Department of Mathematics FAS Harvard University One Oxford Street Cambridge MA 02138 USA Tel: (617) 495-2171 Fax: (617) 495-5132
You've attended all the seminars, gone to Freshman Advising, read the Course Catalogue, and you still have no idea what math class you're really supposed to take this year. Self-determination is great most of the time, but sometimes you just want someone to tell you what to do. Here, compiled from the comments of members of the class of 2006, is an honest assessment of the freshman math offerings.

Note: These opinions are not compiled by the Mathematics department and are intended only as guidance. Exceptions to the generalizations stated below are fairly frequent. When in doubt, ignore your misgivings and talk to the Math Head Tutor. Worst comes to worst, the advice is lousy, but the Head Tutor is then to blame, not you.

An interview with Eli Grigsby

Math 21

A thorough treatment of multi-variable calculus and linear algebra with real-life applications. Assigned problems (especially in 21a) use numbers more often than not, and proofs are limited to non-rigorous examples (basically common-sense proofs). Class meets three hours per week, and homework usually takes three to six hours per week to complete. Many sections are taught by graduate students, which means that it is possible to switch sections (and thus instructors) if necessary, but there may be little contact with professors. This sequence is a very efficient way to learn calculus and linear algebra, but the lack of proofs does not imply that the classes are easy. These courses have ranged from very reasonable (Fall 2002) to hard (Fall 2001). You should take this class if one or more of these describes you:
  • The last math class you took was BC calculus, and you got a 4 or 5 on the AP exam and the Harvard Math Placement exam placed you in Math 21.
  • You liked the way that high school calculus was taught.
  • You are interested in chemistry, physics, or applied sciences and need to learn math that applies directly to other disciplines.
  • You like to see how math applies to real life, or else you don't see a point in learning it.

Math 23

A class that covers linear algebra and multivariable calculus while also teaching proof-writing, starting with the basics. Problems still use numbers but also employ the formal terminology of advanced mathematics. Class meets three hours per week, and problem sets can take anywhere from five to fifteen hours per week. Students usually complete problem sets by pulling all-nighters in groups the night before the sets are due; many students find that working in a group of similarly-abled peers turns out to be one of the most rewarding aspects of this class. This class is always taught by a faculty member; in previous years, the instructor has been someone who enjoys teaching and explains things well. It is not, in principle, supposed to be as homework intensive as Math 25. You should take this class if one or more of these describes you:
  • You got a 4 or 5 on the BC calculus exam, the Harvard Math Placement exam placed you in Math21 and want to learn how to write proofs while learning multivariable calculus and linear algebra.
  • You've already taken plug-and-chug multivariable calculus and linear algebra and want a rigorous treatment of it.
  • You think that you might want to be a math concentrator, but you want to sample proof-based mathematics before you sacrifice your life to it.
  • You are interested in the sciences and want a proof-based class where you can still see the connection to the real world.

Math 25

A rigorous treatment of multivariable calculus, linear algebra, and introductions to other topics in advanced mathematics. This class is a springboard to the study of advanced math; the class thoroughly covers its topics but moves very quickly, and examples tend to be theoretical instead of concrete. Class meets three hours per week and homework can take from seven to fifteen hours per week to complete. The course is taught by a professor; previous instructors have varied greatly in teaching ability and prestige. A previous knowledge of proofs, linear algebra, or multivariable calculus is helpful, but not necessary. However, being 'gung-ho' about mathematics is a definite prerequisite. You should take this class if one or more of these describes you:
  • You completed BC calculus, are very interested in math, and want a class that will allow you, with a lot of hard work, to catch up to the most advanced math students in your class.
  • You have completed multivariable calculus and linear algebra and want a thorough, proof-based review of the topics before moving on to other mathematics.
  • You have completed bits and pieces of advanced mathematics and want to take a class that will patch up the holes in your knowledge.

Math 55

This is probably the most difficult undergraduate math class in the country; a variety of advanced topics in mathematics are covered, and problem sets ask students to prove many fundamental theorems of analysis and linear algebra. Class meets three hours per week, plus one hour of section, and problem sets can take anywhere from 24 to 60 hours to complete. This class is usually small and taught by a well-established and prominent member of the faculty whose teaching ability can vary from year to year. A thorough knowledge of multivariable calculus and linear algebra is almost absolutely required, and any other prior knowledge can only help. Students who benefit the most from this class have taken substantial amounts of advanced mathematics and are fairly fluent in the writing of proofs. Due to the necessity of working in groups and the extensive amount of time spent working together, students usually meet some of their best friends in this class. The difficulty of this class varies with the professor, but the class often contains former members of the International Math Olympiad teams, and in any event, it is designed for people with some years of university level mathematical experience. In order to challenge all students in the class, the professor can opt to make the class very, very difficult. You should take this class if one or more of these describes you:
  • You are fairly certain that you want to be a math concentrator and want to be challenged to your limit.
  • You have a solid base in advanced mathematics and are very comfortable with proofs and rigorous arguments.
  • You want math to be your most important class.

A Few Comments

  • Every year, a few students enter math classes for which they are underprepared and, with a lot of hard work and commitment, manage to succeed. However, many of them comment that although they got a decent grade in the class, they would have learned a lot more had they taken the class appropriate to their level of preparation. Remember, you don't get any extra credit for taking a harder class, so take the course that will teach you the most.
  • Choosing which class to take is not an irreversible decision. If you opt to try a more challenging class, you can always drop to another one after a few weeks, or even after first semester. The proof-based classes are coordinated to run similarly over the first few weeks to give students the opportunity to change classes if necessary. Moreover, Math 55 gives a 'diagnostic' exam right at the beginning that will allow you to judge whether you have the necessary experience to benefit from the course.
  • Finally, your choice of class this year does NOT affect your ability to concentrate in math. Any of these four courses of study will give you the tools you need to continue learning advanced mathematics. If you take 21, you may have to teach yourself how to write proofs independently (or enroll in a proof-writing class like Math 101) before you can take more advanced math classes, but you will be just as capable of learning the advanced material as anyone who took higher-numbered classes.

Compiled by Stephanie Hurder, with input from Emily Riehl, Hans Cutiongco, and Jason Abaluck.