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"do you need to learn how the engine of a car works before you can drive it?"

No, but you'd be a much better driver if you did.

Understanding the basics of a science, especially math, is extremely important to fully grasp the higher level topics. I don't believe one can conceptually understand a topic without understanding the fundamentals that it builds upon.

Unfortunately I see philosophies as you describe in my local school system (Chicago Public Schools), especially in poverty stricken neighborhoods. High school kids are given scientific calculators and taught algebra/pre-calc when they can't even add or multiply single digit numbers. Similarly, there are pushes to move more "computer learning" into grade school, despite extremely poor reading and math proficiency.

I can draw parallels in my job as well - it's remarkable how poor most programmers' basic math skills are. Someone who can do the chain rule is regarded as a genius. With only a strong grasp of fundamentals, I feel that many engineers could be substantially more successful.

> No, but you'd be a much better driver if you did [learn how a car engine works]

Yes, but one simple computer animation could give you the same understanding of the engine as if you had built the engine yourself. Computers can give you an understanding of calculations in addition to doing the work.

> I don't believe one can conceptually understand a topic without understanding the fundamentals that it builds upon.

This is probably the biggest argument against using computers to calculate for you. However, if you haven't already, watch Wolfram's video that I linked to above as he clearly explains how this belief is not necessarily true. The very rudimentary basics will need to be learned by calculating, but anything above the fundamentals can be understood without the calculations.

> Similarly, there are pushes to move more "computer learning" into grade school, despite extremely poor reading and math proficiency.

I think this computer learning is different than the learning I'm talking about. Schools currently use computers as a medium to enhance the current education system. They use them to make calculating more interactive and fun. I think that computers should actually do the calculations, rather than "tricking" the students into doing the work.

> Yes, but one simple computer animation could give you the same understanding of the engine as if you had built the engine yourself.

That must be one impressive animation.

I mean, to what depth do you really need to understand an engine in order to drive it better?

Relevant (but very basic) example of what I think you're describing: http://www.animatedengines.com/

It worked very well for me personally and gave me quite a few "A-HA" moments. I'm not sure it helped with my driving however ;-)

> The very rudimentary basics will need to be learned by calculating, but anything above the fundamentals can be understood without the calculations.

Generally, I agree with this. But if everybody were to adopt that philosophy, eventually we'd get to the point where nobody could actually implement the calculations for those high level concepts. And then what happens when you need to update the software that's been doing all of that for you?

Does everybody need to know how to calculate? Probably not, above arithmetic and some basic algebra. But a lot of people do, and that needs to be included in their education somewhere.

You're probably aware of all that, but I felt it warrants making explicit.

I am aware of this, but still, thank you for bringing it up as it is an important issue. The software engineers and people who have to interact with more calculations than everyone else would at sometime need to learn them - as you said.

I think speleding is onto something, though: http://news.ycombinator.com/item?id=2213985

We all believe that calculations need to be taught first, but maybe it is not necessary. Learning the calculations after the bigger concepts has its own advantages. Everyone could start with general education, and those that need it could be taught the more advanced (or one might say, basic) calculations.

> No, but you'd be a much better driver if you did.

Agreed, but it might still be a good idea to change the order in which you teach them: first learn to drive, then learn how the car works inside. There is ample evidence that the human brain is better capable of understanding if it understands the big picture first (for proof see: "The pyramid principle" by M.Minto)

No, but you'd be a much better driver if you did.

And I think that the time is better spent learning to drive. That's just an unfounded opinion, but at least I can see it as such. Do you have any evidence that understanding engines makes drivers "much better"?

That too much emphasis is put on manual computation is obvious. But taking the opposite tack is an overcorrection. Computing and solving by hand is a font of endless insight in mathematics at any level and should not be left to computers.

Take an elementary matter such as convergence of infinite series. If you've never hand-computed a truncated power series for e^x term by term and noted its rapid convergence rate for positive x, I dare say your understanding of basic calculus is impoverished. Once you've seen the convergence first hand for particular values of x, the proof of its general convergence immediately suggests itself. But if you've only seen the abstract proof, you would probably never have realized that the e^x power series has extremely initial poor convergence for negative x. (Try it for x = -10 to see what I mean. Poor convergence and massive cancellation. That's why numerical analysts always evaluate e^x as 1/e^(-x) for negative x.)

If you were going to major in math, then you may need to go back and learn all the basic calculations in order to build on them. But for anyone else (even scientists), the solution I mentioned above should be enough. Your example of e^x is not closely enough tied to a real world application; therefore it's hard for me to work with it.

> If you were going to major in math, then you may need to go back and learn all the basic calculations in order to build on them.

The way it actually happens is that once you start majoring in mathematics, you will rarely see any computations. It's entirely possible to take a linear algebra course for mathematicians without ever row-reducing a non-trivial matrix by hand.

Back to teaching kids. If you take a constructivist approach to learning then computing by hand simply cannot be avoided. You'll never have a student notice on their own that, for example, you can quickly multiply x by 9 by subtracting x from 10x unless they tinker with numbers directly. You come across patterns like that by doing the tedious work the hard way and looking for short-cuts.

> You come across patterns like that by doing the tedious work the hard way and looking for short-cuts.

The thing is that you come across short-cuts like those on the larger and broader concepts as well. Tinkering around with numbers is great, but perhaps tinkering around with concepts is even better.

There would be downsides to the reform. Students may not know that notice 9x = 10x - x as a calculation shortcut, but they then again, they would never have to calculate 9x past a certain value of x. Computers can easily do that - let the students discover that x needs to be multiplied by 9, and then the let the computer do the straightforward math.

Maybe we're talking past each other. Could you explain your goals with this curriculum? Are you trying to reduce the teaching of mathematics to a bare minimum so people can get on with what you see as more essential subjects?

My stance is that mathematics is first of all useful in the small for normal people. Everyday arithmetic, basic accounting, that sort of thing. But beyond that we need to teach mathematics for the same reason we teach art, music and literature. For achieving this goal, even if what we aim for is an appreciation of general principles, I was arguing that properly directed hand computation and concrete problem solving plays an important part. Concrete tinkering engages your brain in a complementary way to reflecting on nakedly abstract principles.

One of my goals with this curriculum is to make teaching mathematic more efficient. This does not necessarily mean teaching a bare minimum. Rather, students should be taught a great amount more at a younger age. In fact, I think mathematics is probably one of the most essential subjects out there. Unfortunately, the way it is taught now is inefficient and gives it a bad name.

The rudimentary basics are important for everyone, and mostly those who will not pursue math-related careers. However, it is the content that is taught after the basics (middle school and high school curriculum) that could be spent on more advanced areas with less calculation work.

I feel embarrassed expressing all of these ideas with little background to support them. I'm sure you are a lot more knowledgeable on this subject than I am, so thanks for taking the time to acknowledge to my arguments and discuss them with me. ;)

By the time an ambitious student finishes high school, they will have learned some manipulations in linear algebra and calculus by rote but they won't have any real understanding. If they enter college and study some further mathematics, the first task of their professors will be to undo the damage.

We don't need to teach more, we need to teach better. We should not chase nominal accomplishments such as whether students have "covered" differential equations and discriminants by high school's end. There is already too much of that. The same is even more true for the foundational material taught in middle school. If you skate across the basics in an effort to cover more, earlier, you risk serious damage to the students' development in mathematics and science.

The problem all comes down to 10% curriculum and 90% teachers. Curriculum only seriously concerns me when it overconstrains good teachers and prevents them from doing their job.

"The problem all comes down to 10% curriculum and 90% teachers. Curriculum only seriously concerns me when it overconstrains good teachers and prevents them from doing their job."

I agree with this in part but the problem is that relying on good teachers is not something that can be scaled across countries. I'm not sure if the introduction of computers in the national education might help solve this problem, but I'm hopeful.

But beyond that we need to teach mathematics for the same reason we teach art, music and literature.

Perhaps like art and music, we should only be teaching beyond the very basics to those who actively seek it out. We don't require all highshool students to learn how to play the piano, read music, or draw nudes, so why should they all learn how to invert a matrix or find its determinant by hand?

Because mathematics is part of what defines us? You could apply your argument to any topic taught in school: history, literature, physics, ...

>You come across patterns like that by doing the tedious work the hard way and looking for short-cuts.

This doesn't work for me. After a little bit of tedium my mind quickly switches through "just fight your way through this" and no more insight will be gained. All the math shortcuts I know I either read about or discovered while playing with numbers on my own time without any homework deadlines.

Yes, an important point is that everyone learns differently. Aimless computation is the furthest thing from my mind, and I agree that most people wouldn't notice these patterns if left to their own devices. Computation as an accessory to constructivist learning guided by teachers is what I'm suggesting must be part of a solution.

I'm skeptical of curriculum reform that seeks to excise an important source of learning for a sizeable subset of students.

K-12 isn't generally about learning. It is a daycare for children and young adults.

(I am 23 now. I left school when I was 17 to pursue a career in the gamedev industry. Daycare was not the place for me, and I haven't regretted dropping out.)

I know. Talk about a waste of time. Turns out to be 70,000 hours that each person spends in K-12 school. Add any outside school work and that could be closer 90,000 hours.

I thought your number was quite mindblowing but then I worked out that 70000 hours is about 8 years if you were there constantly, 24/7.. 90000 waking hours is over 15 years.

I worked it out at around 15-18k hours (depending on where you live and variations in school day length).

You're right, I was off by quite a bit :) 15k hours seems more like it.

In a way, 70,000 hours is a lot closer to the true number.

(365 days) * (24 hours) * (12 years) = ~105k hours, or 70k waking hours

It isn't until fifth or sixth grade that you can really start finding a passion.

I'd say I managed to spend 5k to 10k hours on learning about technology and programming. I had to completely ignore high school. I literally slept through it. Teachers gave up trying to get me to try; I was also a social outcast.

In the book Why Don't Students Like School ( http://www.amazon.com/Why-Dont-Students-Like-School/dp/04705...) Daniel Willingham argues that people need to have a good vocabulary of mechanics, basic concepts, and skills in order to be able to move on to higher order knowledge. Children should learn how to calculate not necessarily to do calculations instead of a computer, but to help their brains develop.

Besides, until kids hit puberty, their brains aren't ready for higher orders of thinking. What they are good at is memorization and mechanics. The more mechanics they learn and the more they memorize as little kids, the better equipped they will be for higher order thinking as teens and adults.

However, we are still teaching and emphasizing arithmetic and memorization in high school — even in the advanced classes. That's the problem, really.

Against popular belief, concepts can be understood without learning the nitty-gritty mechanics.

I disagree. The problem is that you don't know that you understand a concept until you actually can apply your conceptual understanding to accomplish something. The world is full of people who "understand" quantum mechanics or relativity in terms of half-baked analogies and are unable to apply those concepts to anything but pseudo-scientific discussions.

Real conceptual understanding comes form having worked with the subject area enough that you have built your own conceptual framework. You can't really just teach someone a conceptual framework, because the words you use to describe your framework will likely be interpreted in a completely different way by someone else.

One reason educational material is poor is that it's inevitably written by someone who finds it hard to imagine what it's like not to understand it.

This is why all such material should come with commentaries by thoughtful students which explain it in different ways and link to additional resources. And with commentaries on the commentaries, etc. (We all have different misconceptions.)

There also needs to be a mechanism for rapidly correcting errors. Not every few years but every few days. (Text books are full of errors.)

All of which suggests that we need more educational material which is online, open source and wikified.

Computer-use should not replace the nitty-gritty mechanics; it should change the way students learn the nitty-gritty mechanics.

Instead of a homework assignment like this: 1. compute this integral. 2. compute this integral. 3. compute this integral. 4. compute this integral. 5. compute this integral.

Students should be getting homework assignments like this: 1. compute this integral. 2. write a procedure to compute any integral.

Thats a tricky thing to do though, where do you draw the line?

That's a good question. We need to keep in mind that computers are never going to go backwards in innovation. Once a computer is capable of some calculation, students will never need to have to do it by themselves again. However, they will still need to know how to interact with that calculation and understand the variables/relationships in order to set up the calculation.

Where do you draw the line?

Instead of just one line, you have to draw two lines. One would be on the more elementary end of the spectrum. Students have to learn the rudimentary basic calculations (or memorization) in order to function well in everyday life and higher levels of education. Simple addition, subtract, division, multiplication, estimation, understanding of relationships would be necessary. As much as I adore algebra, it could be left to the computer. Therefore the first line would be drawn at a point where the material does not show up in everyday scenarios.

The next line would be drawn at the opposite end of the spectrum where humans need to come in and do the higher orders of thinking. Given the right software, students would be able to visualize and understand the basics of the calculations and their relationships without actually calculating them. They would therefore be able to use computers in more applied problems - ones that they would be confronted in their careers later on in life. A few experiments could be conducted to find exactly where this line would be drawn. It would be at the point where humans do the smallest amount of calculation work while still having a solid understanding. Computers may be able to work past this line, but they would end up assisting the students to the point where the students lose the understanding of the concepts.

If the US and other countries truly put the time, money, and effort into this reform, I'm sure they could be quite successful.

However, another tricky thing to think about would be transitioning into this new and reformed type of education...

Once a computer is capable of some calculation, students will never need to have to do it by themselves again

However it is important that they know enough to check if the answer they get is at least reasonable and probably correct. So even if they can't solve the problem exactly they really should know how to show that the answer must be between 1/sqrt(2) and 1, for example, so that they know they made a mistake when the computer tells them the answer is 0.3475.

Yes, that goes along with understanding the basic relationships and estimation techniques. The exact calculations can be left to the computer, and the humans can check the solutions and determine what went wrong if the answers don't seem to be what they had expected.