Pretty much nothing works if it's done badly. many things work if they're done well. I claim that when done well, exercising skills by hand that can be done by the computer yields insight in some cases.
the fact that when done badly it rarely yields insight does not necessarily mean it shouldn't be done.
I'm about to make a sweeping generalisation.
The problem is virtually no one in the world can teach math properly. Those who love teaching kids usually hate math and transfer their lack of skills and negative attitude. Those who really, really understand math are usually the type of people who would never teach 3, 4, 5 year olds.
People end up doing math despite the system, rather than because of it. They can usually point to one specific inspirational teacher. My specific inspirational teacher gave me lots of mind-numbing exercises and helped me find the insights.
Rote learning of mechanical processes will rarely, of itself, lead to anything other than a hatred of math and a lack of understanding about what math is. Repetition of mechanical processes, guided by someone who really understands stuff, and asks prompting questions, can lead to discoveries and genuine excitement and engagement - I've seen it happen.
Not to minimize the importance of teaching, but the content is important too. Instead of the content consisting of memorizing and performing algorithms, hoping for students to discover meaningful patterns/concepts/insights or rely on teachers to provide them, maybe the content should be doing this explicitly. One topic in the high school syllabus which actually does this is Euclidean geometry, where everything is coherently derived from basic principles as opposed to a grabbag of techniques. Maybe, one could present arithmetic in the same way, not focussing on the how to do calculations but the patterns in these calculations and a few basic principles too discover/prove them. Alternatively, one could focus on applications of mathematical techniques in toy versions of real world problems.
Arithmetic algorithms still have value - for the insight they give on arithmetic and because following a complex algorithm is itself a skill with value.
But there is no need, as we do now, to insist on performing them so many times, or to do them very quickly in exams.
The main problem with this new approach, I feel, is that it makes learning mathematics harder. Building richer conceptual models which is necessary for both applications or theory is more interesting and meaningful, but also more difficult than following prescribed algorithms. It is harder to test in an exam, and where testable the problems are much harder.
This issue of algorithms vs conceptual understanding, is important at the undergrad level too. Eric Mazur has a nice video about this where he also talks about his way of testing conceptual understanding by asking very simple but illuminating questions - http://www.youtube.com/watch?v=WwslBPj8GgI
Thank you for that Eric Mazur talk -- it really captures what I see. In particular the part where he talks about how his students, at Harvard, were solving triple integrals of complicated bodies to calculate the moment of inertia -- yet they didn't have basic high school level intuition of Newtonian physics.
We're so "drill, drill, drill" focused that we lose sight of why we're doing the drills (Eric Mazur calls it "plug and chug"). It's possible that these drills will in rare instances create the likes of a Colin or an Andrew Wiles, but for virtually everyone else you have a group of students that can solve triple integrals, multiple 5 digit numbers in their head, factor matrices into any form desired -- yet not have a clue why.
the fact that when done badly it rarely yields insight does not necessarily mean it shouldn't be done.
I'm about to make a sweeping generalisation.
The problem is virtually no one in the world can teach math properly. Those who love teaching kids usually hate math and transfer their lack of skills and negative attitude. Those who really, really understand math are usually the type of people who would never teach 3, 4, 5 year olds.
People end up doing math despite the system, rather than because of it. They can usually point to one specific inspirational teacher. My specific inspirational teacher gave me lots of mind-numbing exercises and helped me find the insights.
Rote learning of mechanical processes will rarely, of itself, lead to anything other than a hatred of math and a lack of understanding about what math is. Repetition of mechanical processes, guided by someone who really understands stuff, and asks prompting questions, can lead to discoveries and genuine excitement and engagement - I've seen it happen.
How will we ever get really good math teachers?
There I have no answer.