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xMathFanx:
Yes, it is my experience that effective and systematic training and practice can increase most peoples' computational speed and capacity. I am not convinced however that this will lead to a generally enhanced ability to learn and mahage more complex concepts unless the training is directly related to the new concepts being tackled. So for example, arithmetic training will likely increase a students capacity to learn and manage mathematical concepts that have arithmetic foundations like mastering non-decimal number systems or more easily doing complex algebra but in areas of maths where human arithmetic skills play a small role like topology, fractal systems and the mathematics of infinities, such training would likely play a much more marginal role. An innate talent to visualise and spatially map concepts would be much stronger than such training in arithmetic in such situations.
Taking your example one step further and broadening the skill set being trained, today we train and require students to practice formulating questions, research, learning, analysis, critical thinking and cognitive synthesis as a systematic body of intellectual skills. Being well trained in these skills allows students to better manage the torrential flood of data which comes at them these days and makes managing the tsunami of information which inundates their lives a little easier. But these skills are social skills with built-in social biases and filters which can also blind students to certain types of data which do not fit into the 'box' which our social intelligence based learning biases have inculcated into them. That is when innate intelligence clicks in and allows the rare and gifted student to see 'outside the box' of the received learning process, in order to see patterns and relationships which our biased learning process has filtered out or occluded. These are the 'eureka!-moments' which can shift cognitive paradigms and lead to real progress or even revolutions in human understanding. Unfortunately out-of-the-box thinking can also lead to ruin, madness or societal marginalisation, censure or worse, as such thinking is disruptive and can threaten powerful societal interests in the status quo. So it's a double-edged sword.
The innate intelligence is very powerful and can reshape received knowledge and the social intelligence built upon that foundation of received knowledge, but innate intelligence plays an infrequent and discontinuous role in advancing human understanding. The social intelligence, while limited in its capacity to push boundaries and open new frontiers of inquiry, is the work-horse which pushes human understanding forward by consolidating and institutionalising the great gains of the eureka!-moments into received knowledge and new iterations of social intelligence. At least, that's what I think is happening in human understanding. I could just be a myopic fool who has deluded himself. That's for others to judge.
Cheers.
Evilroddy.
