they actually see those topics introduced in the classroom at age 17, they will think back and say to themselves, "hey, I've seen this before and understand the concept, totally not scary at all, now to just grind through the mechanics of computation for these concepts, which is pretty easy", resulting in complete success at calculus, instead of getting totally intimidated and/or lost due to some crappy teacher teaching the subject and failing miserably at communicating the core concepts of calculus to the students.

Re:Teach Logic instead (0)

Anonymous Coward | about 5 months ago | (#46400507)

You must not know any math.

http://en.wikipedia.org/wiki/M... [wikipedia.org]

Teaching your kids how to get beat up at school (0)

Anonymous Coward | about 5 months ago | (#46400221)

You're doing a great job so far!

How about me? (1)

khr (708262) | about 5 months ago | (#46400277)

I plan to get my children learning the 'advanced' topics as soon as possible. How about you?

I hate children, you insensitive clod!

Trivial (0)

Anonymous Coward | about 5 months ago | (#46400285)

Early exposure helps determine inherent talent, like music, dance, sports or art.

But like most activities, it depends on if the person takes it seriously.

The idea is trivial in sports families. Why not in academic families?

Continued exposure is good (1)

istartedi (132515) | about 5 months ago | (#46400331)

Even if they don't get it the first time, continued exposure is good. I can think of a lot of things in math that didn't "click" until I'd heard it the umpteenth time. For example, how to count to umpteen.

I think a little bit of "modern" math is good but the old stuff still needs to be taught. Rote memorization gets a bad rap; but IMHO the 10X10 multiplication table should be committed to memory just like the alphabet. All else equal, a student with the table in his head will be able to work more quickly and confidently than one without. Notice I said 10X10 table. An odd thing is that they taught us 12X12. I think it's a tradition held over from the English system, where you had 12 inches in a foot. 12*12 even has the name "gross". There's nothing wrong with teaching the traditional table; but it would be nice if they put a red line or something around the 10X10 portion of it so that students understood the significance of that--that 10X10 is the key to unlocking virtually unlimited multiplication abilities with pen, paper, and the simple algorithm that "old math" taught us.

Teaching != Education (1)

dnavid (2842431) | about 5 months ago | (#46400359)

Teaching Calculus to five year olds is stupid. But that's not really what the article describes. I think a critical distinction implied by the article but not stated is that there's a difference between rigorously teaching a topic to the point of mastery, and exposing children to a topic to make them familiar and comfortable with it.

I've always believed that 50% of time in school should be spent rigorous teaching, and 50% of the time should be spent easing students into more complex topics over time. I think its not productive that a student is not exposed in any way to a topic like statistics, say, and then suddenly they land in the class and have to learn the topic from scratch starting with little or no familiarity with the topic at all. I think the article suggests that rather than spend all of your time practicing arithmetic, it would be more beneficial long-term to start introducing complex topics at a level where the students aren't being asked to demonstrate complete mastery. You aren't going to give a five year old calculus homework. But a five year old exposed to mirror books, say, becomes a seven year old that is familiar with the concept of iteration and can be introduced to the notion of infinite series. That seven year old becomes a ten year old that is comfortable with the concept of summation, even if they aren't masters of the formulas. But when geometry comes along at thirteen, they would be a lot more comfortable with construction, with algorithms, with geometric limits.

Basically, these days most schools teach single-point classes. What you learn in this class has little to do with the next class. Learning geometry doesn't help you learn probability, and neither help you much in learning calculus. Its all learn today, forget tomorrow, dive into the deep end of the pool for the next topic next year. But subjects like algebra and calculus and statistics are based on concepts and ways of thinking that are not intuitive or trivial for most people. Taking the long view, and investing time today to make it easier to learn those topics tomorrow is I think what the article is really talking about, and its a notion I happen to agree with 100%.

You can take this to silly levels. Actually trying to teach calculus to a five year old, or even a ten year old, is ludicrous. 1% might get it, the other 99% will just get confused or frustrated, or worse oversimplify to the point of error just to pass the class and then face even worse hardship when they have to learn it "for real." But introduction, familiarization, and slow incremental acclimation without overzealous forcing is probably the best way to both teach and keep interest in many topics, not just math.

Re:Teaching != Education (1)

Anonymous Coward | about 5 months ago | (#46400469)

NOT teaching calculus to 5 year olds is stupid. They are capable of learning it, just like they are capable of learning so many other things (such as language) we do not teach them when their brains are at the ripest age for learning because we're too busy indoctrinating them with one sociopolitical dogma or another.

I was doing both calculus AND speaking functionally in French and Portuguese when I was 7, because the Montessori school offered it, and did not focus on indoctrination with one or another social or political agenda. Teaching me right from wrong was my parents' job.

I taught my kid calculus at 5 (0)

Anonymous Coward | about 5 months ago | (#46400369)

A ream of paper is all you need to get a kit to understand the concept of slicing things up into infinitesimally small pieces.

Here's a ruler. What's the volume of this ream of paper?

What if I halve the ream, measure the volume of each piece, then add them together?

What if I measure the volume of each individual piece of paper and then add it all up?

Congratulations, you just learned what integration is. Now let's look at some other shapes.

It's not that hard to teach a kid calculus once you get by the false notion that kids can't do calculus.

I saw a presentation on Squeak (2)

MpVpRb (1423381) | about 5 months ago | (#46400517)

..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

Stop it. Stop it with Fire. (1)

WillAffleckUW (858324) | about 5 months ago | (#46400551)

Seriously, they are KIDS.

Stop trying to turn them into robots.

Let them PLAY.

This is just beyond the pale.

I remember physics in college (0)

Anonymous Coward | about 5 months ago | (#46400593)

Imagine how astonished I was that all that bullshit they make you do with algebra in high school was rendered trivial by derivatives, integrations and matrices.

Explain it to me like I'm 5 (1)

swb (14022) | about 5 months ago | (#46400737)

In college, I wish someone would have explained Calculus to me like I was five. I might have done better than a C.

Having it taught to me by a passable English speaker would have also helped.

What? (1)

Charliemopps (1157495) | about 5 months ago | (#46400811)

What a shameless and ridiculous headline. 5 year olds can't even usually read... or count above 100. I just got my 6 year old to understand that 0 comes before 1 for gods sakes and he's the smartest kid in his class. If building legos is calculus than I'm a god damned genius. WTF is this even about?

I've always wondered about this (0)

Anonymous Coward | about 5 months ago | (#46400827)

I think kids should definitely learn algebra and geometry from a much earlier age. It helps give context to the numbers.
If children come at problems as understanding what functions and variables are at a deep fundamental level, I think it would open up a whole new world

That said, as the parent of an 8 year old, I can tell you that drilling on times tables and multi digit addition and subtraction, basic simplification problems have lessons of their own... It helps kids develop self discipline, endurance and dexterity.

no thanks (0)

Anonymous Coward | about 5 months ago | (#46401023)

This is called 'Exploratory math' here in GWN and has led to a drastic decline in kids math skills. The only part of the country that has not experienced a regress in this area is QC - they keep using traditional methods.

People are missing the point (5, Insightful)

rabtech (223758) | about 5 months ago | (#46401027)

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.

Drill and Kill (1)

k6mfw (1182893) | about 5 months ago | (#46401301)

that's what teachers call "timed tests." Very popular because easy to prepare, conduct, and grade. But getting into stuff like the number line, proportions, ratios, rates of change, etc. it becomes abstract. However, I wish I was given the number line and also do graphs in elementary school instead of waiting for college. I mean a number line that shows negative numbers. No need to get into complex graphs but can do stuff like plot quantities of stuff compared to other things.

Why Do We Study Math in School? (1)

k6mfw (1182893) | about 5 months ago | (#46401363)

from a elementary school teacher in 1990s:

Because it's hard and we have to learn hard things at school. We learn easy stuff at home like manners.
Corrine, Grade K

Because it always comes after reading.
Roger, Grade 1

Because all the calculators might run out of batteries or something.
Thomas, Grade 1

Because it's important. It's a law from President Clinton and it says so in the Bible on the first page.
Jolene, Grade 2

Because you can drown if you don't.
Amy Beth, Grade K

Because what would you do with your check from work when you grow up?
Brad, Grade 1

Because you have to count if you want to be an astronaut. Like 3... 2... 1... blast off!
Michael, Grade 1

Because you could never find the right page.
Maryanne, Grade 1

Because when you grow up, you couldn't tell if you are rich or not.
Raji, Grade 2

Because my teacher could get sued if we don't. That's what she said. Any subject we don't know--wham! She gets sued. And she's already poor.
Corky, Grade 3

introductory calculus for infants (1)

Anonymous Coward | about 5 months ago | (#46401399)

Introductory Calculus for Infants clearly:

http://www.amazon.com/Introductory-Calculus-For-Infants-Inouye/dp/0987823914/ref=sr_1_1?ie=UTF8&qid=1393968663&sr=8-1&keywords=calculus+for+infants

Rather than Calculus... (1)

mpetch (692893) | about 5 months ago | (#46401493)

I think it might be more beneficial to teach statistics.

3 Words: Life of Fred (3, Informative)

artisteeternite (638994) | about 5 months ago | (#46401495)

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.

Or not teach them maths at all (3, Interesting)

GKThursday (952030) | about 5 months ago | (#46401499)

I recalled an /. article from 4 years ago with a completely different view of maths for children.
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.

Sounds Like Dumbing Down to Me (1)

Zalbik (308903) | about 5 months ago | (#46401501)

Is it just me, or is the education system getting far too concerned with "keeping children engaged" and "making learning fun", than actually teaching concepts.

You don't only teach memorization of addition/multiplication tables in order for the child to know their multiplication tables. You do it because that sort of rote memorization (especially of abstract items) is good for the brain. Children also need to learn that a lot of work is actual work, and some of it involves fairly boring mental drudgery. Is it fun memorizing the difference between (?!) and (?=) in regular expressions? No, but it can be helpful.

This article seems the equivalent of "Little Johnny doesn't like doing push-ups. Can't we just have him play Wii instead? He enjoys playing Wii, and it keeps him totally engaged. And if he plays Guitar Hero, he's learning music at the same time!". Imagine the physically fit musical geniuses we will create if we can get them all to enjoy and appreciate exercise!

Math has been replaced by puzzles. English has been replaced by "multimedia presentations (computer play time)". Phys-ed is now free play. Social studies is "social skills 101 (bullying, including others, fairness, etc)".

I greatly fear we are raising a society of salespeople and telephone sanitizers.

I support many of the activities such as what Khan academy has done to "make math fun". But much of this needs to be an addendum to solid foundational work, not a replacement. The program the article describes seems to replace any rigor with fun, and hopefully children will learn the tough stuff by osmosis or something (or it will be the next school's problem).

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