Foundations for Higher Math – Math 3010

Renato Feres

Fall 2025

Math 3010 – Foundations for Higher Mathematics (Fall 2025)

Section information

  • Instructor: Renato Feres
  • Class time and location:
    • Section 01: M-W-F from 2:00PM to 2:50PM in Wrighton Hall, room 250
    • Section 02: M-W-F from 3:00PM to 3:50PM in Wrighton Hall, room 250
  • Office hours: Wednesday 4:10PM to 6:00PM and Friday 10:00AM to 1:00PM; Cupples I, room 17

Instructor’s email and office address

Please include Math 3010 in the subject line of any email message that pertains to this course. My email address is and my office is in Cupples I, room 17.

Course Mode

In-person

Course Description

An introduction to the rigorous techniques used in more advanced mathematics. Topics include set theoretic methods of proof, counter-examples, basic logic, foundations of mathematics. Use of these methods in areas such as construction of number systems, counting methods, combinatorial arguments and elementary analysis.

Prerequisites

Math 233 - Calculus III

Credit hours

3 credit hours

Learning Goals

This course is designed to introduce you to rigorous mathematical language through a variety of topics in pure mathematics. A large component of the course consists in practicing the skills needed for clearly communicating math ideas, particularly in writing. This is, in essence, what we call proofs! Such skills will be developed as we explore some fundamental (and beautiful) mathematical ideas from basic number theory, algebra, combinatorics, and analysis.

Required text

A Concise Introduction to Pure Mathematics by Martin Liebeck, fourth edition, CRC Press, 2016.

Liebeck’s text contains a large number of relatively short chapters on a variety of topics. I expect we will be able to cover approximate 20 of them. This will be enough to acquire familiarity with some widely used math arguments and ideas, many of which will be developed in greater depth in more advanced courses having Math 3010 as a prerequisite.

Course plan

I hope to cover about twenty chapters in the textbook, although it is possible that we will fall a bit short of this goal. The book consists of short chapters that have some degree of independence, so we have flexibility in which chapters to pick. I plan roughly to follow the order of the book, but not too strictly. My goal is to lead you to a level of mathematical maturity sufficient for succeeding in our upper level classes, rather than to cover specific topics at length. There will be no serious harm done if we do not cover all of the chapters. I will let you know in advance the chapter I intend to cover on each lecture and strongly encourage you to read it ahead of time in preparation for the lecture.

Coursework, exams and grades

There will be two midterm exams during the semester, and a final exam. These will be held as follows:

  • Midterm 1: October 3, Friday, in class.

  • Midterm 2: November 7, Friday, in class.

  • Final exam: December 12, from 8:30PM to 10:30PM; in Wrighton 300.

IMPORTANT: You are expected to take the exams at their scheduled times. If you are away because of a university sporting event or field trip, then you may arrange for your coach or professor to administer the exam. Excused absences may be granted in the case of illness or bereavement, at my discretion. The final exam date cannot be changed for reasons of traveling convenience.

Your final grade will be based on your performance on the three exams and on homework assignments. They will count as follows:

  • Cumulative homework grade: 20%

  • Midterm 1: 20%

  • Midterm 2: 20%

  • Final exam: 40%

Letter grades will be obtained from the total of HW assignments and exams according to the following scale:

  • A (-, plain, +): cumulative score in [90%, 100%]

  • B (-, plain, +): cumulative score in [80%, 90%)

  • C (-, plain, +): cumulative score in [65%, 80%)

  • D: cumulative score in [50%, 65%)

  • F: cumulative score less than 50%.

The cut-offs for the letter grade sign (-, plain, +) will be set at the very end of the course, when all the scores have been computed. Cut-offs will be set so as to make the overall number of -, plain, + roughly equal. (This is not the same as saying that each letter interval will be subdivided into three subintervals of equal length!)

I may modify these cut-off values in the event that, say, an exam turns out to be much harder than expected (not always easy to predict beforehand!), although no changes will be made that would result in a tougher scale than the above.

Resources for students

WashU provides a wealth of support services that address academic, personal, and professional needs. To start exploring resources that can help you along the way, please visit Resources.

Specific to this course:

You are permitted to get help on homework assignment problems from me, other students, or anyone else, consult others books, and research the internet. But the writing of the final product must be your own.

Large language models (LLMs), such as ChatGPT or DeepSeek, are useful tools for exploration and learning and you are free to used them in homework assignments, as well as any other tools you find helpful. However, no such tools will be allowed in exams. Also, keep in mind that LLMs need to be used with great care and, in my experience, often make mistakes in constructing mathematical proofs. I’ll have more to say about this topic in class.

Good writing

This may be the first math course you take in which the writing of mathematics is just as important as knowing the answer to problems. As everything else, writing mathematics well comes from practice (in particular, from the experience of reading math texts and papers), but a few simple recommendations will go a long way. Keep the following in mind when writing your homework assignments:

  • Never submit your first draft! Once you are happy with your solutions, rewrite them in a clean and orderly way. I will ask the grader to subtract points from messy and difficult to read assignments. You may find this course a good opportunity to practice writing in latex (this is neither required nor will give you extra points). On the other hand, it is better to write your assignments nicely by hand rather than use a writing program that does not render math symbols properly.

  • Write with empathy! Put yourself in the shoes of the reader. Are you writing so much that the main points of a proof get lost in the middle of lots of trivial observations, or are you writing so little that the reader won’t find your explanations very helpful?

Homework assignments (IMPORTANT!)

New homework assignments will be posted here (solutions in Canvas under Pages). You will need to visit this page on a regular basis to find the new assignment for the week. Homework submission will be through GRADESCOPE and the solutions are due each Friday by 11:59PM. Don’t wait till the last minute! I may not be up and able to help should a technical issue arise.

If it happens (as it may occasionally) that the email notification from Gradescope to upload your assignment didn’t arrive by Wednesday (simply because I forgot to set it up), please remind me in class and I’ll get to it later in the day.

After uploading your assignment to GRADESCOPE, please make sure that it is readable. An example of what to avoid: say that you upload the entire assignment to the place of exercise number 1, and leave the others blank. The script will be too small to read and I won’t be able to assign points where they should be.

Note: I will rely on the 4th edition of Liebeck’s textbook and will take homework exercises from it. If you have the 3rd edition, look for “Exercises from the textbook 4th edition” pdf file in Pages (Canvas).

Assignments:

University wide policies (not specific to this course)

  • Academic integrity

In all academic work, the ideas and contributions of others (including generative artificial intelligence) must be appropriately acknowledged and work that is presented as original must be, in fact, original. You should familiarize yourself with the appropriate academic integrity policies of your academic program(s).

Specifically concerning Math 3010 (and this instructor), see above remarks in Help Resources.

  • Unauthorized recording and distribution of classroom activities & materials

The following applies to all students in my class: “Except as otherwise expressly authorized by the instructor or the university, students may not record, stream, reproduce, display, publish or further distribute any classroom activities or course materials. This includes lectures, class discussions, advising meetings, office hours, assessments, problems, answers, presentations, slides, screenshots or other materials presented as part of the course. If a student with a disability wishes to request the use of assistive technology as a reasonable accommodation, the student must first contact the Office of Disability Resources to seek approval. If recording is permitted, unauthorized use or distribution of recordings is also prohibited.”

  • Disability resources

WashU supports the right of all enrolled students to an equitable educational opportunity and strives to create an inclusive learning environment. In the event a physical or online environment, learning activity, or learning interaction results in barriers to your inclusion due to a disability, please contact WashU’s Disability Resources (DR) to engage in a process for determining and communicating approved accommodations. As soon as possible after receiving an accommodation from DR, send me your WashU Accommodation Letter. Because accommodations are not applied retroactively, initiate your request to DR prior to, or at the beginning of, the academic term to avoid delays in accessing accommodations once classes begin. https://disability.washu.edu/

Note specific to this course: Special accommodation solutions proposed by the Disability Resources office may not always be the most convenient. For example, I may not be available for questions during an exam if it is going to be given at a location far from our classroom. Before you schedule taking a test at the DR office, come talk to me to discuss whether alternatives are available that you’d find more convenient.

  • Sexual harassment and assault

If you are a victim of sexual discrimination, harassment or violence, we encourage you to speak with someone as soon as possible. Understand that if you choose to speak to me as an instructor, I must report your disclosure to my department chair, dean, or the Gender Equity and Title IX Compliance Officer, which may trigger an investigation into the incident. You may also reach out to the Relationship & Sexual Violence Prevention (RSVP) Center to discuss your rights and your options with individuals who are not mandatory reporters. Here’s a link to Gender Equity and Title IX Compliance WashU page

  • Religious holidays

To ensure that accommodations may be made for students who miss class, assignments, or exams to observe a religious holiday, you must inform me in writing before the end of the third week of class, or as soon as possible if the holiday occurs during the first three weeks of the semester. For more information, please see the university’s Religious Holiday Class Absence Policy.

Note specific to this course: I intend to record lectures, which will be available in Canvas, should you have to miss class due to a religious holiday or illness. On ordinary days, though, I fully expect you to attend in person! You will miss all the fun if you watch a lecture alone in your dorm.

  • Emergency preparedness

Before an emergency affects our class, students can take steps to be prepared by downloading the WashU SAFE App. In addition, each classroom contains a “Quick Guide for Emergencies” near the door.