Widgets Magazine


Math 51: understanding its role

To the editor:

Joe Troderman’s Oct. 23 Op-Ed “Let’s improve introductory chem and Math 51” is timely. The Math department cares very much about providing quality instruction (this is why Math 51 is taught in many sections of moderate size rather than to an audience of hundreds), and in the last couple of years we have recognized that the time is ripe to undertake a review of the 50 series. We will be soliciting feedback from former students of the 50 series as part of that review later this year.

One principle adopted at its inception a generation ago is that since students are drawn from biology, economics, engineering, physics, etc., focusing on examples important to one field would shortchange others.  The aim is to provide a first introduction to the material in a way that can be built upon in work across many fields. Students who complete the Math 50 series deepen their mastery when they meet the ideas in later courses via the viewpoint of their own specialization. (Since multiple CME courses cover material taught in other departments, the overlap of Math 51 with CME mentioned in the Op-Ed is a separate matter.)

Just as one-variable calculus develops basic intuition alongside computational skills without rigorous theory, so it is with the Math 50 series.  But there is conceptual content: students see how linear algebra underlies everything, since Math 51 begins with a half-quarter of linear algebra (not meant to be a full course, and developed further in Math 53 and much further in Math 104 and 113; what is done is much more than a list of algorithms, though it makes substantial contact with that part of the subject). The entire series is about multivariable mathematics, and one needs linear algebra to understand multivariable calculus.  For example, in the absence of linear algebra students grapple with many different-looking formulas called the Chain Rule, but with linear algebra one can learn a single (simple!) universal Chain Rule as in Math 51. The 50 series was innovative by incorporating linear algebra throughout the year, and that approach has since been adopted at other universities.

There is no royal road to multivariable calculus, and the LU decomposition mentioned in the Op-Ed is only one of many ways to deeply understand methods in linear algebra. The general difficulty of learning mathematics has confronted everyone from King Ptolemy I to Einstein, and it is not meaningful to make comparisons with introductory courses in other departments.

All learning is by successive approximation.

Professor Brian Conrad

Professor Brian Conrad is the Director of Undergraduate Studies in the Mathematics Department. He can be contacted at conrad ‘at’ math.stanford.edu.

This post was originally labeled as an Op-Ed submission but has been corrected to reflect that is actually a Letter to the Editor. The Daily regrets this error.

  • victor mehran

    It’s true that incorporating linear algebra throughout the Math 50 series does present the material in an integrated fashion and enables a fundamental understanding of the concepts. However, the teaching in Math 50 series should be improved. The discussion sections need substantial improvement. Moreover, the later courses, such as Math 131P and Math 104 are not taught well at all. These are mostly taught by postdocs with little teaching experience.

  • LT

    Professor Conrad, I wish you had taught my section of 51. As a freshman, I emerged scarred, without understanding any of the underlying meaning of the course, and without faculty or TA resources I felt could adequately explain such topics. I whole-heartedly credit 51 with my conversion from a math lover to someone who fears the discipline and its teaching at Stanford. I know that course should have been deeply useful, but I have since learned much more, much faster, and with better information retainment with Khan Academy. It saddens me that Khan is teaching me better in this regard than the elite group of intellectuals that exist at Stanford.