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Harvard Calculus
Posted:
May 21, 1997 12:36 PM
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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 Hughes-Hallet, 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 thought-provoking for well-prepared
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 de-emphasis 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
well-known 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.
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