If you would like a book that teaches the fundamentals of reading and writing proofs, I recommend the following:

If you would find it helpful to review some of the material from Linear Algebra, please consult my Linear Algebra notes.

• Develop the ability to read mathematics with a critical and inquisitive mind, and use these skills to both refine and strengthen your learning process.
• Develop technical writing skills, including the ability to communicate abstract mathematical ideas correctly, clearly, and efficiently.
• Learn how to think axiomatically in order to understand the power, elegance, and beauty of abstraction.
• Learn some important concepts of algebra, including fundamental properties of some algebraic structures (such as groups, rings, and fields), equivalence relations, quotient constructions, and homomorphisms (structure-preserving functions).

Homework assignments will be due on Tuesdays and Fridays, and will be posted to the course webpage.

• Homework is due at 5:00pm Grinnell time.
• You should submit you solutions as one pdf file to the corresponding homework folder within your course OneDrive folder (that I will share with you).
• Your lowest homework score will be dropped.
• Solutions to the homework will be posted to the course webpage.
• Throughout the semester, you may extend the deadline of at most two homework assignments by (up to) two days. If you choose to extend the deadline of an assignment, you should send me an email by the usual due date. Beyond these two extensions, late homework will not be accepted for credit, unless there is an emergency that you bring to my attention before the due date.

Although there will certainly be some "computational" problems in the course, most of the homework involves writing proofs and/or detailed explanations. This means that there are often many correct answers. This also means that the clarity of exposition and the proper use of mathematical terminology are as vital to your solutions as having the correct idea. A major goal of this course is to learn how to express your mathematical ideas correctly and to write convincing, detailed, and clear proofs. Do not be alarmed if your homework has many comments about how to improve (nobody starts out as an expert).

I strongly recommend that you learn to type your solutions. LaTeX is a wonderful free typesetting system which produces high-quality documents at the cost of only a small amount of additional work. If you plan to do any kind of scientific writing in the future, you will most likely use LaTeX, so taking the time right now to familiarize yourself with it will pay off.

There will be two exams and a final, each of which will be open book and open notes, and which will focus on conceptual problems and proofs.

Exam dates: Friday, September 18 and Tuesday, October 6. The final will be taken during the 2-day final exam period (October 20-21).

In order to learn mathematics effectively, it is essential to constantly ask questions, to isolate what aspects of the material are unclear, and to make conjectures. To help develop these skills, I strongly encourage you to participate in class and also to bring questions to office hours. Furthermore, you should post (at least) one question a week to the Piazza discussion board. For example, you can ask whether a conjecture you have is true, how to overcome an obstacle in a proof, or why a definition takes the form it does. Please try to make your discussion board question as specific as you can. Instead of saying "I don't understand the proof of Theorem 2.4.3", try to explain why you are having difficulty with the proof, isolate where you are getting stuck, and ask a question that might help clarify your understanding.

Posting one question a week to the Piazza discussion board is a baseline. The corresponding part of the grade will also incorporate participation in synchronous settings, attending office hours, and asking questions by email.

Consult the general Grinnell College policy on Academic Honesty and the associated booklet for general information.

Homework: If you enjoy working in groups, I strongly encourage you to work with others in the class to solve the homework problems. If you do collaborative work or receive help form somebody in the course, you must acknowledge this on the corresponding problem(s). Writing "I worked with Sam on this problem" or "Mary helped me with this problem" suffices. You may ask students outside the course for help, but you need to make sure they understand the academic honesty policies for the course and you need to cite their assistance as well. Failing to acknowledge such collaboration or assistance is a violation of academic honesty.

If you work with others, your homework must be written up independently in your own words. You can not write a communal solution and all copy it down. You can not read one person's solution and alter it slightly in notation/exposition. Discussing ideas and/or writing parts of computations together on a shared document is perfectly fine, but you need to take those ideas and write the problem up on your own. Under no circumstances can you look at another student's completed written work.

You may look at other sources, but you must cite other books or online sources if they provide you with an idea that helps you solve a problem. However, you may not specifically look for solutions to homework problems, and you may not solicit help for homework problems from online forums.

Exams and Final: You will be able to use the Course Notes, the homework solutions, and your own written notes and solutions. You can not use any other sources (books, online sites, other people) during the exam period.

• Spend time actively reading the course notes, your notes, and any other sources you choose. I read one chapter of my algebra book nine times during the course of the semester (not counting reading portions of it to help with homework problems). This is normal, and you should not expect to master a section from going to class and reading it once.

• When reading the course notes, do not sit down and read through them like you would a novel. Have a pencil and paper at hand, and work through the arguments and examples on your own. After reading a proof, stop to process it in your own way and then work through it again in your own words. When you see an example illustrating a result, go back through the proof in the context of that one example.

• There will be many new abstract concepts, definitions, and examples introduced throughout the course. In my experience, attempting to memorize them directly is not the most useful way to succeed. Instead, the way to master these ideas is to work with them (both on the homework and on your own) until they because comfortable and natural.

• The material is challenging at both a high abstract level and a detail-oriented technical level. In my opinion, the best way to understand both is to regularly switch back and forth between the two. When trying to understand a difficult theorem, read it several times at different levels (first try to understand the overall structure, then break it down into key points, then work through specific details, and finally repeat the whole process). When working through an example, picture it in the context high level theory we are developing.

I encourage students with documented disabilities to discuss appropriate accommodations with me. You will also need to have a conversation with, and provide documentation of your disability to, the Coordinator of Disability Resources, John Hirschman, located on the third floor of Goodnow Hall (x3089).

I encourage students who plan to observe holy days that coincide with class meetings or assignment due dates to consult with me as soon as possible so that we may reach a mutual understanding of how you can meet the terms of your religious observance and also the requirements for this course.