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Harvard Calculus
Posted:
May 21, 1997 12:36 PM


Hello,
This is my first post. I'd like to respond to the inquiries of Jennifer Nicholls and Phil Larson about the "Harvard Calculus" textbook. My perspective comes from being a math professor at Calif. State University, Northridge. I have been frustrated by the lack of basic calculus and algebra skills that Harvard Calc taught students have. They are ill prepared for subsequent math courses. Here is a list of deficiences of (Harvard) CALCULUS, by Deborah HughesHallet, Andrew Gleason, et al.:
1. The text is virtually useless as a reference book for subsequent courses in science, engineering, and mathematics. The HC text also perpetuates the widespread misunderstanding among students that if a mathematical proposition is true for a finite number of cases, then it is true in general. The text tends to confirm this by the overuse of tables of numbers, followed by general conclusions.
2. Exercises involving algebraic manipulation. The Harvard approach provides students with less practice in standard algebraic manipulations than traditional approaches. The authors state in the preface of the Harvard text:
"We have found this curriculum to be thoughtprovoking for wellprepared students while still accessible to students with weak algebra backgrounds. Providing numerical and graphical approaches as well as the algebraic gives the students several ways of mastering the material. This approach encourages students to persist, thereby lowering failure rates."
The deemphasis of high school level algebra is a disservice to students and is consistent with the "dumbing down" of the California Framework. Lowering failure rates should be secondary to providing good education.
3. The definition of a real power of a positive number and logarithms. Freshman calculus is usually the only course which provides a correct definition of the exponential, logarithmic functions (including the number "e"). The treatment in the Harvard text does not allow the student to ever understand this correctly. Don't we want math and science majors to know what "e to the x" really means? Where else will they ever learn this?
4. The proof of quotient rule for derivatives is incorrect and the fallacy is not acknowledged.
5. Convergence Tests for Series. This topic is so poorly covered in the Harvard approach that once again, Harvard calculus trained students are put at a disadvantage in subsequent differential equations courses.
6. L'Hopital's rule for calculating limits is missing. This is a practical tool for later courses.
7. The intermediate Value Theorem is missing. This is not only important for theoretical reasons, but has practical applications for finding roots of equations.
8. Differentiability implies Continuity. The proof of this theorem is a perfect example of the level of rigor appropriate for first semester calculus. Failure to present this gem to students is the loss of a good opportunity.
9. The Mean Value Theorem. The Harvard text presents this very late, in the context of Taylor series. This theorem is normally used to justify graphing techniques, the Fundamental Theorem of Calculus, and the definition of the general antiderivative, among other results. Harvard calculus trained students have no capacity to prove the following: If the derivative of F is everywhere zero, then F is a constant function.
10. Polar coordinates. This practical topic is missing.
11. Parametric equations. This topic is missing and is important in the development of line integrals and other aspects of pure and applied analysis.
12. Partial Fractions. The failure to adequately develop this topic puts science and engineering majors at a significant disadvantage in subsequent courses. Facility with partial fractions is needed in the study of differential equations (for example, for Laplace Transforms) and complex variables because of its connection to the Laurent series.
13. The definition of limit. This is the theoretical foundation of calculus. Failure to at least expose students to this important concept early on has negative consequences. For example, Harvard calculus trained students suffer a natural disadvantage in subsequent vector calculus courses, complex variables course, and real analysis courses where limits arise in a important ways.
14. Related Rates. This topic is missing and it is important later in applied mathematics courses.
15. The definition of continuity. This is inadequately treated in the Harvard text. It is one of the fundamental ideas in mathematics. While there are differing points of view among mathematicians on how much to emphasize limits and continuity, the absence of material on these topics in the text reduces its utility.
16. The Harvard Calculus definition of the definite integral is not the same as the Riemann Integral. If a function is Riemann integrable, then it satisfies the Harvard Calculus definition of the integral and the integrals coincide. But the converse is false. For example, the indicator function f of the rational numbers has Harvard Calculus integral over [0, 1] equal to zero, but is not Riemann integrable (note that the HC definition uses only regular partitions of the interval). This raises an important question. Can the properties of integrals given in the text be deduced from the definition of the integral given there? If the interval I is a disjoint union of the intervals A and B, then the wellknown theorem given in the Harvard Calculus text that the integral over I equals the integral over A plus the integral over B does not follow, in general, from the definition. The HC text emphasizes continuous functions, so perhaps from a practical point of view, the differences are unimportant. But the authors use the term "Riemann Sum" and so there is a misleading implication that the two concepts are the same. There are no examples in the text of a definite integral evaluated as a limit of Riemann sums. This is understandable since the notion of a limit of a sequence is not provided, but it is a serious omission.



