MATH 239: Introduction to Mathematical Proof

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"It looked absolutely impossible. But it so happens that you go on worrying away at
a problem in science and it seems to get tired, and lies down and lets you catch it."
- William Lawrence Bragg, won the Nobel Prize at age 24.

Course Description:

Content Summary Topics covered: We will cover various techniques of mathematical proof, logic, set theory, functions, and equivalence relations. Topics are subject to change depending on the progress of the class, and various topics may be skipped due to time constraints. We will also have the option to add certain topics based on student interest.

For many of you, this will be the first mathematics course which uses a more mathematically sophisticated approach than that found in your standard calculus courses. At this point in your mathematical careers, you are mostly familiar with computational mathematics but have had very little exposure to theoretical and proof-based mathematics. Students can sometimes get through lower-level courses by imitation, but struggle in upper-level courses that require them to think abstractly, construct logical arguments, and use mathematical language precisely. This is one of the first courses where you will be asked to write an argument in order to solve a problem. That is, you will have to write "proofs". Most of you may have some basic experience with this through your high school geometry course. Others may have some experience from Linear Algebra. In this course, we will be learning a variety of proof techniques, and everyone should be comfortable with the process by the end of the semester.

This course is known as a "transition course." That is, the purpose of the course is to bridge the theoretical gap between lower-level, computation-based math classes and upper-level, proof-based math classes. The topics and tools learned here will permeate the remainder of your mathematical education and will provide you with the necessary skills to become successful mathematicians. Our primary purpose in this class is to help you make progress in developing analytical, critical-reasoning, problem-solving, and communication skills and acquiring mathematical habits of mind. The number of topics we will cover is deliberately kept to a minimum so that you can concentrate on developing careful use of mathematical language, practice logical reasoning skills, and learn theorem-proving skills. The emphasis throughout much of the course is on process rather than on content. Moreover, work with written and oral communication of mathematical ideas and independent learning or reading of mathematical content are essential parts of this course.

Learning Outcomes: Upon successful completion of this course, a student will be able to:

  • Apply the logical structure of proofs and work symbolically with connectives and quantifiers to produce logically valid, correct and clear arguments,
  • Perform set operations on finite and infinite collections of sets and be familiar with properties of set operations,
  • Determine equivalence relations on sets and equivalence classes,
  • Work with functions, in particular bijections, direct and inverse images, and inverse functions,
  • Construct direct and indirect proofs and proofs by induction and determine the appropriateness of each type in a particular setting. Analyze and critique proofs with respect to logic and correctness, and
  • Unravel abstract definitions, create intuition-forming examples or counterexamples, and prove conjectures.

"I hear and I forget. I see and I remember. I do and I understand."
- Chinese proverb

"You can't be a passive learner in a class like this. You can't zone out. You have to come prepared for class because there will be participation, and you'll have to demonstrate you know the material."
- Student


Course Format:

This course is guided by formative assessment - in plain terms, you tell us what you know and what you don't know, and we try to help you gain the skills you need accordingly. Many aspects of our course were funded by a SUNY Innovative Instructional Technology Grant (IITG). The primary content of this course is located at ProofSpace, which best enables us to use a learning technique called "flipping the classroom." In this format, you will acquire the majority of your knowledge outside of class through video lectures and follow-up comprehension problems. Then, you will apply your knowledge to higher-level problems in the classroom. This format has several advantages, including:

  • The instructional team is available when the students most need us; rather than me delivering lectures and then you trying difficult problems by yourself at home, I'm here when you are trying those difficult problems and when you are most likely to ask for help.
  • Class time is used more actively.
  • The instructional team gets to know you better and cater to your individual needs.
  • We can focus more on high-level problems that will help you be more successful throughout your mathematical education and career.
Most importantly, we have tried this format before and found that in this format, we are better able to guide students through the difficulties they face in this course. Furthermore, research shows that many learners forget what they hear in a lecture format faster than in an active learning format. True learning is not simply a consequence of pouring information into a student's head. True learning requires the student to be mentally involved, doing, not just hearing. Explanation and demonstration almost never lead to lasting learning. Active learning places the responsibility of learning on the students, with teachers coaching beside them. Rather than passively listening, students are thinking, working in groups, presenting, questioning, and discovering for themselves.

It's important to note that this course format should be neither more nor less work than a traditional class. It's just a matter of when you're doing what. For obvious reasons, attendance, though not factored in your grade, is vital to your success as a student of mathematics.

We will mostly be working out of video lectures and assignments created specifically for our course. To access these resources, go to the homework page or go to ProofSpace.

Homework: "Homework" consists of two separate things: Video/Reading Assignments and Problem Sets. There will be regular homework assignments which must be completed before class on the due date. Typically, the reading/video assignments and quizzes will be daily, and problem sets will be due weekly. Follow this link for a Description of Homework requirements. Please note that you are STRONGLY ENCOURAGED to WORK IN GROUPS of 2 or 3 students. However, each student is responsible for writing up his or her own homework. When working in groups, please be careful that you are actively participating in the process: many students find that they can understand a problem while they are watching a classmate work through it and explain it, and then conclude that they understand the material well enough. This leads to an unpleasant surprise at test time, when students who "thought they understood" the material find they are unable to work the problems on their own. Please be careful that you are able to work all of the problems on your own before the exam time arrives, with no coaching from a friend. Please use whatever resources aid you in learning the material, including me, other students, professors, other math books, etc. I strongly recommend that you avoid copying from classmates, textbooks, or the internet. This is known as plagiarism, and it is considered cheating.

Day-to-Day Schedule: Before each class meeting time, you will watch some short videos (usually less than 30 total minutes) and try a few problems online. During each class, we will address any important issues or individual questions in a short discussion period at the start of class, discuss any large-scale questions/issues as a class, and work on and discuss problems from the problem set. We will split into pairs to start working on a collection of problems, some of which will be formally written up, handed in at a later time, and graded. During this class time, we can address more individual questions that you may have, but the main focus will be on the problem set. After class, you'll finish up the problem set and hand in the required problems to be evaluated. You'll get them back shortly thereafter with a numerical grade and feedback. Use this as a springboard to meet with your instructor or teaching assistant about your progress and prepare for the comprehensive exams.

Watching the screencasts and trying the problems are required, NOT optional. Your chances of passing this course are infinitesimally small unless you do these things. Because this will form the foundation of your learning for this class, it is HIGHLY RECOMMENDED that you take notes on what you read or watch, just as you would during a standard lecture course. In addition to standard note taking, it may be helpful to keep a separate list of definitions learned and a separate list of theorems learned. These will be the tools you use throughout the semester on problem sets and exams. If you need more assistance than the videos provide, please consider reading the textbook also. Although this is not required, many students find this extra resource exceptionally helpful.

The best way to LEARN mathematics is to DO mathematics. In that sense, I want to be with you as you are learning the material. I will give you some problems to work on in class that are related to the video or reading assignments. This will allow me to be there to answer the typical initial questions you may have and guide you around the mental obstacles that arise when students begin working on problems. It will also allow me to fix any incorrect logic or recurring mistakes early in the assignment, rather than later or after an assignment is completed. Students will present some of their problem solutions or proofs to the class. It does not matter if the presenter is right or wrong; the purpose is for the presenter and audience to learn from both the successes and the mistakes. This is exceedingly valuable to the learning process. We can often learn more from a wrong answer than from a right answer. There will also be extra problems assigned for you to do outside of class. These problems will be due at the beginning of class on the due date, and there will be an optional recitation where you'll be able to get assistance from the TA before the assignment is due.

Before each class, you must complete the video or reading assignment and the comprehension quiz. These quizzes will be completed on WeBWorK BEFORE you come to class. You must understand the material assessed on the quiz before you can begin the problem set during class.

Each problem set is broken up into four sections (see an example here):

  1. Discussed
  2. Evaluated
  3. Supplemental
  4. Advanced

The Discussed Problems are few in number, and we will discuss them as a class. Although you are required to complete these, they will NOT be handed in to be graded. The Evaluated Problems are the only problems on the problem set that will be handed in and graded. These will most likely be completed independently, out of class. They are sometimes easier and sometimes harder than the Discussed Problems. They are almost always related to the corresponding Discussed Problem.

If you'd like more practice or need to review, reference the Supplemental Problems. Each refers to the end-of-section problems in Dr. Sundstrom's book. These problems, together with many of the Discussed and Evaluated Problems, made up the traditional "homework" problem sets that have been assigned in this course in previous semesters.

The Advanced Problems are excellent choices for students who are very comfortable with the material and would like to explore the topics more. As the name suggests, these problems are more challenging and beyond the scope of this course.

The problem sets were very thoughtfully created. No problem should be considered trivial or a waste of time. If there is even one problem you don't understand excellently by the end of each chapter, you should come talk out the issue with me.


Textbook:

We will mostly be working out of video lectures and assignments created specifically for our course. To access these resources, go to the homework page or go to ProofSpace. Each problem set comes with a list of supplemental problems. If you are having trouble understanding a concept, we encourage you to refer to our supplementary textbook:

Mathematical Reasoning: Writing and Proof, by Ted Sundstrom.
A version of the textbook is available for free online.

This text follows roughly the same chronology as our course. It is an excellent resource for extra problems and an alternative explanation of topics. Dr. Sundstrom has generously made the book available for free online. If you choose to download it, we encourage you to e-mail him a word of thanks. The problem numbers in the list of supplemental problems refer to the exercises in the online version. Although the textbook is of the highest quality, well-written and easy to follow, when working with definitions in this course, always refer to the definition given in our videos. The differences between our delivery and Sundstrom's are few, but they are important and deliberate.

Please note that we will work on developing your independent learning skills in Mathematics and your ability to learn and use definitions and theorems. You will be expected to do a fair bit of video watching. The videos and classroom discussion provide a solid foundation, but you should STRONGLY consider reading the book as well. The reading assignments will be on topics related to the videos and problem sets. Unlike in some other courses, the majority of the material covered in the screencasts and reading will not be repeated in class. Thus, failure to complete the out-of-class assignment and comprehension quiz problems will hinder you. This is a course where taking responsibility for your own education will prove quite rewarding. You as students will help decide what we discuss in class, so that we can focus on the topics that give you the most difficulty.


Exams and grading:

There will be regular problem sets, online quizzes, midterm exams, and a final exam. Your grade in this class is based precisely on your attainment of the learning outcomes listed above. We will measure this in a variety of ways, but the majority of our inferences about your knowledge will come from two comprehensive exams and a final. Your overall grade will be determined as follows:

  • 10% - Reading Quizzes and Class Participation
  • 15% - Problem Sets
  • 25% - Exam 1
  • 25% - Exam 2
  • 25% - Final Exam

The letter grade you earn for the class will be based on the following breakdown of number grades:

A...93-100B+...87-89C+...77-79D...60-69
A-...90-92B...83-86C...73-76E...0-59
B-...80-82C-...70-72

In all written work, you must show your work and clearly show the process and reasoning you went through in order to solve the problem. The problems I work for you in class will provide good examples of how your homework and exam problems should be written up. All assessment will be based on your ability to communicate a correct solution and explain your reasoning. It is absolutely essential to write clearly and completely. It is your responsibility to write in a way that tells me that you understand the problem and its solution.

The reading quizzes will be online on WebWorK. The questions are short, and in general the quizzes shouldn't take you any longer than 5 minutes. They are usually 1-4 multiple choice or matching questions. You shouldn't have a problem with them if you watched the videos. In the event that many students struggle with a question, we will discuss it in class. The problem sets will be graded on correctness, but the intent is to give you feedback to correct errors in advance of the exams. The exams are comprehensive, dealing with techniques and ideas covered in the video lectures and problem sets. Any quiz, in-class, out-of-class, or supplemental problem is fair game, as well as totally new problems. (The advanced problems are not fair game for the exam.) The exams will be reasonably challenging. Exams are closed book, closed notes, closed friends, and open brain. Use of calculators, phones and other electronic devices will NOT be permitted during exams. The dates of the exams will be decided a week or two in advance.

Through the first two units (Preliminaries, Proof Techniques), most of your proofs will be graded out of 12 points, on the following rubric:

Rubric 1

After that, most of your proofs will be graded out of 8 points on the following rubric:

Rubric 2

Note that this rubric is used at the grader's discretion and doesn't apply to problems that ask you to list examples, disprove propositions, or state conjectures.


Extra Help:

It is essential not to fall behind because each lecture is based on previous work. If you have trouble with some material, SEEK HELP IMMEDIATELY in the following ways:

  • ASK ME! (either in class or privately),
  • One of the very best resources may be your fellow students, or
  • Talk to your TAs.

I am willing to spend a few minutes in class answering questions about homework problems. However, if you have a lot of questions, I recommend taking advantage of my office hours. I'll say it again…USE MY OFFICE HOURS! My job is to help you -- come to office hours even if you have just a small question. Don't wait until you get too far behind. If my office hours are not convenient for you, make an appointment by sending an email or asking after class. Please come see me as soon as you feel lost -- it is important that I know how you are doing so I can adjust the level of the class if necessary. I WANT to help you, and I WANT everyone to do well.

If you are having any difficulties, seek help immediately - don't wait until it is too late to recover from falling behind or failing to understand a concept!

Accommodations: SUNY Geneseo will make reasonable accommodations for persons with documented physical, emotional, or cognitive disabilities. Accommodations will be made for medical conditions related to pregnancy or parenting. Requests for accommodations including letters or review of existing accommodations should be directed to Ms. Heather Packer in the Office of Disability Services in Erwin Hall 22 or disabilityservices@geneseo.edu or 585-245-5112. Students with letters of accommodations should submit a letter to each faculty member at the beginning of the semester and discuss specific arrangements. Additional information is available at the Office of Disability Services.