Published on *Department of Mathematics* (https://math.yale.edu)

**Table of contents**

- Introduction
- Why are we revising?
- Courses NOT affected.
- What are the main ingredients in our introductory sequences?
- What is the new introductory sequence into the mathematics, and mathematics (intensive) major?
- What about Math 222?
- How do current Math majors transition to the new introductory sequence, if they have already completed some math courses at level 120+?
- What are the new requirements for CPSC + Math major, and how do current students transition into the new system?
- What are the new requirements for Econ + Math major, and how do current students transition into the new system?
- What are the new requirements for Math + Phil major, and how do current students transition into the new system?
- What are the new requirements for Math + Phys major, and how do current students transition into the new system?
- How does the above affect pre-requisites to other math courses? (new)
- Other questions (new)
- What if I have more questions?

Message from the Math department announcing the curriculum changes is linked here [1].

Town hall was held on March 14, and slides are available here [2]. (Everything on them is also on this site.)

The math department has been working on revisions to our introductory course sequences. This page describes the current draft of the changes and how each major will be affected, as well as how current students can transition to the new system.

As of today, March 7, 2021, the changes are scheduled to take place starting in Fall 2021.

Below, you can find a description of the changes. The second half of the document looks at the current requirements of each math / joint-math major, translates them into the new system, and describes a path for current students to complete their requirements depending on courses they have taken so far.

One reason is that our introductory courses, through introduction to analysis (Math 300 / 301) have been getting out of alignment, and we want to align everything better.

We also wanted to make the introductory sequence more inclusive (e.g. by making proof-based courses accessible with less mathematical background), and more flexible (e.g. as far as order of courses, mixing regular and intensive versions in different semesters, or moving past courses taken during high school).

- Current level 100 courses are not affected, they will remain exactly as they are.
- Math 222 is not affected, or any level 200 course except 225, 230, 231, 250.
- Current level 300+ courses are not affected, except 300, 301.

**What are the main ingredients in our introductory sequences? **

They currently are:

- Proof-based linear algebra. (Currently covered in Math 225/230/231.)
- One variable analysis (Currently covered across 230/231/250/300/301.)
- Multivariable calculus / analysis (Currently covered across Math 120/230/231/250.)

In the new system, Math 230, 231, 250, 300, 301 will all be retired. The topics have been arranged into these new classes:

- Linear algebra with proofs: Math 225, and an intensive version Math 226.
- One-variable analysis: Math 255, and an intensive version Math 256.
- Multivariable analysis / calculus: Math 302 (multivariable analysis) / Math 120 (multivariable calculus).

Math 225 has been revised to be accessible as a first-semester course, it will include introduction to proofs and focus on proof-based linear algebra. Topics will include linear transformations, determinants, eigenvalues, inner product spaces, spectral theorem. It will be offered every semester.

Math 226 is an intesive version of Math 225, offered during the Fall semester.

Math 255 will include introduction to proofs, and serve as an introduction to rigorous analysis on the line with some applications. Topics will include limits, sequences and series, differentiation and integration, metric spaces. It will be offered every semester.

Math 256 is an intensive version of Math 255, offered during the Spring semester.

Math 225/226 and Math 255/256 can be taken in either order, though taking linear algebra first is recommended. All four courses will be accessible after integral calculus (e.g. Math 115 or AP Calculus BC).

Math 302 will cover vector analysis, such as differentiation and integration in R^n, inverse function theorem, manifolds in R^n, and generalized Stokes’ theorem. It requires both linear algebra (Math 225 or 226) and real analysis (Math 255 or 256).

**What is the new introductory sequence into the mathematics, and mathematics (intensive) major? **

The new sequence is:

- Linear algebra with proofs: Math 225 or 226.
- One-variable analysis: Math 255 or 256.
- Multivariable analysis / calculus: Math 302 or Math 120. (Note that Math 302 requires Math 225/226 and Math 255/256.)

Math 222 is currently not recommended for mathematics major, and it will continue not to be recommended in the future, because it lacks some conceptual work and preparation for writing mathematical proofs that is important in the math major. For students who complete Math 222 before deciding on a math major, we will continue to accept it in place of Math 225 / 226.

This depends on your introductory coursework completed just before Fall 2021 (i.e. including courses you may complete in Spring 2021 or during Summer 2021).

1. **Full introductory sequence completed** prior to Fall 2021 (meaning, 120 + 225 + 250 or 230 + 231).

No effect: you are done with the introductory requirements.

2. **Completed Math 120 + 225 but NOT 250**, prior to Fall 2021.

You can finish the introductory sequence by completing Math 255 or 256.

3. **Completed Math 120 but NOT 225 or 250**, prior to Fall 2021.

You can finish the introductory sequence by completing Math 225 or 226, and Math 255 or 256.

4. **Completed 230 but NOT 231**, prior to Fall 2021.

With curriculum revisions and remote-learning complications in mind, Math 230-231 in the year 2020-21 was restructured so that most of the analysis was taught in Math 230, and most of the linear algebra left for Math 231. That being so, you can finish the introductory sequence by completing Math 225 or Math 226.

The specific math courses required for the CPSC + Math major are currently:

- Multivariable calculus (120)
- Linear algebra (222 or 225)
- Discrete math (244)

In the new system, all of these courses are still present, so the new specific courses required are very similar, only with an extra option.

The new requirements are:

- Multivariable calculus (120)
- Linear algebra (222 or 225 or 226)
- Discrete math (244)

Students who completed multivariable calculus prior to enrolling at Yale, and may consult the DUS about substituting a higher level course for Math 120.

For current students, there should be no difficulty with transition into the new system, since all current options will continue to exist.

Extra note: Completion of 230 + 231 currently satisfies the multivariable calculus + linear algebra requirement, and this will continue to be the case (for students who complete the sequence before it disappears). If you only completed a part of the sequence, contact Miki and we’ll figure it out.

The specific math courses required for the Econ + Math major are currently:

- Multivariable calculus (120)
- Linear algebra (222 or 225)
- One-variable analysis (Math 300 or 301 - note that these require Math 250, so there are really two courses needed)

In the new system, the specific requirements are basically the same, only they are rearranged a little bit - and they can be done with three courses rather than four as above.

The new requirements are:

- Multivariable calculus (120)
- Linear algebra (222 or 225 or 226)
- One-variable analysis (Math 255 or 256)

Students who completed multivariable calculus prior to enrolling at Yale may consult the DUS about substituting a higher level course for Math 120.

Extra note: Completion of 230 + 231 currently satisfies the multivariable calculus + linear algebra requirement, and this will continue to be the case (for students who complete the sequence before it disappears). If you only completed a part of the sequence, contact Miki and we’ll figure it out.

**Transition for current students: **

If you complete **Math 120 + 222/225 + Math 250 + Math 300/301** prior to Fall 2021, then the changes will have no effect on you.

If you complete **Math 120 + 222/225 + Math 250 but NOT Math 300/301** prior to Fall 2021, you can complete the specifically required courses by taking Math 255 or 256. After consultation with the DUS and the instructor, it may also be possible to use Math 305.

If you complete **Math 120 + 222/225 but NOT Math 250 or Math 300/301** prior to Fall 2021, you can complete the specifically required courses by taking Math 255 or 256.

If you complete **Math 120 but NOT Math 222/225 or Math 250 or Math 300/301** prior to Fall 2021, you can complete the specifically required courses by taking (Math 225 or 226) and (Math 255 or 256).

(If you are in some other situation, please contact Miki and we’ll figure it out.)

The specific math courses required for the Math + Philosophy major are currently:

- Multivariable calculus (120)
- Set theory (270)

In the new system, these courses are still present, so the new specific courses required are the same. The only change is an explicit allowance for students who completed multivariable calculus prior to enrolling at Yale, and may consult the DUS about substituting a higher level course for Math 120.

Extra note: Completion of 230 + 231 currently satisfies the multivariable calculus requirement, and this will continue to be the case (for students who complete the sequence before it disappears). If you only completed a part of the sequence, contact the DUS and we’ll find a way for you to fulfill all the requirements.

The only specific course requirement for the Math + Physics major is currently multivariable calculus (Math 120).

In the new system, the only change is an explicit allowance for students who completed multivariable calculus prior to enrolling at Yale, and may consult the DUS about substituting a higher level course for Math 120.

Extra note: Completion of 230 + 231 currently satisfies the multivariable calculus requirement, and this will continue to be the case (for students who complete the sequence before it disappears). If you only completed a part of the sequence, contact the DUS and we’ll find a way for you to fulfill all the requirements.

How does the above affect pre-requisites to other math courses?

Level 200 courses are unaffected, except that every course that required Math 225 now has Math 226 as an additional option.

During the transition period of 2021-24, the courses level 300+ will continue to list old and new courses for their pre-requisites. Current plan, for courses whose pre-requisites needed updating:

- Math 305: Requires Math 225 or 226, and Math 255 or 256 or 301.

Extra note: students who completed Math 231 or Math 250 may explore the option of taking Math 305, with permission of instructor. - Math 310: Requires Math 225 or 226 or 231, and Math 255 or 256 or 230 or 250, and Math 302 or 120.
- Math 325: After Math 320, or after Math 305 with permission of instructor.
- Math 350: One term of linear algebra and two terms of proof-based mathematics courses. (For example, Math 225 and 255, or Math 225 and 244, or Math 230 and 231, or Math 225 and 250.)
- Math 360: Requires Math 350, and Math 302. With permission of instructor, may be taken after Math 350 and Math 231 or 250.
- Math 430: Requires Math 350, and Math 255 or 256 or 300 or 301.
- Math 435: Requires Math 225 or 226 or 231, and Math 255 or 256 or 230 or 250, and Math 302 or permission of instructor.

1. Is Math 302 worthwhile to take for students who completed Math 230-231 or Math 120-225-250?

Math 302 will include some topics you have seen, and some that you have not (an introduction to these extra topics was previously not an explicit part of any class). If you like analysis and want to learn more about manifolds, it may be good to take Math 302. Or you can read about these, if you prefer!

2. What else might a student take after 231 or 250?

There are a lot of good options! We encourage you to reach out to your faculty adviser or to the Math DUS, we would be happy to consider the possibilities. Recommended core classes include Math 350 and Math 310.

3. How frequently will the new classes be offered?

- Math 225 and Math 255 will be offered every semester.
- Math 226 will be offered in the Fall, and Math 256 in the Spring.
- In 2021-22, Math 302 will be offered in the Fall.
- Starting in Fall 2022, Math 302 will be offered every semester.

4. How should I choose between regular and intensive version of linear algebra (Math 225 / 226) and analysis (Math 255 / 256)?

Both courses will be fun, and both will prepare students well for subsequent courses in the major.

The intensive version will not include an explicit introduction to proof writing, it may go a little deeper into the topics, and ask more challenging questions on problem sets and exams. It may be a good choice for students who have prior exposure to proofs (or are willing to od some extra work to catch up), and are looking for an extra challenge. The regular version may be a good choice for students who wish to see an explicit introduction to proof writing, or are looking to take a less time consuming course.

5. How will you help incoming students make choices about what to take?

We are creating a site that will help incoming students navigate the new system. We also plan to publish a short video, and hold advising sessions during the summer, for example during the Academic Fair and our usual calculus / introductory sequence advising day.

We believe the choice of what to take will not be more complicated than it has been in the past - in fact, in some ways, it carries less pressure now (since one doesn’t have to decide right away whether to commit to a year-long intensive course right at the beginning).

What if I have more questions?

If you have any additional questions, please contact the Mathematics DUS. [3]

**Links:**

[1] https://view.message.yale.edu/?qs=384ce417bbd7f3c7f3bc9b32108e60b533f94d7543bf7df79c5160bfe432f500273ba3180202da6011b3c5b1f8cf63823a7b9b13bd835088c656757a788b8ecced91870ed8f7a28b84845a18c5f31d07

[2] https://gauss.math.yale.edu/~mh225/curriculum_revision.pdf

[3] https://math.yale.edu/director-undergraduate-studies