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Algebraic and calculus concepts may be better way to introduce children to math (theatlantic.com)
161 points by tokenadult on March 3, 2014 | hide | past | favorite | 91 comments

(Disclosure: I'll be working with Maria on the calculus series for kids.)

I'm a big fan of the conceptual approach. One of the largest problems I see with math education is that we don't check if things are really clicking.

I graduated with an engineering degree from a great school, and still didn't have an intuitive understanding for i (the imaginary number) until I was about 26.

Go find your favorite tutorial introducing imaginary numbers. Got it? Ok. It probably defines i, talks about its properties (i = sqrt(-1)) and then gets you cranking on polynomials.

It's the equivalent of teaching someone to read and then having them solve crossword puzzles. It's such a contrived example! (N.B., this anguish forced me to write a tutorial on imaginary numbers with actual, non-polynomial applications, like rotating a shape without needing trig. See https://news.ycombinator.com/item?id=2712575)

Calculus needs these everyday applications and intuitions beyond "Oh, let's pretend we're trying to calculate the trajectory of a moving particle." They're out there: my intuition is that algebra gives a static description (here's the cookie), while calculus describes the process that made it: here's the steps that built the cookie. Calculus is the language of science because we want to know how the outcome was produced, not just the final result. d/dx velocity = acceleration means your speed is built up from a sequence of accelerations.

Man, you're spot on. I've felt that the way math is taught now is akin to spending 10 years on 'if', 'for', 'while', and 'switch' statements (and being able to evaluate them perfectly without a compiler).

Beautiful point. It's along the lines of "How can you learn programming without the details of how to turn a series of tokens into an AST?"

I am so stealing this analogy!

Yes, yes, yes.

I have tried (fitfully) three times to sit down and learn what calculus actually means. I still don't get it. And I suspect that clients and families will push it ever further away.

Yes, give me real world puzzles and applications for learning new mental tools and I will probably love them. Make it abstract enough and I cannot see the value in learning it.

I was the same way :). I have an intuition-first guide to calculus, here:


I think the essentials on how to think with calculus can be conveyed in 1 minute. Let me know if the above helps :).

I see that your examples of teaching fast (one minute, ten minutes) all use discrete models for continuous processes. For people who understand continuousness mathematically, going between discrete models to continuous models and vice versa is easy. However, there are indications that the other way around -- students who perceive these changing situations mathematically as intrinsically discrete -- going from discrete to continuous models might be extreme difficult (due to in-commensurability). Of course, students can learn procedures to make the switch from discrete to continuous models, but even then it is possible that this procedural experience is not build on deeper mathematical understanding. Tricky stuff, learning mathematics :-)

Great point, thanks for the comment.

I think the key missing insight for me was that a continuous process and a discrete process can both point to the same result.

A pixellated word on a screen ("cat") conveys the same meaning as a perfectly smooth vector of the same word. The math idea, to me, is "can a discrete description/process" point at the same result that we get from a continuous one?

The idea of limits, is essentially figuring out when a discrete epsilon can still lead us to the true value of f(x) (if this trick works, we call it a continuous function, and can use calculus with it.)

Yes. You might be interested in Cornu, B. (1991). Limits. In D. Tall (Ed.), Advanced Mathematical Thinking (Vol. 11, pp. 153–166). Retrieved from http://dx.doi.org/10.1007/0-306-47203-1_10 and similar works. Volumes have been written about the limit concept, learning the limit concept, and different approaches to the limit. It is one of the more interesting mathematical concepts one encounters in high school, not in the least because of the ability to use it successfully without understanding the underlying concept and the philosophical implications of the concept.

Unfortunately, I doubt that these implications will be discussed in your average Mathematics classroom. Actually, my experiences with mathematics in primary, secondary and a lot of tertiary education, both as a student, teacher and observator, leads me to believe that conceptual discussion is rare, whilst procedural discussions (how to perform some calculation) are plenty and abound.

I loved the explanation of calculating area of a circle through an unwrapping of infinite rings. brilliant thank you

Try "The Hitchhiker's Guide to Calculus" by Michael Spivak. After reading that, it all just made sense and at the same time made me angry that they didn't teach it like that to us in high school.

Thanks for the suggestion!

For a good motivated treatment of calculus I'd recommend going back to Isaac Newton's Principia. It is surprisingly accessible, and really quite beautiful.

As a math person, I think this attitude is why I hated (and still do) calculus. Calculus is not a language of science, it is a branch of mathematics that is useful for science. It's description is just as static as algebra (not the least because it is algebra, just with two new functions).

Regarding the Gaussian (imaginary) numbers, they have uses which have nothing to do with rotations. My intuition for them is that they are a way of factoring expressions such as A^2 + B^2 as the difference of two squares.

Wrt imaginary numbers, just a thought: capacitance and inductance in an AC circuit.

I don't know what age group you're targeting but I found solving problems in this space gave me a great intuition for imaginary numbers; I found it quite interesting too. Heck, AC circuits are great real-world playgrounds for differential and integral equations - Laplace transforms too.

I've been fretting about how to help my kids get a better understanding of math for years. I've been bouncing between too simple and overly complicated for too long. :-(

I think I'm just going to mandate they spend some amount of time per week using Mathematica and going through some introductory book. At least they will taking baby steps in a direction better than youtube. Grumble...

So I've been fretting on which book. Any suggestions? Heh heh...

Analog electronics is a great domain, and something I want to explore more myself (I did CS, only had 1-2 analog classes, and didn't build an intuitive understanding for electricity, a gap I want to cover).

For books, I don't have a huge amount of experience but I do like the Manga Guides to X (Manga Guide to Physics, etc.). A visual medium encourages the author to rely on metaphors vs. symbolic descriptions, which I prefer in an intro to get the concepts across quickly.

How old are the kids and what do they actually like in this life?

I love it. This really appeals to me because of the all the bad ideas I had to unlearn in my 20s:

- math is about numbers and arithmetic operations on them - being good at math meant you were good at arithmetic - some people (meaning me) just didn't have what it took to be "good" at math. Reinforced by my high school math and physics teachers.

I hated math because I didn't understand that mathematics is a system for representing abstract concepts and manipulating them.

Eventually on my 4th try to get calculus, I took a class from nick fiori(http://www.yale.edu/education/about/faculty/fiori.elw062111).

His teaching methods opened my mind, and I've gone on since then to become an ardent amateur mathematician with minor publications and career tangents in machine learning and data science.

I can only imagine what would have happened if I had been taught math well from an early age.

This is really great and inspiring to hear! I also had a very bad experience to mathematics during my earlier youth.

Relentless studying, book reading and search for motivational information have helped me get over the bullshit I was introduced to.

I love math, and it is ing beautiful.

Good to hear, any math books you'd recommend to someone who's not there yet, or remember from that class?

Sorry, did not learn much from books. The only one of note is the Tom Mitchell Machine Learning Book, which already requires a basic advanced mathematics fluency. I also believe it is a little out of date.

I can recommend a few professors who really opened my mind to how to use math, all at UCSC (CS grad school for me):

- Dave Helmbold (Machine Learning) - Kevin Ross (Operations Research) - Martin Abadi (Security)

Yes yes yes.

The mechanics of basic calculus are remarkably simple. After a while it becomes a game of sorts with a pretty simple set of rules. Once you learn to "see" parts of the equation as symbol blocks (instead of numbers needing computation), you can move them around freely so long as you follow the rules. Algebra is simply a subset of the rules.

Teaching kids to think symbolically will help them in so many other fields.

I'm pretty convinced that with a little thought, you can teach basic derivation and integration to pretty young kids. Carefully craft the problems to avoid difficult division problems, avoid trig, let them use lookup tables for multiplication and you might be able to get kids under 10 to even do some of this.

Then "vertically integrate" other algebraic and trigonometric concepts into this framework, like adding new pieces to the game.

When I started Calculus in high school everything made so much more sense! Trigonometry was just memorizing equations, not having any real idea why the equations were what they were.

With Calculus on the other hand, I could come up with the equations I used in prior classes. I instantly wondered why they didn't start with this?! Trig/Pre Calc was just a waste of my time.

> Trigonometry was just memorizing equations, not having any real idea why the equations were what they were.

The old proofs of those equations are geometric, they are a bit harder to get your head around than learning complex algebra.

Taking Euler's identity:


  > e^(ikx)=cos(kx)+i*sin(kx)=(cos(x)+i*sin(x))^k
Expand the right side of the equation, then split into real and imaginary parts. That is a quick way to determine sin(kx) or cos(kx) if sin(x) or cos(x) are already known and k is a positive integer.

EDIT: bloomin' asterisks.

My favorite is a way to prove the sum of angles formula for sine and cosine.

  cos(x1+x2) + isin(x1+x2) = e^((x1+x2)i)
                           = e^(i*x1) * e^(i*x2)
                           = (cos(x1) + isin(x1)) * (cos(x2) + isin(x2))
                           = cos(x1)cos(x2) - sin(x1)sin(x2)
                             + i*(sin(x1)cos(x2) + sin(x2)cos(x1))
Since the imaginary parts have to be equal and the real parts have to be equal:

  sin(x1+x2) = sin(x1)cos(x2) + sin(x2)cos(x1)
  cos(x1+x2) = cos(x1)cos(x2) - sin(x1)sin(x2)

Yeah, that's a more generic form of what I said.

ok, hold on...

  e^(i*x)= cos(x)+i*sin(x)
  e^(-i*x)= cos(-x)+i*sin(-x)= cos(x)-i*sin(x)
  cos(x)= (1/2) * (e^(i*x)+e^(-i*x))
  sin(x)= (-i/2) * (e^(i*x)-e^(-i*x))
  cos(i*x)= (1/2) * (e^(i*i*x)+e^(-i*i*x))=(1/2) * (e^x+e^(-x))
  sin(i*x)= (-i/2) * (e^(x)-e^(-x))
That's the amusing part.

Hm, you're talking about teaching symbolic manipulation using calculus, but for me the real ideas of calculus are integration and derivation. I think those ideas are very valuable, and symbolic manipulation is just one way to solve those problems.

Teaching integration with various shapes and boxes of ping balls is really fun. Then use smaller balls to get a better estimate. Physical, visceral and chaotic with balls bouncing everywhere. Even counting a hundred-ish balls is hard and methods have to be devised (good way to intro number bases).

Exercise can also be done in 2d with colored squares or circles. Which is a nice segue into guestimation. Cover the entire curve to be measured with a square, value can't be large that as the square totally covers it, ask them to find the lower bounds.

Then get them to make estimates with other things they have around (bananas, shoes, etc), someone will come up with rectangles. Introduce the trapezoidal rule!

Then we are off into history with Newton and Leibnitz and that shit is good drama enjoyable by any age.

None of what I have presented requires counting past 20 (which really most of us can't do anyway).

>but for me the real ideas of calculus are integration and derivation

you mean the ideas of a dual space of a linear space and of tangent bundle to a manifold? :)

Why not? :) All these things are really simple in their essence. They're abstract but an average 14 year old in modern civilisation already has the cognitive machinery necessary to grasp such abstractions. In fact, I think at that age, when they discover abstraction and formal reasoning and what they can do with it (there's a reason teenagers are so full of all kinds of ideas, they're starting to explore the world in full generality), it would be extremely stimulating food for their minds.

Can you provide some examples? To me, the basic mechanics I remember (for simple equations like x^3 d/dx) were almost entirely "solvable" by moving bits of the equation around...

1. move the 3 in front of the x

2. where the 3 was, subtract 1 from it and put that back where the 3 was

but if you think of it as symbols you get more like

1. move the exponent symbol in front of the x symbol

2. subtract 1 from that exponent and put it back where the original exponent symbol was

For me at least, bridging the gap from the specifics of the first example to the more generalized second approach meant that when I was given something like

3x^(y^4) d/dx my mind was absolutely blown because I didn't know if there was some rule I needed to know if the exponent had an exponent or something.

Going from the years of the arithmetic approach to really groking the symbolic manipulation approach was really hard for me, and looking at lots of young kids suffering through algebra it was the same.

Learning how to recognize the symbols through the specificity of the numbers was something I don't think I really got at all until calculus and it's because the mechanics of the rules of simple derivatives and integration really force you to recognize the underlying symbols...more than algebra did (which for me just seems like a really complicated arithmetic rather than what's actually a pretty simple symbol manipulation exercise).

It's odd, because like most technically minded kids, I was already great working with symbols and breaking things down and building them up. I could build huge structures out of various small assemblies with legos, write simple software, crank out papers for English class that would guarantee an A and so on, but understanding that the teachers no longer wanted me to be a human tabulation device but to be a manipulator was something I never really cottoned on to.

I know from spending a little time with nieces and nephews younger then 10 that you can teach basic derivatives in about an hour or two and have them fairly reliably doing simple work like the above in an afternoon if you don't worry them too much about simplification of the result or all the arithmetic.

My gut says that if you start adding more rules, like what do you do when you have something like (x^3)/(x^2) and how exponents should be subtracted, and with a little care in the examples you show them of this rule in action, you end up showing them all of algebra, fractions, exponents and later trig, logs, etc. while building up and hanging all of these concepts off of calculus.

TBH I don't know enough about early childhood education to know if this should be used instead of basic arithmetic or not, but I bet if you use shapes and colors instead of letters and numbers and start to teach basic rules, you can just slowly introduce numeracy later anyways when their brains are more developed.

No offense, but what you're proposing would be a huge step backwards. All of those examples you've given are extremely special cases of a general concept, where the concept is relatively easy to understand (two points make a line, we only have one point, so take the limit) although the implementation is complex (requires either understanding algebra or knowing lots of rules.)

If you try to teach a little kid the power rule right off the bat, they will both be turned off of math (because it's just pushing x around for no clear reason) and have learned absolutely nothing of the important bits of calculus.

> where the concept is relatively easy to understand (two points make a line, we only have one point, so take the limit) although the implementation is complex (requires either understanding algebra or knowing lots of rules.)

The pedagogy of Mathematics education is fundamentally broken and we've lost generations of math users because of it.

Approaching it from this sense "two points make a line, we only have one point, so take the limit" is the method today and with all respect, it's been a terrible terrible failure...even if it is "correct".

A few kids will grok it and turn into computer scientists or physicists or mathematicians of some sort, and the other 98% will take the bare minimum to get their high school education, and if they go on to college see which degree programs require math and which don't and select the B.A. degree that doesn't.

Anecdotal, but I don't know a single person who enjoyed their K-12 math education. I know lots of people who enjoy math, and found that joy in college or later, but found the educational experience of K-12 so abysmal and torturous that they completely swore off even pursuing fields with heavy Math components. There are lots of artists, musicians, writers and historians who would probably be great Mathematicians, Physicists and Scientists were it not for the piss poor job we do indoctrinating kids into math.

It's not just that students fail to learn mathematics, it's that the pedagogical experience is so negative that they swear off ever even trying. And IMHO, a very big part of that is the (to the student) endlessly pointless jargon filled inapplicable overly rigorous and formal mess that is Math education today.

I've sat in the audience on some very heated round tables about promoting STEM education in my region and I've come away convinced that getting more people into STEM is critical to long-term economic success, but students are not only not drawn into STEM, they're actively driven away from it by the pedagogical approach of what amounts to a single class every year. Students love science classes and science labs, they love shop class and learning engineering, they love computers and everything else to do with STEM except for the K-12 Mathematics education. And that loses them, it simply drives them away from all of these great (and often lucrative) fields.

More importantly, I think fields which are not traditionally Math focused, could benefit greatly from a better general Math education. Read a social sciences academic paper and see what I mean for a quick example.

> If you try to teach a little kid the power rule right off the bat, they will both be turned off of math (because it's just pushing x around for no clear reason) and have learned absolutely nothing of the important bits of calculus.

I want to agree particularly with this, "it's just pushing x around for no clear reason". The fundamental problem with learning Mathematics in K-12 is applicability. Beyond arithmetic, students have almost zero examples of why they should bother learning anything else. Most of the population gets by just fine without anything more complex than arithmetic and figuring out percentages (and even then that's a stretch).

When students go to their parents to seek guidance, their parents also don't have any idea why they have to learn all this stuff. They certainly don't need it in their day jobs and can't provide a hint as to why it's important other than the student needs to study it for the grades.

And to be honest, even if their parents do use Mathematics in their work and can provide examples, it's likely that the student's ability to relate to that work is very limited. The reinforcement that all this time spent learning Math is pointless is much stronger than the reinforcement that it's important or useful. Getting a kid to crank through 20 or 30 algebra problems is much harder than say, a 5 paragraph essay for English class because the entire time they're doing this they're saying to themselves "why am I doing this? Math is useless! At least learning to write a little has some kind of use!"

But children will play "pointless" games for hours and hours and hours - and not even ones they're especially having fun with, just ones that hold their interest (if you've ever watched a 9 year old vent frustration at their Xbox you'll know what I mean). If we can turn Mathematics education into a kind of "game", then fill in the details and formal bits as they age, they'll at least be able to relate to it even if they don't understand the application or relevancy.

And the truth is, once you get to Calculus and get it, it's actually pretty fun and pretty easy. That's a high enough discipline for most STEM jobs and I firmly believe that every K-12 student should be able to do what we call "college level" Calculus by the 10th grade. So why not try to capture the things that make Mathematics at that level fun and easy, and I think that's the symbolic manipulation, even if it is hard to establish relevancy, and get them used to doing it from a very early age. Kids can move blocks around and stack them before they can walk, why can't the blocks be bits of equations? And why can't moving the blocks have little game-like rules they can learn?

>Read a social sciences academic paper and see what I mean for a quick example.

I have actually done some graduate-level work in sociology and history, and the papers and books I read were mostly examples of very good statistical work and well-thought-out process analysis. You can put your STEM-master-race badge away.

>Beyond arithmetic, students have almost zero examples of why they should bother learning anything else.

When I say "for no reason," I don't mean "for no day-to-day practical reason." Playing with abstract concepts is and should be its own reward; that was the whole point of TFA. Mechanically memorizing how to take the derivatives of polynomials is neither a fun abstract concept nor a boring-but-necessary practical skill.

>If we can turn Mathematics education into a kind of "game", then fill in the details and formal bits as they age, they'll at least be able to relate to it even if they don't understand the application or relevancy.

"Gamification" as a cynical ploy to get kids to sit still long enough memorize their times tables may or may not work. But even if it does, it's only gotten them to play the game long enough to pass them to the next level; it has deliberately shifted their interest away from the joy of learning for its own sake. That is not what the article is about, and it's not helpful in the long run.

>Approaching it from this sense "two points make a line, we only have one point, so take the limit" is the method today and with all respect, it's been a terrible terrible failure...even if it is "correct".

No... no, it isn't. The approach today, for the majority of students, is to learn the bare basics so that you can plug them into an equation and find out what the marginal cost of widgets will be next year given a certain set of equations. And in any case, it comes so late that kids have been taught that "math" is something that actually is boring and useless.

> Playing with abstract concepts is and should be its own reward

I'm sorry, but you're just simply wrong on all points. Promoting the status quo in math education, as you are doing, has been, is, and will continue to be a failure that drives kids away from learning. There are now decades of evidence of the failures of k-12 education to address this need and I find it unbelievable that you haven't gotten the picture yet.

I'm not saying that what I'm proposing is correct, but continuing the very poor pedagogical approach that you support is not going to solve the educational failures that we're experiencing today. What we need are fundamentally new approaches to Math education. You are not providing any insight into what those approaches should be.

I'm sorry, but in a discussion to fix and change what is obviously utterly broken in k-12 maths education, suggesting to just continue the course is not a helpful contribution and is simply part of perpetuating the problem.

This has been recognized for so long, that it has finally percolated out of educational establishment, which has failed to address the problem with undereducated and unqualified teachers, student motivation, repeated failures in curriculum development (Common Core is simply the latest joke of a curriculum), and has reached levels as high as the White House for targeting. You have to know that the current approaches are failing if the President has to get involved.

From the "Report to the President...K-12 education in STEM" http://www.whitehouse.gov/sites/default/files/microsites/ost...

"Schools often lack teachers who know how to teach science and mathematics effectively, and who know and love their subject well enough to inspire their students. Teachers lack adequate support, including appropriate professional development as well as interesting and intriguing curricula."

"As a result, too many American students conclude early in their education that STEM subjects are boring, too difficult, or unwelcoming, leaving them ill-prepared to meet the challenges that will face their generation, their country, and the world."

"Put together, this body of evidence suggests that grade-school children do not think as simplistically about STEM subjects as conventional curricula assume. They are capable of grasping both concrete examples and abstract concepts at remarkably early ages. Conventional approaches to teaching science and math have sometimes been shaped by misconceptions about what children cannot learn rather than focusing on students’ innate curiosity, reasoning skills, and intimate observations of the natural world."


"The first principle of How People Learn emphasizes both the need to build on existing knowledge and the need to engage students' preconceptions -- particularly when they interfere with learning. In mathematics, certain preconceptions that are often fostered early on in school settings are in fact counterproductive. Students who believe them can easily conclude that the study of mathematics is 'not for them' and should be avoided if at all possible."

I differ from this report in that they continue to promote a bottom-up approach to maths education. I think math should be taught from a top-down approach, like nearly every other discipline. You don't learn to bake a cake by first spending 10 years learning about chemistry, agriculture, nutrition, animal husbandry, distillation etc. You say "I want to bake a cake" and you start with a simple cake recipe. Then the next time you say "I want to bake a different cake" and you use a more complicated recipe. And so on and so forth until you don't need a recipe and are putting together your own cakes from scratch.

The "cake" I'm proposing is calculus...and I believe, from experience teaching basic calculus to kids under 10, that this is realizable and beneficial.

The end result is that I don't expect kids to come out of K-12 baking cakes that look like this http://www.askmamaz.com/wp-content/uploads/2013/01/Most-Beau...

when the average STEM work-a-day mathematics really requires them to do this.


Right now, your approach is producing students who don't even bother with cake baking, and when pressed into service produce this http://fc05.deviantart.net/fs70/f/2012/070/1/b/ugliest_cake_...

This needs to change.

Well as a simple example, you can draw a curve on the board, then lay a ruler up against it and estimate the slope of the curve at that point. Now you have the value of the derivative at that point. Or use a car's speedometer to show the rate that your position is changing. This is an easy example for integration as well: given 60 MPH for 5 minutes, how far did we go? I'm having a hard time thinking of general solutions instead of single points, but I remember we did several in class.

You are confusing the "derivation rules" with the "derivation concept".

The rules are an easy way to calculate the derivative of a function, for example (f+g)'=f'+g'. They are important, because using the definition is tiresome and error prone.

The main idea of derivation is that each function f has an associated function f' that is the slope of the tangent lines. This is new function is useful to describe some properties of the original function. (Approximation by the tangent line, increasing and decreasing intervals, ...)

It interesting to use the intuitive idea of derivation to understand some of the rules, for example (f+g)'=f'+g'. Something like:

If when x increase 1 unit, f increases approximately 3 units and g increases approximately 2 units, then the sum increases approximately 5 units. If f or g are too curved we need to analyze smaller steps like increasing x in 0.1 units. Now f increases only approximately 0.3 units, but the rate of increment is still approximately 3, ... (It's much easier with a graph and hand waving.)

And in an advanced course it's important to transform this intuitive idea into a formal proof.

Another interesting example is the discussion of where is the minimum of x^2+16/x. The general form of x^2 is well known (graph) and also the general form of 16/x (graph) and then it's possible to draw x^2+16/x approximately (graph). The minimum is where the rate of increment of x^2 is equal to the rate of decrement of 16/x, so x^2+16/x is "horizontal" in that point. This is an intuitive idea, and is exactly what 0=(f+g)'=f'+g' says. In a class you can do the formal calculation of the derivative of x^2+16/x. Then find the exact value of x. Show it in the graph. (Explain that the graphs are approximations, so it's not strange that the minimum in the graph is slightly bigger. Ask if they have a graphic calculator?) If there is enough time evaluate x^2+16/x in a few points near the minimum to see "experimentally" that it's really the minimum.

> And in an advanced course

Definitely not talking about this as an advanced course, but a paradigm shift in Mathematics education. Teaching Calculus as early as possible and build the rest of the K-12 Mathematics education around it.

Nearly none of what you explained here would be interesting or relevant for teaching say, a 9 year old the basic mechanics of computing a derivative. As they age and their minds mature, you can start filling in the gaps and details with these important examples.

I wrote more here https://news.ycombinator.com/item?id=7337778

> The mechanics of basic calculus are remarkably simple.

Interesting comment, calculus in its modern form took around two hundred years to get a firm logical foundation.

> Interesting comment, calculus in its modern form took around two hundred years to get a firm logical foundation.

That's true, but while it languished in limbo because of some technical misunderstandings, it still produced practical and useful results, and people who used it could accurately say they understood the topic.

Calculus was eventually analyzed in depth and the various issues and objections were addressed, but these changes had little effect on those who applied it to technical problems.

A big secret in mathematics is that many ideas in mathematics are rather simple and straight forward.

Formulating and proving these ideas in a rigorous and logical manner, however, is the difficult part.

Explains my joy in puzzles and sequences. Sometimes I don't know why the next number in a series is. I used my intuition, arrived at the correct answer, and now the hard part is explaining myself.

This has, to some extent, been known for a long time. How many stories have you heard about math prodigies? I think the problem, though, is in that very sentence. Rather than considering whether this material might be more accessible than traditionally thought, we just assert outright that the child is a prodigy for learning calculus at a young age and then learn nothing from the observation.

I learned calculus in sixth grade and was and am very bad at arithmetic. It was easier for me to understand the fundamental theorem of calculus than it was for me to do long division in my head. I don't think I'm alone. I think I'm just among the few who, for one reason or another, actually made a serious attempt to learn the subject at the time.

> It was easier for me to understand the fundamental theorem of calculus than it was for me to do long division in my head. I don't think I'm alone.

You're not. I had the same experience. I have severe problems with mundane arithmetic, my mind just locks up when faced with concrete numbers and even simple numerical calculations. OTOH, no problem whatsoever with geometry, (abstract) algebra, calculus etc. I remember when I first got calculus in high school, how natural and easy it felt, how it simply made sense and how all around pleasant and interesting it was, and it struck me how different this experience was to the other more "basic and mundane" areas I struggled with during school. A few years ago I was amused when I saw an interview with an astrophysics PhD who struggled with doing basic arithmetic. It's a source of amusement for her now, but you can imagine the dread she had to endure in "school". So, no, we're not alone.

What the article says when it compares what children face in school under the label of "mathematics" to torture akin to digging a trench with a spoon sounds very true to me.

The article headline is a little sensationalist. It's very rudimentary, and it's more about learning limits than equations. That said, games like DragonBox highlight that kids are capable of being introduced to these concepts well before schools get around to it.

My 5-yr-old played DragonBox a bit - first for the colourful critters and powering up her monster, but then for the problems.

It was pretty neat when I wrote down 2 + x = 7 and she was able to rearrange the equation to solve for x. (The later stages get considerably more complicated.) It's fantastic that kids think in terms of symbolic manipulation. Most adults still don't get that, and think maths is just about coming up with numbers.

I haven't figure out how to convert the on-line intuition into pen and paper. I saw a paper on their website. It still seems like a leap, when the kids haven't formally been taught multiplication, division and fractions. I'm glad that it worked in your case.

Hmm, have you tried having them do the actual problems on paper? No numbers or letters, just copy the screen onto paper:

Like write out:

  Side 1         |     Side 2
    Fly     Box  |   Mushroom
And have them solve into:

  Side 1         |     Side 2
    Box          |   Mushroom    Inverted Color Fly ("Dark Fly")
From there it's just a visual replacement to turn Box into x, Mushroom into a 7, and Fly into a 2. The game introduces "inverse color = -", so going from 2 to -2 should be a visual translation that doesn't actually require them to understand subtraction or inverse.

After they're all done, they might notice "hey, wait, I recognize 7 -2 and it can also be written as 5!" But that's not relevant to manipulating the equation, is it?

Sure, without arithmetic they won't be able to calculate the most simple form (7 + -2 -> 5 or 2/4 + 3/6 -> 1) -- but that's a different skill. And people seem to have more issues with handling the equation than actually doing the arithmetic part.

Does anyone know of any resources to introduce math to a 6yo? I have apps like Dragonbox to introduce algebra to my son. He played with it, completed it, but now is back to the crap IAP games typical on tablets these days. I wouldn't mind a few more like Dragonbox.

I also use flashcards to help with the rote math homework.

I don't ever use flashcards for mathematics learning. Not once, not ever. (I hardly use flashcards for foreign language vocabulary either, not even for Chinese characters, except for recognition of single characters in isolation, because there isn't any one-to-one correspondence of words between languages.) Working a lot of problems[1] eventually makes the math facts second-nature, without flash cards.

I do recommend to my children and my students that they consider doing one thing I did as a kid, namely fill out a multiplication table (the one I did in childhood was 30 × 30) by hand by hand calculation. I did that in odd moments during the school day and kept the table stashed in my desk between moments of working on it. Filling in a multiplication table helps learners notice number patterns (it made me very conscious of perfect squares, for example) and produces a tangible accomplishment at the end.

For a child the age of your child, I enthusiastically recommend the Miquon Math materials,[2] which are inexpensive and take a thoughtful, playful approach informed by higher mathematics for learning much of the content of the first several years of elementary mathematics. All you need to use with the Miquon Math books are Cuisenaire rods,[3] which are available from various sellers and are a fun plaything in themselves.

[1] http://www.epsiloncamp.org/ProblemsversusExercises.php

[2] http://miquonmath.com/

[3] https://en.wikipedia.org/wiki/Cuisenaire_rods

A fun one is to fill out a multiplication table of different sizes and bases, especially once kids reach about 5th or 6th grade. I remember reading about how Sumer did math in base 12, and so I tried to replicate a base-12 table, and noticed the way numbers lined up was really different.

Brilliant! thanks.

I did the multiplication as a child myself, only 20x20. I've also tasked him with the table, 10x10.

I've used flash cards (SuperMemo and Anki) for mathematics and do not see anything wrong with using flash cards, memorizing the solutions to 500 complex analysis qualifier exam questions and using spaced repetition algorithms. I would prefer to have a perfect memory and not to (have to) rely on such techniques, but life is short. J.F. Jardine once told me that "there is no right way to do mathematics."

But I enjoy solving problems, e.g., http://publicsphere.org/2014/02/07/exercise/ http://publicsphere.org/2014/01/04/fibrations/ http://publicsphere.org/2013/12/23/markets-and-market-games/ http://publicsphere.org/2013/12/14/elementary-equivalence/ http://publicsphere.org/2013/12/12/another-problem/ http://publicsphere.org/2013/12/08/problems/

I have many more in password-protected blogs, notes, etc.

This morning a friend informed me about Oppia https://www.oppia.org/. I took a few minutes to solve their introductory example, Euler Project Problem 1 (Question: What is the sum of all the multiples of 3 or 5 below 1000?). I did this in Mathematica by combining inclusion-exclusion (expressed via indicator functions if you like) with little Gauss's formula for summing the integers from 1 to n.

SumN35[n_] := 3SumN[Floor[n/3]] + 5SumN[Floor[n/5]] - 15*SumN[Floor[n/15]] where SumN[n_] := Binomial[n + 1, 2]

I got SumN35[999] = 233168.

[OT lamentation ahead] Unfortunately, in my experience, employers don't seem to value this activity--at least I tend not to impress them as a confident person who will passionately defend his answers until shown wrong. I don't have the personality, patience and energy for this kind of bravado. (I have an acquaintance who has the requisite unaesthetic attitude, who is not more intelligent that I am and who landed a job at Google.) I tend to have inspirations that come from nowhere that I have to go back and reason about. The act of convincing yourself that a correct answer is really correct can sound like a lack of confidence to some individuals. To me, projecting confidence requires extraneous cognitive load, but I can be confident for an additional charge.]

If you tell me that using flash cards is morally wrong, or if you have a theory of learning that invalidates their use, I won't argue.

I see the article title (which I put on the submission, and one comment commented on) was replaced by an HN moderator with a rewording of the article subtitle. I guess that's okay.

The article was especially eye-catching for me, because it was first posted among our mutual Facebook friends by my mentor in homeschooling mathematics, the mother of the first United States woman to win a gold medal in the International Mathematical Olympiad. The teacher featured in the article as the developer of innovative teaching methods for young learners is also a Facebook friend of mine, and we are part of a network of parents and teachers who are curious about how to stimulate interest in mathematics among young learners by doing things differently from the United States typical school curriculum in mathematics. I'm grateful that my children have been exposed to approaches like that (which I follow imperfectly in my homeschooling my children and in teaching other learners in classroom courses for my occupation), as that has helped my oldest son launch into the adult world with good problem-solving skills for hacking at the startup where he works.

There is a continual tension in mathematics education between teaching topics in their logical order as we understand mathematical topics in light of modern higher mathematics and teaching topics in their historical order of development (which is largely what happens in school mathematics courses). Children need some concrete, tangible experience with counting and with shapes to understand much about mathematics. The cool abstractions that motivate higher mathematics may be inaccessible to children who have no experience with tangible examples of those absractions. (See


an article submitted to Hacker News yesterday, for more about this.) But it looks like we can gain in mathematics instruction by letting children play with more sophisticated representations of mathematics than most children get to play with. Games are great learning tools. One of favorite games for teaching mathematics I learned about from John Holt's book How Children Fail (originally published 1964, which I read in 1971 on the advice of my school's assistant principal). The twenty questions game asks children to find a number from 1 to 1,000,000 by asking twenty yes/no questions, which the person who chose the unknown number must answer truthfully. It is an interesting mathematical exercise to show that twenty questions is (barely) enough for finding one number in one million if the twenty questions are used optimally. I have played this game many times with the children in my classes, and the opportunity to play this game again at the end of class is one of my strongest incentives for the children to stay focused during a lesson on a Saturday morning to get through the lesson content efficiently.

Playing with challenging problems appears to be the royal road for learning mathematics.


I hope ideas like this spread through many communities in the United States and the English-speaking world, so that more young people gain more opportunity to learn to enjoy challenging mathematics early and often.

As for math games, I can recall a few interesting examples:

Algebra for children: http://www.wired.com/geekdad/2012/06/dragonbox/all/

Interactive tutorial for sequent calculus: http://logitext.mit.edu/logitext.fcgi/tutorial

Ancient Greek Geography: http://sciencevsmagic.net/geo/

DragonBox on iOS is pretty good at teaching algebra. http://dragonboxapp.com

++ Dragonbox. also on android

I have an intuition and a fear.

My intuition is that it is worthwhile for me to try and add in the best of homeschooling to my children's time after school / weekends, and that is probably linked to some of the great experiments of science (https://github.com/mikadosoftware/importantexperiments4kids)

My fear - well, that even at four years old my son seems to be labelled as falling behind, and that my own attempts to persuade him that large chunks of life do not flow from school may not be enough. I doubt very much that there is one true way to learn any subject - those of us who have mostly lucked into the royal road as above. There are many many of us who stumbled and were simply left, and that I suspect is the fear and motivation of most Tiger mums. It certainly is mine, as a tiger dad.

Why is he even thinking of school at age 4?

And how can you know if the great experiments of science are important for his schooling if you haven't tested them with him? I think there are many ways to learn about engineering and science. Looking at historical experiments is just one of them. Perhaps your son will enjoys other things better (like building Lego models)?

I hope you can have some fun together.

What ways of exploring science would you enjoy?

Maybe he doesn't like engineering, but likes biology? There are so many things in the world...

I think I was over reacting (we had some feedback from his teachers - yes he is at reception class age 4)

he loves Lego and I encourage it. I have not tried to calculate the size of the earth with him yet but it's experiments I want to be able to do so I can guide him.

I am reluctant to "just chill" because if I do he might not be encourGed or helped at the right moment - even writing that down I realise how silly it sounds

Didn't know school starts so early where you live. In my country it starts at 5, or preferably 6.

I probably overreacted, too. It's hard to take care of a kid. My kid is only 3, but I know some "genius" kids started at age 4 (like Mozart, or Tiger Woods or the Polgar sisters). Nobody wants to talk about it, but my impression is that it is possible to train for special talent. I don't want to spoil my kid's childhood, though, so I only want to encourage such things if he enjoys them (and also I am not really qualified for teaching many things).

Also, I'd say that school is overrated...

By the time kids are three or four, they learn a large part of their native language. It's actually pretty complex - at least on par with algebra - but they learn it, because other people around it do that. Imagine the world richer in math, where young kids are immersed in it from young age...

The best I can say is ... apprently parents have quite a small influence on the kids adult success - genetics, peer groups, and I suspect luck have a bigger role than us.

I am not fully convinced. Parents try to influence peer groups (by choosing schools and neighborhoods), and no matter what role genetics play, you still want the best outcome the genes allow. Luck is of course a big factor.

I am not advocating "genius training" because I am not sure what it does to the kids. But if they enjoy it, why not.

Thanks for your interesting comment! I really love John Holt's writings about children and learning.

I know that when I learned calculus the hardest part was reconciling the notion of infinitesimals with algebra. The delta/epsilon definition of a limit was unsettling. It's a shame. Much of calculus is easy to grasp intuitively - Riemann integration is easy to see and explain.

Yes, the concepts of limits and derivatives are simple and intuitive, but the concept of the epsilon/delta proof just completely soars over students heads. I didn't learn the epsilon/delta stuff until after all that other stuff, and I don't feel like it was the wrong order of things.

That and the prof I had never just really explained that epsilon just means "really insanely small" which probably would've helped.

I had to learn the epsilon-delta definition before I really understood limits. As far as insanely small goes, as you must know, the definition is equivalent to "for every epsilon < M, there exists a delta >0, ..." where M is some positive constant (it could be 1 or 1/10^100...). Sometimes it helps to play with the quantifiers...

In my university, the first calculus course for most of the people uses only an intuitive definition of limit (without epsilon/delta or epsilon/n_0). Also the definition of derivative and integral is intuitive. The idea is that the students can do the calculations following some rules. And then see some applications, like the tangent line of a function or the area below a function.

The first calculus course for engineers and exact science students is more difficult, but the discussion of the technical details of the limit is very small. Perhaps calculate the limit by definition only for 2 or 3 easy functions, and then just use the algebraic properties of the limit.

The second course for Math and Physics students has more time assigned to the calculation of limits by definition.

Actually, limits and infinitesimals are two very different explanations/foundations for calculus. Infinitesimals are much more intuitive.

Up to last decade, limits were the only rigorous enough explanation. Until someone invented an equally rigorous reconstruction of calculus with infinitesimals.

Relevant anecdote: For a high school ICT project my class had to make an educational game in Visual Basic. I made a 'math invaders' clone where you answer simple algebra sums (e.g. 5 + x = 12) to defeat the relentless hordes of aliens. At home my 8yo brother watched me play for about 5 minutes, then exclaimed 'Oh I get it, the letters are like numbers that you have to find!', then went on to play the game himself for hours. He learnt algebra about 6 years before any of his classmates through gamification, and years down the track enjoys math more than many of his friends.

This looks extremely interesting, I have a 5 year old daughter that I want to see the playful side of mathematics before she hits the typical US Maths curriculum. One of my guiding documents has been "Lockhart's Lament" [1], but trying to figure out how to make that a reality for her has been difficult for someone like me with so little time! I am going to check this out tonight!

[1] http://en.wikipedia.org/wiki/A_Mathematician%27s_Lament

If you're interested in this kind of work, and especially if you're also interested in using computers to open up new kinds of mathematical play and learning, you should read Seymour Papert's "Mindstorms" book.

The ideas are powerful and radical. Radical enough to explain why run-of-the-mill schools never successfully integrated awesome tools like LOGO and its descendants. They can't deal with learning that's so child-driven and free.

There was a study done in a place quite unexpected to me now, and interesting back then when I was in school myself.

My Math Teacher told there were some schools where a pilot was conducted teaching calculus concepts from 5th,6th std students and were monitored till they did their graduations and career paths.

Interestingly the people who were chosen randomly, performed equally well and their decisions in addition to math were more logical.

This study was done in India :)

"Revolutionizing the way math is taught" and not "memorizing multiplication tables as individual facts rather than patterns" sound like very worthy and important goals. At the same time, it's hard to judge this particular initiative based on the article; it might be Hippie Math for Rich Kids Who Will Study Humanities Subjects and Become Trustafarians by a Berkelely Yoga Instructor"

> But they also need to see meaningful (to them) people doing meaningful things with math and enjoying the experience.

I was wondering if this was essential for the learning process of five or six year olds.

Making maths enjoyable for young children would be the primary concern, but showing something meaningful (not necessarily to them) being done with maths by someone meaningful (to them)? It would work better if:

The kids require to see something that is relevant to their interests or needs being solved with maths by any old codger, as long as said codger can adequately manipulate algebra.

By the way, who is going to teach these children algebra? A fair number of teachers won't be up to the task. Reminds me of the learn to code buzz-phrase being passed around - first off you need to spend money and time on teaching teachers how to code.

EDIT: bloomin' asterisks.

http://siterecruit.comscore.com/sr/atlantic/broker.js is holding up the entire site for me. Async, anyone?

  “What is learned without play is qualitatively different. It helps
   with test taking and mundane exercises, but it does nothing for
   logical thinking and problem solving. These things are separate,
   and you can’t get here from there.”
This is the core of what people mean when they — legitimately — talk about ‘passion’. Did you learn computing (or any field) for fun, or alternatively because you thought it paid well, or your parents wanted you to do it, or whatever.

(Not to be confused with ‘passion’ as mbaspeak for ‘naïvité enough to work ridiculous hours for peanuts’.)

Maria co-authored a great book aimed at making higher level math concepts fun for schoolchildren titled Moebius Noodles. The site appears to be down now, but here's the FB page: https://www.facebook.com/moebiusnoodles

I highly recommend it, as well as signing up to receive periodic emails from Yelena/Maria with additional activities, tips and information.

Obligatory Star Trek reference: http://www.youtube.com/watch?v=ETt8GJRbqLc

I've thought for a long time that an abstract approach might work better than a concrete approach for introducing children to math.

I remember when my kids were in kindergarten, and their math education started with estimation. My first thought was, "Great. Now they've turned math into a touchy-feely all-answers-are-right-answers nonsense subject!"

I don't want to turn math into a touchy-feely all-answers-are-right-answers nonsense subject, but estimation is a good skill in math, engineering, and computing.

Two examples: Fermi problems[1] and Jon Bentley's "back of the envelope"[2] estimates.

[1] http://en.wikipedia.org/wiki/Fermi_problem [2] http://www.csie.fju.edu.tw/~yeh/research/papers/os-reading-l... and http://www.eecs.harvard.edu/cs261/background/p176-bentley.pd...

I agree that estimation is important. But so important that we start children with it? That smells wrong to me. Estimation is about intuitive thinking. Mathematics is usually a child's first introduction to formal, rigorous thought. Don't go soft on it.

I think estimation is an important early concept - often I'll give my lad a question and he'll respond with a first answer that's ludicrous. "Now think about it," I'll say. "If you've only got 30 sweets, how can all 7 people get more than 10, does that sound right? Everyone has to get less than 30, surely?".

It's about getting a feel for quantities and units of measure.

I definitely appreciate that math is much more than numbers and equations...

On the other hand teaching a kid math has to at some level at some point revolve around taking some numbers and getting other numbers out...

I cant really see how one can do algebra without knowing arithematic. algebra is fundamentally a different beast but you just cant do it without some kinda arithmatic

> On the other hand teaching a kid math has to at some level at some point revolve around taking some numbers and getting other numbers out...

Yes, but perhaps not the very first thing. To me, the first thing should be interesting, aesthetically pleasing patterns -- images -- that have mathematical meaning (http://arachnoid.com/mandelbrot_set/graphics/title_image.jpg). Then, when the student's curiosity has been aroused, we can "open the machine" and show its inner workings.

> I cant really see how one can do algebra without knowing arithematic.

But that's easy. I'm not necessarily recommending it, but one can easily have algebra without numbers. Remember that the most useful algebraic equations are those in which the numerical values are irrelevant.

Ever since I took Discrete Math in College, I have thought it would be a great way to introduce school kids to Math.

Does anyone else find math like trig and calculus dramatically easier to comprehend than stuff like combinatorics?

Combinatorics has always been easier to comprehend for me, but the problems have always been harder. That's because calculus depends on higher-level concepts but is otherwise rote, whereas combinatorics depends on only the most elementary concepts but the problems quickly become very complex. I think this is why combinatorics features so much more strongly in math competitions than does calculus.

Am I the only one that enjoyed math more earlier than now? I loved calculus and geometry in high school and undergrad, but now in graduate school I can't follow the physicists at all, and my eyes just glaze over.

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