>I mean, problem by type, as long as it aims at mathematics, >is way more important.
I have a very hard time with seeing this as more important than almost anything you care to mention. I am sorry but learning problems by type is creating a situation of trying to fit some very interesting non textbook type problems into some nice little box we call problem type, kind of unfair and extremely unrealistic to the student.
Before students can master the "well defined" >"mathematics oriented" problems by type, even a well designed "real world", >"open ended" problem can be very confusing and harmful for students' >mathematics growth, nonetheless to say those illy designed non-mathematical >so called "real world" "endlessly open" problems. >
I don't see these as being harmful at all. Bringing common sense into the mix(which open ended problems do and encourage as a rule) tends to not only engage a far greater majority of the student population but also motivate some pretty interesting thinking that WOULD NOT occur in the context of cooking cutter problem(by type) solving. If anything I have seen much more harm done from this learning then from the most ambiguous problem you care to mention.
Scott Powell University of Hawaii University Labratory School 1776 University Ave. Honolulu, Hi. 96822 firstname.lastname@example.org