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Re: [CALCREFORM:3028] Riemann Sums
Posted:
Nov 28, 1995 1:33 PM


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 >xaxis 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 >(WileyInterscience,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://wwwcm.math.uiuc.edu > >"The real problem in speech is not precise language. The problem is clear >language." . . . Richard Feynman



