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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 en

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.)

solipsist 16y ago
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.
- psykotic 16y ago
  > 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.
  - solipsist 16y ago
    > 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.
    - psykotic 16y ago
      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.
      - solipsist 16y ago
        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. ;)

solipsist 16y ago

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.

psykotic 16y ago
> 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.
- solipsist 16y ago
  > 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.
  - psykotic 16y ago
    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.
    - solipsist 16y ago
      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. ;)