Course Description
How do mathematicians think? Why are the mathematical facts learned in school true? In this course students will explore higher-level mathematical thinking and will gain familiarity with a crucial aspect of mathematics: achieving certainty via mathematical proofs, a creative activity of figuring out what should be true and why. Through this course, students will prove some mathematical statements with which they may already be familiar; introduce new concepts and ways of thinking that illuminate known facts and help us to explore further; and learn how to carry out careful logical arguments and write proofs. The two projects will involve writing up, in stages, a logically more complicated proof or sequence of proofs.
This course is ideal for students who would like to learn about the reasoning underlying mathematical results, but hope to do so at a less intense pace and level of abstraction than Math 61CM/DM offers, as a consequence benefiting from additional opportunity to explore the reasoning. Familiarity with one-variable calculus is useful since a significant part of the course develops it systematically from a small list of axioms. We also address linear algebra from the viewpoint of a mathematician, illuminating algebraic notions such as groups, rings, and fields. This course may be paired with Math 51 although that course is not a pre- or co-requisite.
This course is a descendant of the freshman seminar, Math 83N, and is expected to take a similar format to the extent possible.
Satisfies Ways requirement for Formal Reasoning (FR).
Per University policy, your decision to take the course implies that you agree to these requirements and to the grading policies spelled out here; so be sure to read everything on these pages.
Schedule:
This course is structured with the expectation that you will attend every lecture. We plan on having class
activities that will be graded mostly for participation, not so much for accuracy.
Of course, sometimes an absence is necessary. In such a situation, you should contact a classmate to get notes and other information for the class you missed.
We will have 30 lectures in total. A tentative schedule is posted at this link, but may be adjusted as the quarter goes on.
Text:
There are two textbooks for this course:
R. Hammack book
We recommend that you print out the relevant pages, since we find that students more deeply engage with printed rather than electronic materials.
Course website:
Most of the materials will be posted on this website: math56.stanford.edu
Some course announcements will be posted through Canvas.
Grades:
-
Grades will be based on the following components:
- Homeworks: 15% + 5% for presenting one or two problems at office hours
- Class worksheets: 40%
- Two writing projects: 20% each
Students who elect credit/no credit grading basis need to obtain the grade of C- or better to receive
credit.
Typical week:
This class started as a seminar. When the enrollment reahed nearly 50 students,
some modifications had to be made. For this quarter my goal is to restore some of
the elements of a seminar.
To make this class more participatory and engaging during each class we will have a problem or two solved in small groups.
I will collect those worksheets as evidence of your class participation.
In addition to that
you will have to present one or two homework problems either in the class or at a meeting in my office. This mini-presentation will count for 5% of your grade.
Supplemental Textbooks and other Resources:
Stqrting with the homework, we encoursage you to write your work in TeX. Here are some good resources for LaTeX:
In addition to the required textbooks, the following textbook gives a systematic approach for teaching students how to read, understand, think about, and do proofs. We recommend it! The approach is to categorize, identify, and explain the various techniques that are used repeatedly in all proofs. The sections on mathematical quantifiers (∀, ∃) may be relevant for our section on real analysis. The book is available in a limited simultaneous online format at searchworks.stanford.edu/view/13884217 but one might want to find one's own copy from a (perhaps online) bookstore.
How to Read and Do Proofs: An Introduction to Mathematical Thought Processes by Daniel Solow (6th edition, ISBN: 1118164024)
You are encouraged to attend the
office hours provided by the instructors and course assistant.
Another resource which may be of use is Counseling and Psychological Services. See vaden.stanford.edu/caps-and-wellness.
Access and Accommodations
- If you experience disability, please register with the Office of Accessible Education
(OAE).
Professional staff will evaluate your needs, support appropriate and reasonable accommodations, and prepare an Academic Accommodation Letter for faculty. To get started, or to re-initiate services, please visit
oae.stanford.edu. The "OAE" is located at 563 Salvatierra Walk (phone: 1-650-723-1066).
- To receive academic accommodations in this course, you must provide an accommodation letter from the OAE, dated in the current quarter, by filling out the
ACCOMMODATIONS & FLEXIBILITY FORM (hosted at http://goto.stanford.edu/math56oae) to detail the specific accommodations you will need in this course.
- Letters are preferred by the end of week 2, and at least two weeks in advance of any exam for us to have adequate time to arrange the accommodations. All accommodation requests for the final exam must be submitter
no later than November 17, 2025, at 5:00 pm.
- New accommodation letters, or revised letters, can be submitted throughout the quarter; please note that there may be constraints in fulfilling last-minute requests. Alerting the course staff of Math 56 is not equivalent to filling out the ACCOMMODATIONS & FLEXIBILITY FORM and may result in you
not receiving accommodations.
- You need only submit your letter once per quarter. For urgent OAE-related accommodations needs that arise after the deadline, please consult your OAE advisor.
You need only submit your letter once per quarter. For urgent OAE-related accommodations needs that arise after the deadline, please consult your OAE advisor.
Special situations adaptations
As standard practice, lectures and discussion sections in Mathematics
courses are taught in-person. As such, Zoom links will not be provided. Additionally, in-person
lectures and discussion sections will not be recorded.
Students who miss class due to illness (including COVID-19) should
make arrangements to obtain lecture notes from other students
in the class. As standard practice, there are no make-up exams or
remote exams. If you will miss an exam due to illness, please
reach out to your instructor for more information.
Fall 2025
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