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Topic: Trade-off Between Proficiency and Comprehension (Part 5 of
Open Letter)

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Frank Allen

Posts: 17
Registered: 12/6/04
Trade-off Between Proficiency and Comprehension (Part 5 of
Open Letter)

Posted: Dec 7, 1995 1:25 PM
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5. In their zeal to cope with the" drill-master'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
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 trade-off 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
trade-off is obviously not either-or; 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 de-emphasis 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 position--and many undoubtedly do--they 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, well-trained, well-intentioned teachers
put such a premium on developing students' skill in the routings of
arithmetic and algebra despite decades of advice to the contrary from
so-called 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 9-12 Standards are perhaps a bit
too sweeping--a bit too general. Certainly they require and deserve
intensive, impartial scrutiny, to which, so far, they have not been subjected.

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