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Five-Year-Olds Can Learn Calculus (2014)

theatlantic.com

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Re: Five-Year-Olds Can Learn Calculus (2014)

#111

Post by kazinator »

iovrthoughtthis wrote: post #103
occultist_throw wrote: post #67

Earlier quoted context omitted.

And you succinctly put why I despise "School". I love learning new areas and new techniques. And when Im teaching myself, I can fail and I go back a step or 2. Not the end of the world. If I do similar in a class, then I set myself further back with a slowly growing chance of doing badly (D or F). And in College, then its another 1-4 credit hours to pay for the privilege. They not only fail me, but demand more money.…

I must have attempted to implement a lisp many times before I got it "right", or a "simple" perceptron etc. Without the pressure I'm not scared of revisiting hard things, I'm driven to actually. Edit: to indicate that a "simple" perceptron isn't actually that simple.

> I must have attempted to implement a lisp many times before I got it "right"

How did you know it wasn't right? Did you run into a problem trying to implement requirements which you knew all along and accepted? (As in, you started coding without planning for some of these requirements ahead of time, then got stuck?)

Or did you learn about and adopt new requirements which weren't easy to retrofit into the existing work in progress?

Re: Five-Year-Olds Can Learn Calculus (2014)

#112

Post by zaptheimpaler »

Ntrails wrote: post #74
eecc wrote: post #65

Earlier quoted context omitted.

No, you're straying away from the original point which was: there's no need to make rote memorization of arithmetic tricks and multiplication tables a hard prerequisite to everything else about math. Nobody claimed that arithmetic is absolutely useless

It is never hard blocked on arithmetic though (in the UK at least). A bunch of simple numerical solving will be harder without being able to do multiplication in your head - but certainly someone could get through without. I'm honestly dubious of anyone who claims that the reason people hate maths is because of times tables.

Well a lot of people who dislike math do so because they think its all about rote calculation. Not just times tables, even algebra can be seen as a completely syntactic procedure - memorizing a bunch of rules. Another example - if you are taught 10 different "templates" of problems and solutions in calculus class and memorize which equations to apply where, you can solve a lot of problems without ever building any intuition.

If someone thinks of math that way, I can understand why it would be boring to them. Thats what I mean by blocked on arithmetic. If every thing is taught as a cookie-cutter process to follow its not surprising that people associate math with blindly following processes.

Re: Five-Year-Olds Can Learn Calculus (2014)

#113

Post by mannykannot »

mclehman wrote: post #110
RobertoG wrote: post #97

Earlier quoted context omitted.

I would say that the fundamental idea to understand is the concept of changing ratios. Limits seems to me to be more a tool in order to calculate the ratios than a fundamental concept.

I agree with that. You can hand limits as a tool to someone already comfortable with the concepts and present them as a way to quantify their intuitive understanding.

But without limits, you cannot actually do much with the concepts. Whether you call limits a concept or a tool, they are essential to the practice of calculus, and, historically, to its emergence (and not trivially so, either.)

Re: Five-Year-Olds Can Learn Calculus (2014)

#114

Post by njarboe »

zaptheimpaler wrote: post #41
chii wrote: post #26

Earlier quoted context omitted.

There are some aspects of doing math that isn't fun, because it is grinding. Learning the time tables for example - you just have to do lots of exercises to get used to multiplying. No way around that. Just like in sports, where you must also do a lot of basic practise to get good. Making the practise like a game can be done to done degree, but ultimately, a kid is going to see through that after a while, since it ge…

Thats the kind of BS math they shouldn't bother teaching. Most of math is about concepts, and if 5 year olds can grapple with super-basic calculus how about teaching 10 year olds proof-based math and concepts (closer to college level) than memorization? Its kind of dumb how arithmetic is seen as a precursor to math when its really not. You could do an ENTIRE college-level course in linear algebra or calculus or graph…

If you don't know up to 9x9, then doing basic algebraic problems become vary cumbersome. Factoring and simplification becomes a chore and one can't get into a good flow if you are typing in 2x5 into a calculator.

On the other hand, I can't imagine anywhere that would expect one to know 11x15 off the top of ones head.

Re: Five-Year-Olds Can Learn Calculus (2014)

#115

Post by thomastjeffery »

roceasta wrote: post #21
jonnybgood wrote: post #2

I have no doubt they can. There's so much negative cultural conditioning towards mathematics by eliciting negative emotions that it's ridiculous (i.e. math is scary, math is hard, math is for smart people, etc.). I believe kids who excel at math is not due to them being smarter but due to the bypassing of the cultural conditioning. The bypassing can be due to the parents, teachers, some mentor, or their particular fa…

The fact that school math is thought to be 'scary' and 'hard' is not the issue. Plenty of video games are known to be scary and hard. For instance, they contain monsters and are difficult to complete. They even have objective grades (scores). What makes the difference is that video games aren't compulsory. So they represent true play . Which is what makes them educational, unlike a school curriculum, however much it…

> math is thought to be 'scary' and 'hard'

> Plenty of video games are known to be scary and hard

Those statements seem related because of the ambiguous definition of "scary". In relation to math, "scary" causes anxiety, whereas in relation to video games, "scary" causes excitement.

As soon as you grasp the different meanings, you can clearly see that even though both statements are about the same word they are still orthogonal.

Re: Five-Year-Olds Can Learn Calculus (2014)

#116

Post by thomastjeffery »

jonnybgood wrote: post #2

I have no doubt they can. There's so much negative cultural conditioning towards mathematics by eliciting negative emotions that it's ridiculous (i.e. math is scary, math is hard, math is for smart people, etc.). I believe kids who excel at math is not due to them being smarter but due to the bypassing of the cultural conditioning. The bypassing can be due to the parents, teachers, some mentor, or their particular fa…

In my opinion, the biggest problem is the focus on learning math subjects as dependencies for learning the next math subjects. This problem is exasperated by tying a subject to semester or school year.

If I'm not supposed to learn Calculus until I learn Algebra, that must mean I can't learn Calculus until I learn Algebra.

If my Algebra class lasts for a year, that means it must take an entire year to learn Algebra.

If I am supposed to learn Algebra first, then I must not be able to sufficiently wrap my head around anything related to Calculus.

That means that whenever I hear about "limits", "d/dx", I should immediately lose attention; after all, those are Calculus-related things, and my underdeveloped brain will not be able to comprehend such subjects for another year, and I shouldn't even try.

Of course, none of those are true. If I wanted, I could

* learn at my own pace * learn in the wrong order * wrap my head around limits, d/dx, L'Hopital's rule, etc. before understanding factoring, trigonometric functions, etc.

In fact, any of these would have put me far ahead in my education.

"Traditional" education requires a student to be lazy, or hold [him/her]self behind academically. To make matters worse, it requires that student to be a specific, quantified amount of lazy/held back.

For me personally, I struggled to keep up with the obscene amount of expected rote homework, while also feeling miles ahead of lectures, even though the homework and lectures were from the same class.

When I started learning about programming on my own, I suddenly had the freedom to do all of the things I mentioned above. Programming, as a subject, is especially applicable: There are abstractions, paradigms, languages, etc. You can jump into any aspect headfirst, and still get your bearings.

Re: Five-Year-Olds Can Learn Calculus (2014)

#117

Post by thomastjeffery »

roel_v wrote: post #31

Anyone commenting on this (whether saying 'yes' or 'no'), please first read the article. What it considers 'calculus', none of us here would consider calculus. Because very few 5 year olds are capable of understanding even something as basic as a line plot (or '2d graph' or however you want to call it). They simple do not have the neural matter to comprehend fairly advanced (from an evolutionary point of view) abstra…

> abstractions like the relationship between a value on an x and a y axis.

That's entirely the point. You can understand calculus without understanding graphs.

It may sound crazy, but it's true. You can understand a subject without all of the dependencies academia tends to categorize Maths into.

Re: Five-Year-Olds Can Learn Calculus (2014)

#118

Post by thomastjeffery »

The way math is presented to children is like so:

Here's a function, now practice evaluating it on this list of 200 inputs. Due tomorrow.

I would prefer:

Here's a concept. Play around with it. Tell me its limits, and I'll tell you another concept that breaks them.

Re: Five-Year-Olds Can Learn Calculus (2014)

#119

Post by roel_v »

sp332 wrote: post #81
roel_v wrote: post #71

Earlier quoted context omitted.

True; but do you have anything in mind that is less abstract than a graph and still expresses the continuous (or not, depending on what you're looking at...) nature of a relationship between two variables? Because I don't know of any way that is easier and more intuitive to understand than that. But maybe I'm just projecting based on my own preferences.

I think you have to understand how two things can change "with" each other before you can understand how to read a chart. Personally I think seeing something vary with time or distance is more immediately understandable than reading a chart.

Reading a chart yes, but that's because (IMO) reading is harder than making one. For example, no matter how much someone would explain, I could never understand the relationship between an angle and a sine wave, until I had a teacher made me draw a circle and an x/y axis alongside each other and project the line in the circle onto that axis. I still don't think one can really explain this without resorting to describing what happens visually. But again, maybe that's just me.

Re: Five-Year-Olds Can Learn Calculus (2014)

#120

Post by Ntrails »

zaptheimpaler wrote: post #112
Ntrails wrote: post #74

Earlier quoted context omitted.

It is never hard blocked on arithmetic though (in the UK at least). A bunch of simple numerical solving will be harder without being able to do multiplication in your head - but certainly someone could get through without. I'm honestly dubious of anyone who claims that the reason people hate maths is because of times tables.

Well a lot of people who dislike math do so because they think its all about rote calculation. Not just times tables, even algebra can be seen as a completely syntactic procedure - memorizing a bunch of rules. Another example - if you are taught 10 different "templates" of problems and solutions in calculus class and memorize which equations to apply where, you can solve a lot of problems without ever building any in…

> If every thing is taught as a cookie-cutter process to follow its not surprising that people associate math with blindly following processes.

A hammer and a chisel are tools. You can use them to shave a bit of wood off the door, or you can use them to carve a beautiful sculpture. You can do neither without knowing how to use them.

I agree that too much of maths teaching is "here are 100 quadratics, go solve them" - which is obnoxious and dull. But "blindly following process" (matrix multiplication, change of variables/base, sum to infinity etc) happened throughout my degree and is a component of higher maths as well. Heck, most of the proofs I remember use "tricks" that any mathematician is expected to just know.

I sure as hell don't think children are taught maths well. I remember being taught how to take the derivative of x^2 without the teacher bothering to show where that came from or what it really meant. Heck - children are given the quadratic formula to memorise and basically told "it's magic, learn it" instead of showing them how you can trivially create it by completing the square.

But mathematics does involves a ton of following some process or other to get the problem into a format you can do something with. Probably by following a different process that you've done before. I'm not sold that you can simply eliminate that aspect.

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