CRN 11174
From Classes
Tiger gotta hunt. Bird gotta fly.
Man gotta sit and wonder why, why, why.
Tiger gotta sleep. Bird gotta land.
Man gotta tell himself he understand.
Kurt Vonnegut Jr.
Syllabus under Construction
- Topic. Introduction to Higher Mathematics.
- Time and Place. This course is completely online.
- Instructor. Helmut Knaust, hknaust@utep.edu
- Office Hours. I will offer frequent 30-minute virtual office hour slots on Zoom. Reserve a time here, one day in advance. Additionally, I hold physical office hours in Bell Hall 219 on Tuesdays and Thursdays at 13:30-15:00.
- Textbook and materials. Mount Holyoke College. Laboratories in Mathematical Experimentation. A Bridge to Higher Mathematics. The textbook is available for download here.
- Co-requisite. Calculus I (Math 1411).
- Course Description. An introduction to mathematical problem solving, experimentation, and proof writing, and the relationship among all three. The course will be built around a series of in-depth problems from a variety of areas of higher mathematics, especially those not encountered in pre-calculus and calculus courses.
- Course Objectives. This course is built on the proposition that you learn mathematics, and how to construct mathematical proofs, better when you formulate the questions and discover the answers yourself. Upon successful completion of the course, you will be able to investigate mathematical questions, big and small, both experimentally and theoretically. This is very different from courses like pre-calculus, calculus and differential equations, which are primarily focused on computations. Although there are computations in this course, they are a tool for discovering, and proving, more general mathematical truths.
- Laboratories. You will work exclusively in small teams on lab projects. You can meet with your teammates on MS Teams (MS Teams tab on the BlackBoard page for our class). Each lab will start with a video, giving a brief explanation of the question or problem to be explored. Your team will perform experiments with the computer software program Mathematica and gather data. The data will lead you to make your own conjectures, which you will then test and refine by further experimentation. Finally, when you are more certain of your conjectures, you will prove them carefully. (In practice, this process is rarely as straightforward and linear as outlined here. You will often revisit earlier steps as you carry out later steps.) After about two weeks of work on a project you will write up your discoveries, both experimental and theoretical, into a clearly-written report. (Grading criteria are below.) The reports are written jointly by the members of your group. After each report is graded and returned to you, you will have approximately one more week to revise your report for a better grade, if you like.
- Grades. Each lab will be graded based on the following criteria: (1) Experimental design, (2) Organization and presentation of data, (3) Analysis of data, (4) Statement of conjectures, and most importantly (5) Mathematical analysis (including proofs) of conjectures (see p. xvii of the text). The final grade for each lab will be the average of the grades you receive on your initial report, and on your revision. If you do not turn in a revision, it will simply be the grade of your initial report. Your grade for the course will be the average of the final grades for each of the labs. Deadlines for the various assignments can be found below on the calendar. A late submission of an assignment will result in a grade of zero.
- Mathematica. All of the projects will use the computer algebra system Mathematica. You need to request a license for "Mathematica Online" at https://www.utep.edu/technologysupport/ServiceCatalog/SOFTWARE_PAGES/soft_mathematica.html. Follow the instructions in Access Mathematica Online. Learning how to code is not required, but if you want to learn more about coding in Mathematica, a nice introduction to Mathematica can be found at An Elementary Introduction to the Wolfram Language, by Stephen Wolfram.
- Time Requirement. I expect that you spend an absolute minimum of nine hours a week on the projects. Not surprisingly, it has been my experience that there is a strong correlation between class grade and study time.
- Drop Policy. To be fair to your team members, please drop the course only immediately after at a project due date and notify your team and me. The class schedule lists Friday, November 3, as the last day to drop with an automatic "W". After the deadline, I can only drop you from the course with a grade of "F".
- Academic Integrity. All students must abide by UTEP's academic integrity policies. For detailed information visit the Office of Student Conduct and Conflict Resolution (OSCCR) website. Academic Integrity is a commitment to fundamental values. From these values flow principles of behavior that enable academic communities to translate ideals into action.” Specifically, these values are defined as follows:
- Honesty: advances the quest for truth and knowledge by requiring intellectual and personal honesty in learning, teaching, research, and service.
- Trust: fosters a climate of mutual trust, encourages the free exchange of ideas, and enables all to reach their highest potential.
- Fairness: establishes clear standards, practices, and procedures and expects fairness in the interaction of students, faculty, and administrators.
- Respect: recognizes the participatory nature of the learning process and honors and respects a wide range of opinions and ideas.
- Responsibility: upholds personal responsibility and depends upon action in the face of wrongdoing.
- Military Service. If you are a military student with the potential of being called to military service and/or training during the course of the semester, you are encouraged to contact the instructor as soon as possible.
- Counseling Center. You are encouraged to go to Counseling and Psychological Services (202 Union West) for personal assistance as you work through personal concerns. Confidential counseling services are offered in English or in Spanish.
- Disabilities. If you have a disability and need special accommodation, please contact the Center for Accommodations and Support Services (CASS). The Center aspires to provide students accommodations and support services to help them pursue their academic, graduation, and career goals. Phone 747-5148. E-mail: cass@utep.edu.
Project 6 (Chapter 8)
- 9. Prelude: The real numbers (17 min.)
- 10. p-adic norms (11 min.)
- Things to do: Work all the exercises and answer all the questions in the chapter.
- A recent popular science article about p-adics: Kelsey Houston-Edwards, An Infinite Universe of Number Systems, Quanta Magazine (10/19/2020)
- Reflection on Project 5
Project 5 (Chapter 14)
- Mathematica Notebook(s): 5QuadIter.nb | 501Compositions.nb | 502Repeller.nb
- 7. Introduction to Project 5 (30 minutes):
- Things to do: Answer all questions in Sections 14.1-14.3 and 14.5.
- 8. Two more Mathematica Notebooks (8 min.):
Project 4 (Chapter 11)
- Reflection on Project 4.
- A sample report from the past.
- Mathematica Notebook: 4SeqSer.nb
- 6. Introduction to Project 4 (15 min.) - Error: The last line I write in the video should be \(\sum_{n=1}^k \frac{1}{n}\approx\ln(k)\).
- Things to do: Read and answer all the questions and exercises in Sections 11.1-11.5. Do not do Section 11.6.
- I know that quite a few of you want to become teachers. The learning theory behind a class like ours was first articulated by the Soviet psychologist Lev Vygotsky and centers around the concept of Zone of Proximal Development. If the textbook does not suffice as MKO: your instructor is just a Zoom screen away.
- Reflection for Project 3
Project 3 (Chapter 9)
- Mathematica Notebook: 3Parametric.nb (There is now a little "reset" button at the upper right corner of each animation.)
- Trigonometric Identities
- How to Write Mathematics, by Martin Erickson
- 5. Introduction to Project 3 (14 min.):
- There are lots of definitions in the text. Make sure you understand all definitions.
- Things to do: Exercises 12-14, Questions 13-18, 9.5.3. (If you have the book: Exercise 2-4, Questions 1-6, Question 9.5.3.)
- Make lots of conjectures about symmetries, etc, and prove as many conjectures of yours as possible.
- I am part of a team that plans to issue a second edition of the textbook. If you encounter any errors or typos, or if there are parts in a chapter that should be rewritten for clarification, please let me know.
Project 2 (Chapter 3)
- Mathematica Notebook: 2Euclid.nb
- 3. Introduction to Project 2 (18 min.):
- Things to do:
- Explain how and why the EA works.
- Investigate Questions 1-6. What are your conjectures? Why are your conjectures true?
- (Skip Section 3.4.)
- Investigate the questions posed in Section 3.5: Are there GCD and EA for polynomials?
- 4. Speed test: EA vs. PF (<2 min., interesting, but not relevant for the project):
Project 1 (Chapter 1)
- I have assigned you to a team on Blackboard. Please contact your other team members via Blackboard and start working on the first project.
- 1. Introduction and Syllabus:
- 2. Intro to the Project and Mathematica (16 min.):
- Mathematica Notebook: 1Iteration.nb
- Things to do for Project 1: Answer all questions in the chapter. Questions 6 and 8 are central! Section 1.5 may help with understanding what is going on. Remember that answering "why" is always the most important thing in Mathematics.
- Here are guidelines for writing your project reports.
Calendar
Tuesday, September 12 | Project 1 (Chapter 1) due |
Thursday, September 28 | Project 2 due |
Thursday, October 5 | Project 1 revision due |
Thursday, October 12 | Project 3 due |
Tuesday, October 24 | Project 2 revision due |
Tuesday, October 31 | Project 4 due |
Tuesday, November 7 | Project 3 revision due |
Thursday, November 16 | Project 5 due |
Tuesday, December 5 | Project 4 revision due |
Tuesday, December 12 | Project 5 revision due |
Thursday, December 7 | Project 6 due (no revision) |