Many first rate universities have experimented with this approach and subsequently rejected it, including, for example, UCLA and USC. However, the Harvard Calculus approach, under the banner of reform, has widespread use just at a time when growing numbers of minority students are enrolling in colleges and universities.

California State University, Northridge, for example, incorporates it in its math/science major sequence. It appears that the Math Education Reformers don't believe that the Nation's students can learn real calculus; instead they promote a watered down version. The text, CALCULUS, by Deborah Hughes-Hallet, Andrew Gleason, et al. is based on this approach. We direct our criticisms below to this textbook.

Many future high school teachers are not required to take advanced calculus or real analysis as part of their university preparation (e.g. California State University, Northridge). Therefore, secondary teachers who learn calculus through the Harvard approach are poorly prepared to answer questions by their own future students, especially concerning the foundations of calculus, e.g. limits and continuity (see 3 and 4 below).

The Harvard Calculus approach is deficient in the following areas :

• 1. 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.

• 2. 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 give the student the option to 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?

• 3. The definition of limit. This is the theoretical foundation of calculus. Failure to allow students to be exposed to the definition of a limit 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 important ways.

• 4. The definition of continuity. This is inadequately treated in the Harvard text. It is one of the fundamental ideas in mathematics. While there are different points of view among mathematicians on how much to emphasize limits and continuity in a first year calculus course, the absence of standard material on these topics severely limits the utility of the text. It is consistent with the "dumbing down" trend in the reform movement.

• 5. L'Hopital's rule for calculating limits is missing. This is a practical tool for later courses.

• 6. The intermediate Value Theorem is missing. This is not only important for theoretical reasons, but has practical applications for finding roots of functions.

• 7. The proof of quotient rule for derivatives is incorrect and the fallacy is not acknowledged.

• 8. Differentiability implies Continuity is not proved. The proof of this standard theorem is a perfect example of the level of rigor appropriate for first semester calculus.

• 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). It is helpful in later complex variables courses because it helps students to appreciate and understand the Laurent series.

• 13. 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.

• 14. 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.

• 15. Related Rates. This topic is missing and it is important later in applied mathematics courses.

• 16. 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.

The Jan. 1995 issue of UME Trends (Undergraduate Mathematics Education) includes an article by Barry Cipra entitled, "The Bumpy Road to Reform" which talks about Harvard Calculus at UCLA. Here is the relevant passage:

Likewise, UCLA abandoned the Harvard Consortium text in part because of negative evaluations from students (the other part was faculty skepticism). According to Thomas Liggett, chair of the UCLA math department at the time, out of approximately 100 comments about the Harvard Materials, 'only one was positive, all the others were negative.' UCLA is now trying other approaches. 'We certainly haven't given up,' Liggett insists. 'We tried one thing which we had high hopes for, and it didn't really work out so for the time being we're going back to the more traditional thing while we work out some other plan of attack.'