Edit: also on computational thinking, I'm not sure that's what people are arguing for http://worrydream.com/MeanwhileAtCodeOrg/
A famous counter-example is an abstract selection task which adults fail with high probability. Yet, when a structurally identical set of rules are given to ~9 years olds but couched in the language of permission, the children are able to pass with high probability. Stating the rules in terms of what the individual has experience with (or altering the linguistic phrasing) significantly increases the probability of solving much more than adjusting the age. So we have that on one hand, people develop capacities sooner (and more unevenly) than his theory supposed and that on another, some stages of reasoning aren't ever reached.
amasad is wrong to imply that the field has not progressed beyond the ideas of Piaget. Although, people do often get the important parts of Papert's ideas wrong by focusing too much on the computer part of computer-based learning. The computer should be a means to an end, a tool to help the learner explore more possibilities, make the abstract more graspable and amplify one's ability to ask better questions.
> .. the most important thing about any communications medium is that message receipt is really message recovery: anyone who wishes to receive a message embedded in a medium must first have internalized the medium so it can be “subtracted” out to leave the message behind.
(a tangent, but perhaps interesting)
As for block-based programming: that seems like something we can test empirically, and indeed there’s quite a bit of literature about it (which I have not read and am not familiar with, sorry).
Child centered, problem centered is vague. More practically, the big things are: you want material that's novel to the learner but also somewhat familiar (this is true, regardless of age). Children's attention is more distributed and worse at blocking out irrelevancies. Therefore, more complex tasks (such as math) should be presented in a way such that the signal meant to be learned can be extracted with minimal ambiguity (less extraneous information in presentation). Few other things have replicated.