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Tradeoff Between Proficiency and Comprehension (Part 5 of Open Letter)
Posted:
Dec 7, 1995 1:25 PM


5. In their zeal to cope with the" drillmaster's" approach to teaching mathematics which relies largely on developing manipulative skills, the writers of the Standards may be dangerously over simplifying a very important issue: the "trade off between proficiency and comprehension." (See Enclosure 5 Kilpatrick's statement)
"A group completely under the sway Of theoriticians, far away From schoolroom events of everyday."
Enclosure 5 Kilpatrick's statement
Readers will find the Standards replete with statements like these: "A shift of emphasis from a curriculum dominated by an emphasis on memorization of isolated facts and procedures" "Computational facility is no longer the sole expectation." "Algebra curriculum also will move away from a tight focus on manipulative facility to include a greater emphasis on conceptual understanding." To these readers I commend the following statement by Jeremy Kilpatrick, which is also found in the July 1988 issue of our Research Journal.
"One of the most venerable and vexing issues in mathematics education concerns the tradeoff between proficiency and comprehension, between promoting the smooth performance of a mathematical procedure and developing an understanding of how and why that procedure works and what it means. The tradeoff is obviously not eitheror; rather as William Brownell pointed out over 30 years ago, some balance need to be found between meaning and skill. Amid today's arguments that technology has modified, and sometimes supplanted, the skills students need, the issue has grown into not just achieving a balance but finding a balance point. The working draft of the NCTM's Curriculum and Evaluation Standards for School Mathematics argues forcefully for a deemphasis in skill instruction and for a change in the apparently widespread view that proficiency nneds to precede, and perhaps to dominate, comprehension and problem solving. Although researchers may agree with the draft positionand many undoubtedly dothey should not dismiss too lightly the questions of how and where skill development fits into the school mathematics curriculum. Recent research in cognitive science suggests that a strong knowledge base is needed for problem solving, and surely some of that base should be composed of procedural knowledge. Furthermore, conceptual knowledge both supports and is supported by what Brownell termed "meaningful habituation," the almost automatic performance of a routine that is based on understanding. A neglected yet critical item both in implementing the NCTM standards and in gaining a better grasp of the role skill development plays in learning mathematics concerns the folk wisdom in today's school practice. Why is it that so many intelligent, welltrained, wellintentioned teachers put such a premium on developing students' skill in the routings of arithmetic and algebra despite decades of advice to the contrary from socalled experts? What is it the teachers know that the others do not? What we often forget when we look at classrooms is that they are a place in which teachers too develop mathematical meanings. Although teachers often teach as they have been taught. At least some of our research needs to take them seriously as informants on the wisdom of practice." When we consider this thoughtful, balanced statement which applies to the final draft as much as to the working draft, we have to wonder if some of the recommendations hadned down in the 912 Standards are perhaps a bit too sweepinga bit too general. Certainly they require and deserve intensive, impartial scrutiny, to which, so far, they have not been subjected.



