Mathematical Reasoning: Course syllabus

Course description #

Catalog: An introduction to logical deductive reasoning and mathematical proof through diverse topics in higher mathematics. Specific topics include set and function theory, modular arithmetic, proof by induction, and the cardinality of infinite sets.

Reality: Who wrote that description above? What were they thinking? You have taken so much calculus! Math 2020 is an introduction to what mathematics is outside of calculus. Along the way you will learn how to make a clear mathematical argument – this is a skill, not an innate ability. You will see that mathematics has its own language, which you will learn like any other language- slowly! You will become a creative and bold problem solver, and think about math in completely new ways.

Learning goals: #

  • You will gain an appreciation for how much mathematics there is past calculus! You will begin to see that there are interesting and important connections between different areas of mathematics.

  • Mathematics is a language – it’s just not English! You will learn how to represent both mathematical and non-mathematical statements using symbolic notation. Concurrently you will learn to take a symbolic expression and understand what statement it represents.

  • You will learn how to make a rigorous mathematical argument. What needs to be said to justify a statement? What doesn’t need to be said? What logical rules govern mathematical deduction?

  • You will learn how to read a detailed mathematical argument, by isolating the main steps and understanding the justification of each.

  • You will become a creative problem solver, and you will learn how to be wrong. To be mathematically fearless involves much experimentation, and rarely do we find the correct solution on our first try. You will learn that an incorrect approach often gives you the tools you need to solve a problem correctly. (If it makes you feel better, Prof. Taback is wrong all the time!)

Some perspective: Making a precise mathematical argument is a learned skill, not an innate ability. Everyone is starting from zero in this class! We will all struggle at the beginning, and our abilities will grow over the semester. Like any skill, the more you practice the faster you will improve!

It’s also important to know that even professional mathematicians are still improving at writing mathematical proofs! This is an endeavor that we will begin together in Math 2020, and you will work on for your mathematical career. It will get easier over time and Math 2020 will provide you with the tools you need to succeed.

Book and Materials #

The textbook for the class is Introduction to Mathematical Structures and Proofs, by Larry J. Gerstein. It is available for free from the Bowdoin library. You must have the second edition so that you do the correct problems for your homework!

I will use additional materials for this course. All will be available on this website.

Course meetings and office hours #

We will meet in Searles 213 every Tuesday and Thursday at 8:30am. I will hold formal office hours on

Mondays and Wednesdays at 4:15pm. Office hours will be held in Searles 205.

There may be times in the semester where I will have to reschedule office hours due to a conflict. I will let you know by email beforehand. I am available throughout the week for additional meetings. To schedule either individually or as a group, please send me an email with a couple of times that will work. And please don’t hesitate to set these up, I am very happy to see you.

Reading and watching mathematics #

There will be times this semester where you will be asked to read a section or watch a short video about a topic related to the course. Learning mathematics is not a spectator sport. Reading mathematics is not like reading a novel; watching mathematics is not like watching an action thriller. Some paragraphs are easy to digest, but you may find yourself looking at one line of text for five or more minutes trying to understand what the author is trying to say. Use the pause button when watching a video. As you read or watch, take notes, just as you do in class. This is crucial! If questions arise, write them down and ask during office hours.

Homework #

Homework will be due weekly at 11:00pm on Tuesday evenings, and is submitted through Gradescope. Homework must be submitted with the Math 2020 cover sheet. Each submission will consist of three parts:

  • Cover sheet: In addition to your name, I will also ask you to recognize individuals you worked with and sources you used to complete your work. There will also be a space to briefly discuss your weekly group meeting.

  • Class notes: As you participate in class, I expect you to actively take notes. You will submit them as part of the homework each week. They will be graded generously, but should be complete and legible.

  • Homework problems: Problems will be assigned during every lecture and formally posted on this website.

Your homework will be graded using GradeScope. To begin, you will need to set up an account.  I will send you the code for our class by email.  Each homework assignment will need to be submitted as a .pdf file.  If you edit your homework electronically, make sure you can save or export your work in this format.  If you write-up your homework the old-fashioned way using pencil and paper, use a scanner or a phone scanner app.  See the help document for a list of suggestions. Please let me know if this does not work for you; I will come up with an alternative.

Collaboration and groups: Throughout the course of the class, you will be a part of a group. While your work will be written-up individually, I would like all of you to check in with each other and discuss the course material and homework at least once a week outside of class. Homework which is not neat and legible will not be graded. Late homework will be graded at the discretion of the grader.

Additional lectures #

Parts of the goals of this course is to acquaint you with the mathematical world beyond calculus and to show you how mathematical reasoning is used in real life. As part of the class, you will be required to go to three math department lectures this semester. After each, you will submit a short response paper. Submission of three responses will count as a completed homework assignment. You may email your responses directly to me.

Exams and a quiz #

There will be two mid-term exams, a quiz, and a final exam in the course. A few details about them. Both mid-term exams will consist of two parts: a take-home portion and an in-class portion.

A quiz

  • There will be a quiz on February 17. It will count as a homework assignment for grading purposes. It will be available on Gradescope and you will take 30 minutes to complete it at your own convenience.

First Exam:

  • Part 1: Open book, open notes, take-home exam, available February 26 at 8:00am and due on March 1 at 11:00pm.
  • Part 2: Closed book, February 26 from 7:00pm-9:00pm.

Second Exam:

  • Part 1: Open book, open notes, take-home exam, available April 16 at 8:00am and due on April 18 at 11:00pm.
  • Part 2: Closed book, April 16 from 7:00pm-9:00pm.

Final Exam: The final exam will be an open book, open notes, take-home exam available after our last class meeting and due one week later. More specific details to come.

Grades #

Your grade will be computed as follows:

  • Homework: 30%
  • Exams: 50%
  • Final Exam: 20%

Some axioms #

Federico Ardila enunciated the following axioms in his Todos Cuendan. They form a lens through with I view both teaching and doing mathematics.

  • Mathematical potential is distributed equally among different groups, irrespective of geographic, demographic, and economic boundaries.
  • Everyone can have joyful, meaningful, and empowering mathematical experiences.
  • Mathematics is a powerful, malleable tool that can be shaped and used differently by various communities to serve their needs.
  • Every student deserves to be treated with dignity and respect.