Humm..(the sound my brain makes when it is thinking:) There is always a risk of oversimplification with any systemic or cyclic approach to logic or reasoning, yet with students coming to class failing college math two and three times, this framework has helped them succeed by taking some of therror out of the picture. The context of a logical oversimplification doesn't seem to apply. I am realtivly sure these student will not make any great theoretical contributions to the discipline, yet the fact that for this one semester they used an oversimplification, passed the class, and graduated matters on some cosmic level. As semester after semester you see these students who have spent years and thousands of hour trying to pass one math class... If you find, as many of our math professors did, that you notice the way you teach the different types of thinkers, the concept was worth the effort. From the standpoint of those same professors, they also noticed a direct correlation between the semester they began using the simple questions and an increase in their traditionally low student evaluation scores. Additionally in the vast majority of cases pass rates increased that same semester for all students, not just repeat students. Using the same exit exams. If you are teaching a math theory class this is not to meaningful, but if you are teaching college math grad assistants or adjunct math instructors it might be woth considering. My best to all, Eldon L. McMurray Director Faculty Center for Teaching Excellence Assistant Professor College Success & Academic Literacy Utah Valley State College 800 West University Parkway Orem, UT 84058 (801) 863-8550 Dr. McMurray, I recognize that it's possible to see the practices I listed as subsets of your categories; however, I don't think of them that way. (And note that the practices I listed were intended only as examples of a larger set of mathematical practices one might want students to have experience with / develop beliefs about.) Consider the practice of defining, for example. Mathematicians certainly master existing definitions, but they also encounter the need to define new constructs, and there may be questions about how best to define something, and some definitions change over time, etc. If one wants students to experience any of these things (e.g., to have a felt need for defining something), then one structures one's teaching in a different way (e.g., one doesn't define every concept in anticipation of its need). Also, in this case, defining isn't just a matter of what, but also of why (e.g., why is this concept useful / worth naming?) and perhaps other categories as well. Dara _______________________________________________ PSTUM-list mailing list PSTUM-list(a)lists.fas.harvard.edu http://lists.fas.harvard.edu/mailman/listinfo/pstum-list