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.
- Topic. Introduction to Higher Mathematics.
- Time and Place. TR 15:00-16:20 in LART 210
- Instructor. Helmut Knaust, Bell Hall 219, firstname.lastname@example.org, 747-7002
- Office Hours. TR 12:00-13:20, or by appointment.
- for download here. Please bring a laptop to all class meetings.
- 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. Each lab will start with a brief explanation of the question or problem to be explored. You will perform experiments (usually with a computer or programmable calculator) 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.) You will work in small groups in class (as well as outside of class). There will also be whole-class discussions about your experimental and theoretical discoveries. After about two weeks of work on a project your team 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, your team 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 will be announced in class. A late submission of an assignment will result in a grade of zero.
- Mathematica. Almost all projects will use the computer algebra system Mathematica. You need to request a home license (see https://www.utep.edu/science/math/mathematica/). 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 six hours a week outside of class. Not surprisingly, it has been my experience that there is a strong correlation between class grade and study time.
- Attendance. You are strongly encouraged to attend every class meeting. Students with six absences (excused or unexcused) will be dropped from the course with a grade of "F".
- Drop Policy. The class schedule lists Friday, October 28, as the last day to drop with an automatic "W". Please only drop after an assignment has been completed. 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 747-948. E-mail: email@example.com.
Project 1 = Chapter 1 Day 1
|August 25||Project 1 Day 2|
|August 30||Project 1 Day 3||September 1||Project 1 Day 4|
|September 6||Project 1 Day 5||September 8||Project 1 due|
Project 2 = Chapter 3 Day 1
|September 13||Project 2 Day 2||September 15||Project 2 Day 3|
|September 20||Project 2 Day 4||September 22||Project 2 due|
Project 3 = Chapter 9 Day 1
|September 27||Project 3 Day 2||September 29||Project 1 revision due|
Project 3 Day 3
|October 4||Project 3 Day 4||October 6||Project 3 Day 5|
|October 11||Project 3 due|
Project 4 Day 1
|October 13||Project 4 Day 2|
|October 18||Project 2 revision due|
Project 4 Day 3
|October 20||Project 4 Day 4|
|October 25||Project 4 Day 5||October 27||Project 4 due|
Project 5 Day 1
|November 1||Project 5 Day 2||November 3||Project 3 revision due|
Project 5 Day 3
|November 8||Project 5 Day 4||November 10||Project 5 Day 5|
|November 15||Project 5 due|
Project 6 Day 1
|November 17||Project 6 Day 2|
|November 22||Project 4 revision due|
Project 6 Day 3
|November 29||Project 6 Day 4||December 1||Project 6 Day 5|
|December 6||Project 6 due|
Project 5 revision due
(no class meeting)
Projects and Mathematica Notebooks
Project 6 - Chapter 8
- Mathematica Notebook: 6Padic.nb
- 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)
Project 5 - Chapter 14
- Mathematica Notebook(s): 5QuadIter.nb | 501Compositions.nb | 502Repeller.nb
- Things to do: Answer all questions in Sections 14.1 - 14.3 and 14.5.
Project 4 - Chapter 11
- Mathematica Notebook: 4SeqSer.nb
- Things to do: Read and answer all the questions and exercises in Sections 11.1-11.5. Do not do Section 11.6. Don't forget Question 4.
- I know that many 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. Remember I am trying to be your MKO! You may also want to check out the home page of the Academy for Inquiry Based Learning.
Project 3 - Chapter 9
- 3Parametric.nb | 301CurveSalad.nb | 302ProofwithoutWords.nb
- 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.
- You may need Trigonometric Identities
- How to Write Mathematics, by Martin Erickson
- Wheels on Wheels on Wheels, by Frank A. Farris, Mathematics Magazine 96(3), June 1996.
Project 2 - Chapter 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?
- Speed test: EA vs. PF (<2 min., interesting, but not really relevant):
Project 1 - Chapter 1
- 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.