I just finished reading three selections from Alan Schoenfeld's "Mathematical Thinking and Problem Solving." The book is incredible and I would highly recommend it. It features eight different papers that are each counterbalanced by a written response or discussion. This may be a lengthy post but I wanted to record my thoughts here for later reference. In an effort to promote conversation about these, I would encourage you to find at least one quote that is interesting to you and post a thought or question in the comments. |
Research and Reforming Mathematics Education (Schwartz)
1. "If the important idea that one devises is to find its way into the way mathematics is taught, learned, and made in schools, then it is important that it not appear to threaten current practice." (p. 3)
2. "It should appear to augment, rather than replace." (p. 4)
REFORM CANNOT BE 'TEACHER-PROOF'
"However, let it be clearly stated and recognized, that any curricular development that takes place without the continuing central and focal involvement of teachers is almost certain to fall short of its potential."
RESISTANCE FROM LACK OF CLARITY
"Less visible, but equally if not more important, is the role of the social network in legitimizing the adoption of new habits of mind. It is difficult for students well into a school career in which mathematics has been an endless series of incompletely understood calculation and manipulation ceremonies to shift gears and to exercise in class their curiosity and inventiveness. It is difficult for teachers and principles, who will be held accountable to superintendents and school boards to imagine mathematics classes in which mathematics is 'discovered' rather than 'covered.' It is difficult for school boards to imagine that the mathematics they learned and the way they learned it is possible not universal or eternal in its importance. All of these groups need support as they develop new 'habits of mind.'" (p. 6)
Doing and Teaching Mathematics (Schoenfeld)
1. "In turn, our classrooms are the primary source of mathematical experiences (as they perceive them) for our students, the experiential base from which they abstract their sense of what mathematics is all about." (p. 53)
2. "The activities in our mathematics classrooms can and must reflect and foster the understandings that we want students to develop with and about mathematics." (p. 60)
3. "When mathematics is taught as received knowledge rather than as something that (a) should fit together meaningfully, and (b) should be shared, students neither try to use it for sense-making nor develop a means of communicating with it." (p. 57)
'DOING' MATHEMATICS IN THE CLASSROOM
1. "The implicit but widespread presumption in the mathematical community is that an extensive background is required before one can do mathematics." (p. 65)
2. "I work to make my problem-solving courses 'microcosms of selected aspects of mathematical practice and culture,' in that the classroom practices reflect (some of) the values of the mathematical community at large." (p. 66)
3. "Mathematics is the science of patterns, and relevant mathematical activities - looking to perceive structure, seeing connections, capturing patterns symbolically, conjecturing and proving, and abstracting and generalizing - all are valued." (p. 68)
Classroom Instruction That Fosters Mathematical Thinking and Problem Solving (Romberg)
1. "What is constructed by an individual depends to some extent on what is brought to the situation, where the current activity fits in a sequence leading toward a goal, and how it relates to mathematical knowledge." (p. 300)
2. "The 'intended' curriculum can only include our best guesses about what will both interest students and lead all toward development of mathematical power. At the same time, the 'actual' curriculum depends on teacher choice, and the 'achieved' curriculum depends on each student's interest and prior knowledge." (p. 300)
3. "Nussbaum and Novick suggest a three-part instructional sequence designed to encourage students to make the desired conceptual changes. They propose the use of an exposing event, which encourages students to use and explore their own conceptions in an effort to understand the event. This is followed by a discrepant event, which serves as an anomaly and produces cognitive conflict. It is hoped that this will lead the students to a state of dissatisfaction with current conceptions. A period of resolution follows, in which the alternative conceptions are made plausible and intelligible to students, and in which students are encouraged to make the desired conceptual shift." (p. 298)
CURRICULUM DESIGN PRINCIPLES
1. "Classrooms should be places where interesting problems are explored using important mathematical ideas…This vision sees students studying much of the same mathematics currently taught, but with quite a different emphasis."
2. Romberg calls for a problem-based approach centered around his five principles of curriculum design (conceptual domains should be specified, domains should be segmented into 2-4 week units, students are exposed to domains as they arise naturally in problem situations, activities are related to how students process information, and curriculum units should always be considered problematic).