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.
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.
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:
EDIT: bloomin' asterisks.
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))
sin(x1+x2) = sin(x1)cos(x2) + sin(x2)cos(x1)
cos(x1+x2) = cos(x1)cos(x2) - sin(x1)sin(x2)
ok, hold on...
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))
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).
you mean the ideas of a dual space of a linear space and of tangent bundle to a manifold? :)
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.
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.
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?
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.
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"
"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.
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.
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
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.