Slashdot Mirror
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.
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.
'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.
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.
Knowledge is how to play a game, intelligence is how to win, wisdom is knowing what game to play.
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 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
Teach the children Art, and Music.
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.
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?
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.
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.
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).
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.
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.
Life is like untied shoe laces; it always tripping you up and getting in your way.
I've said it before, but kids already do simple algebra in elementary school.
3 + [] = 5
and you fill in the box.
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...
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.
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.
Eat sleep die
You're doing a great job so far!
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.
I plan to get my children learning the 'advanced' topics as soon as possible. How about you?
I hate children, you insensitive clod!
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?
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.
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.
For all intensive purposes, "whom" is no longer a word. That begs the question, "who cares"?
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.
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.
..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
Seriously, they are KIDS.
Stop trying to turn them into robots.
Let them PLAY.
This is just beyond the pale.
-- Tigger warning: This post may contain tiggers! --
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.
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 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 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.
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.
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)
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%.
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.
mfwright@batnet.com
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
mfwright@batnet.com
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
I think it might be more beneficial to teach statistics.
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.
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).
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.
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.
Not necessarily. It might not be "before age 30" but "less than 20 years after exposure to X". If everyone is currently exposed to X round about the age of 10, it'll look the same.
So exposing them to it earlier just changes 30 to 25.
Confucius say, "Find worm in apple - bad. Find half a worm - worse."
My father taught 5th grade. In the late 60s, in this school, that was when they worked mainly on word problems. My father decided to teach them all (not just gifted students) basic algebra having to do with subtraction, addition, multiplication and division in order to solve these word problems. He presented it as playing games with the numbers. There were several interesting results with his experiment. First, most of the 5th graders quickly grasped the concepts when presented this way. Second, the students began solving these word problems more quickly and more accurately as compared with classes of previous years. Third, the students enjoyed the insights. Fourth, the concepts seem to help the "slowest" students the most. Fifth, creative teaching, regardless of its success, was frowned upon by the school administrators and he was forced to stop the program.
Anyone notice one of the books mentioned in the article, “Calculus by and for Young People", is selling on Amazon for $693 and up (used)? It's 119 pages and the reviews note it's about 3 inches by five inches.
I'm curious now what they made that book out of.
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DERRRRRRR
MATH!!!
If you're trying to get kids interested in the concepts of higher math, Vi Hart has some fascinating videos.
Oliver's law of assumed responsibility: If you're seen fixing it, you will be blamed for breaking it.
Five years is too young for renal calculus.
Calculus is a topic that is taught very badly in High School. Too many "teachers" make it hard - they believe it's hard to teach, so it becomes hard to learn. I've seen a trainee teacher stressing out at the term "limiting slope of the secant" because "I've never learned about secants". In my experience (only 30 years), calculus is easy to teach, and easy to learn, as long as you leave formal treatments of limits 'till later. The formal statement of the meaning of a limit is a scary sentence to most students, so let them get a good intuitive feel for what it means first, let them get the "rate of change" and "slope of the tangent" ideas firmly consolidated, then the formalisation makes sense.
Abstract algebra is another one that most teachers shy away from. I've taught it to 11 year olds though, by simply using something they should be familiar, the good old analogue clock, where 12 is a zero, and 3 times 4 makes zero. It really opens their minds to realise there are ways of looking at number that break the mold they've been used to. There is no need to make a beautiful, elegant and logical subject hard, regardless of how "advanced" it is.
If you want to prepare children for higher levels of mathematics or logic, please start with life.
Asking a child to understand 1/4 or All S is not P, is simply ridiculous.
But cutting an apple into quarters, that they can understand. Ask if all fruit are apples. And it's amazing how correct and logically sound kids can be.
The article is saying that we're beating our kids to death with rote math, and not letting them enjoy the math that exists in the world around them.
The purpose of calculus was to provide a way of deriving answers that would otherwise take thousands of calculations to do. Now that we have computers that can do millions of calculations in a second, then why do calculus?
I love these examples as bridges to algebra. "Two hours from now" can land on different times, because "now" is variable. You can tackle these problems as arithmetic, or you can tackle them as algebraic.
Cut a piece of paper in half... and again... and again... Kids can understand that you can keep halving the paper. They also realize that at some point you'll have to stop, because your scissors are too large for tiny papers. But then they can imagine how you COULD keep going and going and going - in your mind.
I am aggregating some (healthy, not mnemonic) ideas for memorizing times tables efficiently, based on patterns. This will be one of the next projects for Natural Math.
"Inspired by calculus" is what I like to call our activities. Both because the goal is to get kids inspired by calculus, and to distinguish the activities themselves from formal calculus courses.
I try to mention free play multiple times everywhere I talk about math. It's very important. It's the foundation.
>I have always wondered why puzzles were never included in any educational system.
They have been in many educational systems. Merchant problems were part of traditional math education in Hungary and Russia.
It's not just young children. My youngest brother was homeschooled and had no interest in math beyond basic arithmetic until he was 16 or so, when he found he needed integral calculus to solve a problem he was working on -- at which point my parents managed to fit twelve years of math into six months of education.
:The other thing worth noting is how most mathematicians make their breakthrough discoveries before age 30.
Most people don't have major responsibilities between their early 20s and 30, which gives them time to think & ponder.
Physics should be done very soon after algebra also. It would help people see the practicality of mathematics in their environment.
http://www.dragonboxapp.com/
Teaches algebra in a very subtle way.
This is why I don't practice my scales.
Pardon my cynicism but this is just another opportunity to confuse children even further. 'Math teacher' is generally an oxymoron. The real problem in math education is that the system requires math teachers to be 'math people'. Nerds who do math but can't communicate worth spit. In the elementary grades, we need artists who can teach math. They're rare but they do exist.
And I'm the other way around. Introduce something as theory and proofs, and I'll get it (and then I can extrapolate it to examples). Introduce it as examples without theory, tho, and I may never understand it. This is why I did well in math all the way up to college calculus, which was presented as examples. All my prior math had been theory and proofs type stuff.
~REZ~ #43301. Who'd fake being me anyway?
Re: Funny story.
Totally. It's like they make you do it the hard way a million times and only then do they show you the shortcut. But you are never told that that's how it works.
-l
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I asked my teachers to teach me advanced math in grade school. They didn't like it, it was a 'problem' because it would take up too much time and resources. Extra time and resources already spent in copious amounts on a couple of retards. One grew up to be that guy you see picking his nose while he bags your groceries.