Wednesday, June 19, 2013

Paths through Caltech Ma1a

 Quote: (emphasis added)

Ma1a. Introduction to the Mathematical method via one‐variable Calculus
(From Dinakar Ramakrishnan)

Goal: Develop the central results of one‐variable Calculus, explaining why they hold, and under which hypotheses, illustrated with examples; also delineate how to write logically correct arguments.

Emphasize critical thinking.

This course forms the basis of all the Math courses; AP Calculus‐BC is no substitute.


There are two main Paths in Ma1a. Path 1 can only be taken by those students who pass the Diagnostic test, while Path 2 will be for those who either don’t pass the Diagnostic test (by either not taking it or not doing sufficiently well in it) or else just want to see the material covered at a slower pace with more examples.

In addition, there are auxiliary paths for those who place out of Math 1a.

Path 1. Calculus of One and Several Variables (9 units; 3 lectures + 1 recitation section per week)

This path is currently taken by approximately 80% of incoming students, but we foresee the numbers lessening in the succeeding years as the (unfortunate) stigma attached to the second, more leisurely Path 2 (see below) is steadily overcome.

Topics covered: (weekly)
1. How to write proofs from first principles, involving induction, real numbers, rational approximations
2. Sequences and series: absolute and conditional convergence, Power series, tests
3. Continuous functions: Existence of extrema on closed intervals, small span, examples
4. Derivatives: Mean Value theorem, critical points, max/min, curve sketching
5. Integral Calculus: Fundamental theorems, primitives (anti‐derivatives), substitution
6. Integration by parts, logarithm and exponentials
7. Polynomial approximations, Taylor and MacLaurin series
8. Indeterminacies, L’Hopital’s rule, Limits
9. Improper integrals, Stirling’s formula
10. Complex numbers, sequences, differentiability, factoring

The key changes which have been made since the Fall of 2010:

1. During the first week of classes it is explained, with many types of examples, how one writes proofs in mathematics. (In addition, the prefrosh are encouraged to try Ma 0 online during the summer before starting here, thereby getting an introduction to the fundamentals.)

2. Some material used in the previous years (in Ma1a) on the abstract side has been reduced or eliminated, partly to be able to give more examples and partly for being able to spend the first week explaining the writing of proofs.

3. Lecture Notes are posted, which are used by many who may have difficulty reading the text; right now more than half the students seem to read the notes. The plan is to add figures and (more) examples to the notes next year.

4. Care is taken to make the class appealing to the general Caltech freshman, not mainly to those for whom mathematical thinking comes easily. At the same time, enrichment notes are posted online, which are not (at all) needed for the course, but which appeal to some 20% of the class wanting to learn more than what is discussed in class. This allows the Professor to relax and not
rush in class (as it may have at times been done in the past) to try to do something sophisticated to keep the interests of the top tier.

5. During Fall 2010, the class lectures were videotaped, and viewing them gives the students a way to go over the material at leisure.

6. The homework problems are chosen to reflect the basic things the students must know, and which they can do by following the lectures. Every week, there is one problem which is slightly more difficult (than the others), but is provided with hints, and the students are in any case encouraged to collaborate with others. In the future, care will also be taken to ensure that the (midterm and final) exams reflect correctly what the students have learnt to do.

7. Care is exercised in choosing very good Teaching Assistants for the course, and the students are encouraged to go to recitations, even more than one if needed, to learn the subject matter regularly. They can go to any recitation they want (as invariably some TAs are better at explaining), but they need to turn in the homework assignments to the appropriate TA monitoring their section. The homework problems are graded uniformly, meaning each problem is graded by the same (two) TAs for the entire class, so there is no bias in favor of one section or another.

8. The TAs and the Professor teaching the course maintain weekly office hours, and detailed reviews are given before the midterm and the final exam. The students can also go to as many different office hours as they like.

9. Students are encouraged to sit in, or register for, the problem solving class Ma8, which can be a helpful supplement to the course. This is especially recommended for the students who do not excel in the diagnostic test given in August (before they arrive).

Path 2. Freshman Mathematics (12 units; 4 lectures + 1 recitation section per week)

This path is currently taken by about 10% of incoming students. As mentioned above, we expect this percentage to go up some over the next few years, as students come to realize that it is advantageous to learn things in math a bit slowly to gain a thorough understanding without pressure.

It is intended for students who may not have a thorough background in Mathematics or else may desire a more relaxed treatment of One‐variable Calculus.

The lectures cover all the topics from Path 1 above except for those dealing with infinite series, but with more detailed explanations and examples. This is a kinder, gentler form of Path 1.

Students who are in this path are asked to take, in addition, Ma1d (5 units), which is taught in the Winter quarter for seven weeks, which will complete all the relevant topics, including infinite series,
(from Path 1) needed to be able to take Ma 1c (Vector Calculus) later in the Spring quarter .

Care has been exercised to choose excellent instructors for this course. It was a resounding success during the Fall of 2011, when Dr. Andrei Jorza taught this path.

Auxiliary Paths. These are intended for students with stronger preparation in mathematics, as indicated by their placing out of Ma 1a (by passing the appropriate Placement Exam). About 10% of the incoming students may be in this category.

There are three possible auxiliary paths for such students:

1. They can take Ma2a (Differential Equations) during the Fall of their Freshman year.

2. If they manage to place out of Ma2a as well (which will mean that they place out of Ma1b,c in addition), they can get placed in a higher level Math course such as Ma 108 (Introduction to Mathematical Analysis). Or they can place into Ma 6 (Discrete Math) or Ma 5 (Algebra). Such students may also take ACM 95.

3. If the students place out of Ma 1a‐c and Ma 2b without placing out of Ma 2a, they can’t take Ma 108, but they can take Ma 2a (if interested) or get placed into Ma6 or Ma 5.

Problem solving: The recitations will focus on problem solving and on understanding logical arguments.

The students who want more exposure to the former may audit (or take for credit) the following course, which is not part of the core but could be very helpful to some:

Ma8. Problem Solving in Calculus (3 units)

This is a support class for Ma 1a, taught in the Fall of the Freshman year and takes a hands‐on approach. The students will learn in great detail how to look for the answer, find limits, write a precise argument, etc. The course will also illustrate concepts with interesting examples.