MAT 351 Euclidean and Non-Euclidean Geometry

Welcome to the home page of Prof. Xiao Xiao’s MAT 351 Euclidean and Non-Euclidean Geometry course at Utica University. You can find all the information for this course on this page.

Important Dates

• Add/Drop deadline: 01/20/23
• Spring break: 03/13/23 - 03/17/23
• Withdraw deadline: 03/27/22
• SOOT: 04/24/23 - 05/01/23
• Final exam: 05/02/23 - 05/06/23

Instructor Information

• Instructor: Prof. Xiao Xiao
• Email: [email protected]
• Office: White Hall 255
• Office hour: Tuesdays, Thursdays 1-2 pm, Wednesday 10-11 am

An Important Course Policy

I pride myself on having a good environment for working and learning. It is very important to me that we all treat each other with care and respect, in equal measure. I know that I ask students to take risks in class almost every day, and this can be challenging for many. I ask that you help me keep our classroom a supportive place for each of the people in it. Each of us deserves the space to bring our full, authentic selves to class and be comfortable. (Adapted from T.J. Hitchman.)

General Course Information and Policies

• Course name: MAT 351 Euclidean and Non-Euclidean Geometry

• Course credit hours: 3 credit

• Course Prerequisite: MAT 151

• Class time and location: Tuesdays and Thursdays 11:30 am - 12:45 pm at Hubbard 210

• Textbook: Euclidean Geometry, A Guided Inquiry Approach, by David M. Clark

• Course description (Catalog): Euclidean geometry examined as a system of carefully formulated axioms, precise definitions, and rigorous proofs of theorems in plane and solid geometry. Additional topics may include history, foundation, and application of non-Euclidean geometries.

• Program learning goals: In accordance to the learning goals of the Department of Mathematics of Utica College, MAT 351 will introduce and reinforce students’ ability of:
  • (PLG1) Reading and analyzing mathematical proofs.
  • (PLG2) Writing mathematical proofs.
  • (PLG5) Communicating mathematics in written form.

• Class organization: This course will likely be different from any other math course you have taken before. As an instructor, I will not be lecturing most of the time although I love lecturing very much. Scientific research shows that most people do not learn mathematics by listening, instead, they learn by doing it! I am sure you have said to yourself before “It looked so easy when the professor was doing it, but now I am confused when I have to do it by myself.” Why? Because the knowledge belongs to your professor and does not belong to you. You do not learn the knowledge simply by hearing it once or twice from somebody else. In order for you to have a more thorough understanding of the knowledge, we will use a pedagogical practice called inquiry-based learning. Most of the time during the class, students will be working in groups and presenting solutions that they have produced by themselves and not by other people or textbooks.

• You should not look to resources outside the context of this course for help. That is, you should not be consulting the web, other texts, other faculty, or students outside of our course. On the other hand, you may use each other in this course, the course notes, me, and your own intuition.

• Regular attendance is mandatory and is vital to success in this course, but you will not explicitly be graded on attendance. Yet, repeated absences may impact your participation grade.

• You need to get a set of straight and compass (e.g., like this). We will do a lot of straight and compass constructions during this course.

Homework

• Daily Preparation and Research Notebook

  After each lesson, I will post a list of the problems that we are working from the textbook. You will be expected to read and solve (or make as much progress as possible) these problems before walking into the next class period. This will ensure that you are ready to take an active part in our class presentations and discussions. The research notebook is a place to keep and organize your notes on all of these problems. Submitted work should be carefully, clearly, and cleanly written. Among other things, this means your work should include proper grammar, punctuation and spelling. You will almost always write a draft before you write down the final argument, so do yourself a favor and get in the habit of differentiating your scratch work from your submitted assignment. Also, keep your mistakes in your research notebook. The process of understanding and attempting to solve new problems will involve many dead ends. Do not erase your mistakes. Write a note next the mistake about why you believe it is wrong. These ideas might be useful in the future. Please use filler paper a 3-ring binder as your research notebook.

  Each Daily Preparation will be submitted twice. The first time you will submit in Google Classroom before 3 pm on the day before we meet in class. For example, a Daily Homework assigned on Thursday will be discussed on Tuesday, so you should submit your draft by Monday 3pm at Google Classroom. I will review your draft and give suggestions and comments as soon as I can. You can read the comments and make necessary changes before you come to class on the next day. Your Daily Preparation work will usually be finished by hand and paper so the best way to do this is to take a picture of your writings. Please make sure that all pictures are properly oriented before submitting them. During the class, you will use felt tip color pens (provided by me) to take notes and make edits based on the presentations and class discussions on your Daily Preparation work. After the class, I will collect and grade them based on whether your pre-class work shows that you made a genuine effort before class. Incorrect but well-thought-out answers are fine because the goal is to be prepared, not be correct every single time!

• Weekly Homework

  In addition to the Daily Homework, you will also be required to submit two formally written problems each week. You may choose any two problems marked with * that were turned in during a given week to submit the following week. The Weekly Homework assignments are subject to the following rubric:

  Grade	Criteria
  E: Excellent	Correct, complete, convincing, and clear. Writing is clear and easy to follow. All reasoning is provided. Any errors are trivial.
  M: Mastered	Demonstrates fundamental understanding of the relevant concepts with correct reasoning and explanation. May include minor errors and a few parts may be poorly explained or difficult to follow.
  P: Progressing	Demonstrates partial understanding and useful progress, but with some major gaps including math and/or logical errors or missing steps. You should review and resubmit your work.
  X: Not assessable	Fragmentary or no response. Insubstantial attempt. Too many errors to correct or uses an inappropriate method or tool. Must be redone and you should come to see!

  Any Weekly Homework problems that receive a grade below mastered can be resubmitted once for reassessment. Please label the assignment as “Resubmission” on top of any problem that you are resubmitting and keep separate from any other problems that you are turning in.

  You are allowed and encouraged to work together on homework. However, each student is expected to turn in their own work. In general, late homework will not be accepted. However, you are allowed to turn in up to 3 homework assignments (daily or weekly) late with no questions asked. Unless you have made arrangements in advance with me, homework turned in after class will be considered late.

  All of your Weekly Homework must be typed using LaTeX. LaTeX is the industrial standard for typing scientific works in mathematics, physics, computer sciences, among others. The best way to learn how to use LaTeX is just like how you learn everything else: by using it! Fortunately, there is a website called Overleaf so you can use LaTeX online for free without having to install any software. I have also created a template for your Weekly Homework that should make things much easier for you. I will try to schedule a training session during week 1 to prepare you with some basics.

Presentations

• Though the atmosphere in this class should be informal and friendly, what we do in the class is serious business. In particular, the presentations made by students are to be taken very seriously since they spearhead the work of the class.

• The problems chosen for presentations will come from the Daily Preparation. After a student has presented a problem that the class agrees is sufficient, I will often call upon another student in the audience to recap what has happened in the proof and to emphasize the salient points.

• In order to make presentations go smoothly, presenters need to write out the proof in detail and go over the major ideas and transitions, so that he or she can make the proof clear to others.

• The purpose of presentations is not to prove to me that the presenter has done the problem. It is to make the ideas of the proof or the solution clear to the other students.

• Presenters need to write in complete sentences, using proper English and mathematical grammar.

• Fellow students are allowed to ask questions at any point and it is the responsibility of the presenter to answer those questions to the best of their ability.

• Since the presentation is directed at the students, the presenter should frequently make eye contact with the students in order to address questions when they arise and also be able to see how well the other students are following the presentation.

• Confusions and mistakes are very common when learning new mathematics and they should be handled positively to stimulate your thinking. In light of this idea, in each presentation, the presenter must do one or both of the following:
  • ask at least one question that they do not know the answer and then the whole class will discuss about it or
  • talk about at least one mistake that they made while working on the problem, how they understand that mistake, and then how to correct the mistake to enlighten the final solution of the problem.

• It is also uncommon for someone to get stuck during a presentation because a major mistake is discovered. The presenter can try to resolve the mistake on the spot at the board, or they can hit the “pause” button. That means they will end the presentation and the come back during the next class to finish the presentation.

• If you have an idea about a proof that you would like to present, but you are worried that your proof is incomplete or you are not confident your proof is correct, you are strongly encourage to present! You will be rewarded for being courageous and sharing your creative ideas! Yet, you should not come to the board to present unless you have spent time thinking about the problem and have something meaningful to contribute.

• In each class, a sorted class list produced by a computer program will be shown before presentations. Students whose ranks are high in the list have higher priority to choose problems. The sorted list is not produced randomly. It takes three factors into consideration:
  • The number of past presentations: the more one presented in the past, the lower one is on the list.
  • Recentness of past presentations: the more recently one has presented, the lower one is on the list.
• A student can choose not to present on a day when they have a high rank in the sorted list. This is called a “pass”. No one shall “pass” for two consecutive classes. If you need help to prepare presentations, see me during office hours as soon as possible.

Portfolio

The portfolio is a curated collection of your work that tells a story of you as a mathematician and student of geometry from the first day of class to the last. The portfolio is due at the start of the final exam, but you should work on assembling this portfolio throughout the semester. You are welcome to visit me and ask for feedback. Here are the main parts of your portfolio.

• Growth: Describe how you’ve improved as a mathematician and student of geometry this semester. Focus on how you have changed or grown over the course of this semester. Write 2-3 paragraphs that describe your growth, both in terms of geometric knowledge and in terms of your understand of how mathematics works. Include at least three specific examples of work that illustrate this growth and explain why they show what you’ve described.
• Productive Failure: Show me a failure that ended up supporting and fostering your learning. Write a description of an important mistake or error that you made (in a presentation, a homework problem, or an exam). Clearly describe your error. Then describe how this mistake was productive – what you learned or how you grew as a result of this mistake. Use specific examples and include work to describe the growth. 2-3 paragraphs total.
• Process: Select a problem from this semester that required you to do a significant amount of scratch work. Include the scratch work and annotate it (using different colors) to indicate what insights or key ideas each piece represents. Be sure to annotate errors, dead-ends, ideas that didn’t work out. INclude your final proof that should be written as good as you can. Write a narrative (1-2 paragraphs) describing the process you underwent to get from reading the theorem to writing your finished proof.
• Assumptions and Axioms: Choose a proof from this semester that illustrates the importance and use of assumptions in mathematics. Annotate (using different colors) your work as follows: Point out where you make explicit assumptions and then also point out where you use them in the proof. Point out where you use implicit assumptions, such as axioms, that are not explicitly assumed in your writing. Be clear and be specific. Write a clear, thorough, and general explanation of the role of assumptions and axioms in mathematics. (2-3 paragraphs)
• Overall reflection: This component is a self-evaluation of your performance in this course. Write thorough but concise answers, 2 pages total. I am interested in your growth throughout the semester so please be honest. This part is graded for completion and thoughtful effort only. Here are some important criteria for evaluating your performance. Briefly answer each of these questions:
  • Daily Preparation and Homework. Did you complete these? On time? How much time and effort did you spend? Was your homework correct? Did you take extra time to think about difficult problems, or do the bare minimum?
  • Attendance, Participation, and Collaboration. Did you attend class regularly> Were you an active participant during the class? (Such as presenting, asking questions, helping your peers?) In what ways did you collaborate with others, both in and our of class?
  • Development of proof, presentation, and writing abilities: How did (or didn’t) your skills improve in discovering, presenting, and writing proofs? Please address all three separate things. What other skills have you developed, and how did you demonstrate them?
  • Are there other important criteria that I should use to evaluate your performance? Explain what they are and why they are useful criteria. Then state how well you met them.
  • Based on the above criteria, give yourself a grade for this course. Explain and be honest. I am very interested in your reasoning.
  • Fill in the blank with another student’s name: “If ___ doesn’t get an A, then nobody should.” Explain and I am very interested in your reasoning.
  • What was the most difficult topic in class for you? What was the easiest? Which topic surprised or interested you the most?

Evaluation

In this class, we will use a system known as standards-based grading. In order to achieve a certain letter, you have to meet all the criteria on that row. For example, if satisfies the criteria for daily preparation, presentation, weekly homework, and exams for A but only satisfies the criteria for porfolio for B, then your final grade will be B.

If you do not meet or exceed all of the criteria for a D, your grade will be an F. The +/- grades based on how close you are to the next higher or lower letter grade and similar criteria.

Academic Integrity and Collaboration

Your primary goal in this course is to develop a deep personal understanding and expertise in geometry. Collaboration and cooperation are extremely helpful in the learning process and encouraged in most circumstances! However you may only collaborate with students currently enrolled in the same section of the course. When collaboration has occurred, you must acknowledge this clearly (state the name(s) of the person(s) you collaborated with on each problem.) You are permitted to collaborate on big ideas and hints with classmates, however, you must work independently when writing up solutions. All collaborations should occur when your collaborator is at essentially the same stage of the problem solution as yourself. In particular, if you have not yet started a problem and you ask a friend who has already completed it, how do you do this problem. This counts as plagiarism. The resulting work is not and cannot be considered your own. Regarding to outside resources: all work (including daily, weekly etc.), unless directly stated otherwise, the only resources you may use are our course textbook and your class notes. You are not permitted to go looking for completed solutions to problems in other texts or resources. In particular, use of internet resources is completely off limits for homework problems. If you see a solution, there is no way that you can claim to have an original solution. Evidence of using internet sources in your work will result in a minimum penalty of failing the assignment. Copying a solution, or any part of a solution, from any source (friend, internet, book, etc.) in any setting, constitutes plagiarism. You may not seek the help of an instructor or tutor (other than me) unless you first discuss this with me in advance. If you do not verify that this is acceptable before seeking help, this will be considered plagiarism as well. I am always willing to discuss any aspect of the course with you. Evidence of dishonest behavior on any assignment will be grounds for a minimum penalty of failing the entire assignment. In severe cases, the minimum penalty will be failure of the course. Peers who willingly assist others in acts of plagiarism are equally guilty, and will suffer similar penalties. All academic dishonesty will be reported to the Academic Standards Committee. There might be additional sanctions by the Academic Standards Committee such as dismissal from the university. See Utica University official page for Academic Honesty for more details.

Special Accommodation

The stuff just below is the University approved language, and is a bit… “legalese:. The point is, if you need accommodations to succeed in this course, talk to me and we can make sure you get what you need. And the social environment of this course is important to me, too. Let’s work together to make a welcoming and affirming space for everyone.

Any student who has need of special accommodations in this class due to a documented disability should speak with me as soon as possible, preferably within the first two weeks of class. You should also contact Judy Borner, Director of Learning Services in the Academic Support Services Center (315-792-3032 or [email protected] ) in order to determine eligibility for services and to receive an accommodation letter. We will work with you to help you in your efforts to master the course content in an effective and appropriate way. See Utica University official page for Office of Learning Services.

Disclaimer

It is the students’ responsibility to keep informed of all announcements, syllabus adjustments, or policy changes during the semester via this web page or via school emails. The author of this syllabus reserves the right to change it with notice at any time during the semester.