Here's a diagram which I offer as a tool for thinking. All diagrams are partial. They are only in 2-dimensions after all.
In the traditional paradigm, the maths to be learned in a lesson is selected from the mathematics which has been proved and written down.
In the constructivist paradigm, however, the focus is instead on nurturing the way that students mathematise the world around them. Because the focus is on the thinking processes to be developed, the maths to be learned is not clearly specified. Hence it becomes irrelevant as to whether it is mathematics which has been previously proven or not. In my opinion and experience substantial amounts of mathemics is learned during these lessons, but that mathematics is most powerfully learned when it is backed up with some traditional teaching at another time. There is also a job to be done to fill in the gaps of core mathematical techniques which have not been learned.
But - oh - the magic of invention. When students have aha moments. Which a child expresses something incredible that you've never imagined before. The intellectual journeys you take as a teacher. Please excuse me while I get a little misty eyed. I can understand why some teachers who discover the constructivist paradigms for themselves can become rather messianic about it. I wouldn't want to teach without it being part of my teaching. It inspires me. My students inspire me.