Beyond Math 1:
Which math course is for you?
If you have completed the Math 1a/1b sequence at Harvard or if you have had the equivalent material elsewhere, you may be wondering which course is for you. The mathematics department provides a variety of options which you should consider based on your academic interests and your background. With exceedingly rare exceptions, students in your position are advised to take one (or more) of Math 18, 19a, 19b, 21a, 21b, 23a, 25a, 55a, or 101. (The School of Engineering and Applied Sciences also offers Applied Math 21a,b which covers selected topics from Math 21.) This pamphlet describes the Mathematics Department's offerings and should help you decide which course is for you.
Math 21a covers multivariable calculus, while Math 21b is a one-semester introduction to linear algebra and differential equations. First-year students who had an equivalent of Math 21a in high school often take this course in the fall of their freshman year. The students with such background who intend to major in math or theoretical physics should also look into Math 23, Math 25, or Math 55. Those who are considering a concentration in mathematics may want to take Math 101 concurrently with either Math 21a or b.
Math 101 is a one-semester introduction to the three main branches of modern mathematics (algebra, analysis, and geometry) and to the methodology used in higher mathematics. It has no official prerequisites. In this course students learn to write rigorous proofs and encounter fundamental concepts which are further developed in other 100-level courses. Math 101 is intended both for those who wish to concentrate in mathematics and for those in other fields (related or not) who have an interest in learning what higher math is all about. Students often take it concurrently with or right after Math 21. Those who are taking or have taken Math 23, 25, or 55 should not take 101. In 2015-16, Math 101 will be given in the spring, but not in the fall semester.
Math 25 and 55 are both full-year advanced courses designed for students with a very strong interest in theoretical mathematics. Each covers multivariable calculus, linear algebra, and some additional topics from a rigorous and advanced point of view. The students in these courses are frequently committed to concentrating in mathematics and are asked to put in extensive work outside the classroom. Many have had more than one year of college mathematics while in high school or have participated in various summer math programs. However, it is not necessary to have had multivariable calculus before taking 25 or 55. Although the syllabus of Math 25 is similar to that of Math 23, students will usually have had more preparation in math.
Math 55 is a faster paced course and covers topics more deeply. It is designed for students who arrive at Harvard with an extensive background in college level math. Math 25 and 55 differ from Math 23 in the level of outside work required: homework assignments in Math 25 and 55 are typically very time consuming. Math 23, 25 or 55 all provide an excellent foundation for further study of mathematics.
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