Revision of the Introductory Courses

March 11, 2021

Dear Math majors and Friends of math,

We write to announce the long-awaited revision of the introductory sequences, through real analysis at the level of Math 300 and 301. 

Briefly, the main changes are:

  1. The material taught in the existing introductory sequences has been restructured, and Math 230/231, 250, 300, 301 are being phased out. 
  2. The new introductory sequence will include proof-based linear algebra (Math 225/226), and real analysis (Math 255/256). 
  3. There is a new course (Math 302) focusing on a rigorous treatment of multivariate analysis (inverse/implicit function theorems, smooth manifolds in R^n, differential forms, etc.)

For more details, please visit .  The site includes notes about the revisions, descriptions of the new courses, and information about transitioning to the new curriculum, for all math and joint-math majors who need to complete the introductory sequence requirements. The old sequences (Math 230+231 or Math 120+225+250) will continue to satisfy the introductory requirements for students who have already completed the courses. 

The new curriculum is the result of several months of work by many members of the math department. We believe that the new sequences will be easier to navigate, make the major more easily accessible to students with less mathematical background, and add flexibility for incoming students in terms of course placement as well as the ability to mix regular and intensive courses according to personal preferences and time constraints. 

We realize that these are extensive changes, particularly as some of our old course numbers are disappearing. We feel that everything that was wonderful about the current courses will remain in the new ones, so that nothing will be lost, and a lot will be gained in our new and exciting courses!

We expect that many of you will have questions about the new curriculum. In addition to the site linked above, Yair, Pat, and Miki will hold a town hall next Sunday, March 14, at 1pm ET. The link has been sent in an email to current and prospective majors.  We hope that many of you will attend to ask questions and share thoughts about the changes. 

Lastly, we include below a message from Pat Devlin, who has been teaching Math 230/231 for several years, and shares his thoughts about the new curriculum.  

Best wishes,

Yifeng and Miki

A word from Pat Devlin (current instructor of 230/231)

Hello all!

As noted above, we are restructuring some of our introductory courses to make our course catalog easier to navigate, to allow for greater flexibility in students’ scheduling, and to make it easier for folks of all backgrounds to explore the awesome courses the department has to offer. 

I am very excited for these changes.  Regardless of their calculus backgrounds in high school, all students will be able to take fun proof-based math starting in their very first semester.  Our newly designed linear algebra course (225) as well as our new introductory analysis course (255) are both deliberately structured to emphasize a careful introduction to proof-writing, which makes them perfect for any student hoping to explore the major, and these courses can be taken in either order (though linear algebra is recommended first).  Students already familiar with proofs (or who might want a somewhat more time-intensive option) could choose to replace either [or both] of these courses by taking a slightly more work-heavy flavor (226 or 256), and in fact students will be able to mix and match these flavors however they see fit while still fulfilling all the requirements for the major.  Regardless of which combination of flavors for linear algebra and analysis students might choose to take, they can expect that after just two classes, they will be well-equipped budding mathematicians ready to dive into exciting classes at the 300-level and beyond!

One of my favorite things about this restructuring plan is the flexibility it provides our students.  For instance, perhaps a student is curious about proof-based math and they decide to take the linear algebra course 225, which includes a solid built-in introduction to proof writing.  The student enjoys the course, and they decide they would like to take analysis next—to further explore the major and to finish the introductory track.  The student now has complete flexibility to take either 255 [with a greater emphasis on the mechanics of proof-writing] or 256 [which assumes those skills already].  This choice is entirely based on student preference of what course might better fit their goals and schedule, and in either case, they will have fulfilled the two introductory courses required of the major, and they will be ready to take abstract algebra (350), measure theory (305), or whatever.

Or, a student might start out taking the more time-intensive 226 linear algebra course, and then in their next semester they could follow it up with either 255 or 256 based on whatever they feel like!  And the best part is that regardless of the introductory path chosen by students, they will all be on a level playing field without needing to take any additional classes that might otherwise slow their progress in the major.

On a more personal note…  For the past eight semesters, I have had the genuine delight of teaching 230/231.  There is something inherently wonderful that happens as students are welcomed into the exciting new world of proof-based mathematics, and the communities that develop from our shared scholarly pursuits are what I cherish most in my work.  The course name “230/231” is being phased out in favor of numbers that more accurately reflect the new structure of our introductory sequence, but in fact the spirit of separating the course content into two standalone semesters is essentially identical to how I am currently teaching the material.  I am thrilled that this transition will provide students with more options and greater flexibility in how they choose to explore the major, and I am particularly excited to see the communities of budding Yale mathematicians that will benefit for years to come.