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?
>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.
