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Topic: Re: [CALC-REFORM:3028] Riemann Sums
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woods debra j

Posts: 7
Registered: 12/6/04
Re: [CALC-REFORM:3028] Riemann Sums
Posted: Nov 28, 1995 1:33 PM
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Jerry,
I very much enjoyed reading this response. Thank you.
Debra

>A grad student named Daniel Walker sent me the following inquiry.
>His question has been asked many other times; so I decided to post his
>question and my reply:
>

>>
>> Dr. Uhl,
>>
>> I am a graduate student at the University of Wisconsin at Eau
>>Claire. As an independent study, I'm doing some research on calculus reform
>>projects. A report from the University of Illinois on 'Calculus and
>>Mathematica' appears in "Priming the Calculus Pump"(1990). The report
>>indicates that the Riemann sum definition of the integral is amoung the list
>>of topics that are excluded from the C&M curriculum. When I give my
>>presentation, I'm certain that someone will doubt that this key definition
>>was omitted.
>>
>>So, help me out here-
>>
>>Please advise as to how I explain the justification.

>
>->Uhl's Reply:
>
>The "justification" is simple. For strong pedagogical reasons, we have
>chosen not to define area but to measure it. As a result the integral of
>f[x] from x = a to x = b measures the signed area between the f[x] curve
>x-axis for
>a <= x <= b.
>
>This is the way Newton thought of the definite integral. And this is the
>way almost everyone thought of the definite integral until Thomas's
>calculus book appeared in the 1950's
>
>
>This is in harmony with the definitive calculus treatise by Richard Courant
>(Wiley-Interscience,1949) which defines the definite integral this way this
>way (page 77). Peter Lax, past president of the American Mathematical
>Society and father of calculus reform, agrees that Riemann sums should be
>deemphasized in reform calculus. Our project is the only reform project
>that took this seriously.
>
>More observations:
>
> Neither of the commonly used calculus texts Granville,Smith and Longley
>(Ginn ,1934) nor Sherwood and Taylor (Prentice Hall, 1954) even mention the
>term "Riemann sum."
>
> Looking at Emil Artin's famous Princeton Honors calculus course (1955),
>one finds no mention of the term "Riemann sum."
>
>All of these texts mention approximating the integral with sums of
>rectangles as does Calculus&Mathematica to underscore the fact that
>integration is a process of accumulation.
>
>To sum it up: The calculus malaise felt by many has increased in direct
>proportion to the the number of years integrals have been defined by
>Riemann sums in standard calculus courses. The giants whose shoulders we
>stand on did not see the need for formal study of Riemann sums. Why should
>we?
>
>Another point:
>Throwing Riemann sums at first semester calculus students is to throw a
>huge bureaucracy of unfamiliar notation and all the baggage that goes with
>it. Riemann sums are a big calculus roadblock. Many students never survive
>and fail to grasp the meaning of the definite integral.
>
>Everyone can grasp the idea of integral if it is defined as an area
>measurement. The students who try to run the Riemann sum gauntlet are
>usually left only with mindless symbolic manipulation without meaning
>(MSM).
>
>-Jerry Uhl
>
>----------------------------------------------------------------------
>Jerry Uhl juhl@ncsa.uiuc.edu
>Professor of Mathematics 1409 West Green Street
>University of Illinois Urbana,Illinois 61801
>Calculus&Mathematica Development Team
>http://www-cm.math.uiuc.edu
>
>"The real problem in speech is not precise language. The problem is clear
>language." . . . Richard Feynman







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