When I was growing up there was computer boom in Iran. So, almost every students in the major cities was aware of computer and its abilities. Or at least they had some advanced calculators to numerically or symbolically calculate equations and integrals.
It was since the high school where I started to hear this nagging noise complaining that if we can solve these equations or integrations by a computer why we should do it? Honestly, I was fan of solving integrals, and equations analytically and making relevant approximations myself. I never bought and used calculator for calculations to the end of my university degree, at least for tractable problems, although I was constantly using computer to double check my solutions or to create graphs and figures. However, I could not give a clear answer to those classmates who were constantly complaining about that and refusing to learn advanced calculus, since it seemed unnecessary.
After some years, now I believe that I have an answer to this complaint. Setting the beauty of finding solution and [temporary] feeling of achievements aside, solving some challenging but tractable equations, or integrations analytically, first it helps to appreciate how the mathematics practically works. Second, it improves our brains imagination by closely observing the dynamics between similarity and differences which leads to the final solution. By gaining this ability we would be able to also observe similarity and differences in facing real world problems.
In my opinion [and many others], our general world view is directly related on our perception of happiness and life. How we live depends on how we think we should live. All these questions would obtain different solutions if our brains are trained to see many different possibilities by looking at a certain problem.
In conclusion, I believe that, solving the tractable problems using our brain not only helps to be innovative in facing new mathematical problems in our future careers, but also it would have life changing consequences, since we can see life with higher resolutions. Relying on computer too much, particularly, for kids would lead to replace our dynamic and innovative brain which is capable of creating art, music, math, etc, to some body part which only controls our involuntary moves.
For me the late Richard Feynman was one of those people whom solving simple yet chanllenging mathematics gave him different views in life. Some of his views about the future of science and technology is started to come to reality just recently. I encourage all of you to the following BBC interview with Feynman: http://www.youtube.com/watch?v=EKWGGDXe5MA