The MOST effective style for math and science.
If this NEW forum responds to such a question, I would expect some very creative posts. Yet, the essence of "child led" learning may dissuade any response, as I suspect a month of minimal activity in this forum indicates.
As exemplified by the response to the science of flying post, I would expect many resourceful responses to the challenge of keeping a child's interest engaged in a topic.
In a general sense, I would recommend the Montessori style - the offering of a limited set of choices (any of which would yield skill development) and enthusiastically cheerleading whatever choice they make.
Though the following comment may be a bit ascerbic, if a child were given complete freedom to choose ANYthing they wanted, they would probably soon become obese diabetic alcoholic illiterate experts of cartoons and video games (as much of post industrial populations have become). And if that is what their custodians wanted, then it would be a very effective style. One of the things I want for my children is that they learn the beauty of math and science.
So, anybody have any suggestions for specific math/science engagement?