>I had some problems this year teaching integers using these new methods >(alge-tiles and other, number-line - based manipulatives), to the point that >the kids themselves said "Can't you just teach us the rules and give us the >homework? - this is too confusing!". I ended up using both and, as an >experiment (loose use of the term) gave them the same test their teacher had >used last year. The marks were an average of 18% higher, and the lowest score >was a C+. I think this actually bolsters what you were saying - that new >methods often overlook focus and practice. In this I am in total agreement >with you. Yet in spite of their professed confusion, the kids actually >discovered the rules for themselves - by the time I told them they all either >knew them or had some intuitive idea of them.
The response Richard got from his students is certainly one we hear frequently. I think it is important to listen "between the lines" of what students are saying. Jere Confrey's comment seems especially appropriate:
"Doyle, Sanford and Emmer (1983) examined students' views on the "academic work" in traditional classrooms and found that, as students convince teachers to be more direct and to lower the ambiguity and risk in classroom tasks, the teacher may inadvertently mediate against the development of higher cognitive skills."-What constructivism implies for teaching. Journal for Research in Mathematics Education. (1990)
I'm glad Richard didn't allow his students to convince him to be completely direct.
I think, however, that Richard (and Dan & many others) have misinterpreted the Standards and other materials of the reform movement when they say such documents "overlook focus and practice". The practice is there, but it is not in the form we are used to seeing it. Practice and focus do not have to be routine, dull, or meaningless (without context) to be effective. A fairly good example of this is the "24 game" where students are given four numbers and must use the operations and each of the four numbers to get a total of 24.
H^2 Assistant Professor of Mathematics Western Illinois University