As for constructivism--I agree that math instruction should be grounded in developing conceptual understanding (this is why I love the Singapore series). However, I don't think it is necessary for a child to "discover" every last concept and algorithm for himself. One can "construct" knowledge without "discovering" it.
I would agree with this. There are many aspects of constructivist approaches that I have come to appreciate relative to the traditional, algorithim/property memorization style of instruction I recieved, but there are some significant weaknesses as well. One of the greatest weaknesses I see is that students don't learn the same appreciation for precision and efficiency. It's interesting to me that constructivist approaches in all areas emphasize focusing on process vs. content without recognizing that learning and applying rules is itself a process that needs to be developed. I definitely favor a combined approach to instruction in all areas. As it relates to math specifically, I think that constructivism--at least as it is applied--overlooks the fact that while some students/adults work best by gathering parts and constructing a whole understanding, other students and adults work best by deconstructing the whole,examining the parts, and then imagining or exploring different ways to work those parts to get the same, or a similar, result.
For sure we don't want students to look at problems like 101-99 and set it up with a standard algorithm (something I too have seen students do on multiple occasions), but I think we do want students to have efficient tools for the computations that aren't easily done without pencil and paper. I like to work on having students identifying for themselves problems that they need algorithms for and problems they don't need algorithms for so that they are in the practice of considering the tools they have and choosing the best tool for the job.
Re: fractions. Fractional understanding is probably the area in which I most support an almost universal constructivist approach, at least with initial concepts. If a student can't grasp the relative size/meaning of a fraction, I question the wisdom of teaching them to solve equations which include fractions.
