Prerequisites: Math 111; EHD 50 (may be enrolled concurrently; or permission of instructor).

Hours: 3.

Content: Topics to be covered include: brief review of logic, types of proofs, and principle of mathematical induction; Dirichlet's box principle (also called pigeonhole principle), number theory, case analysis, invariants, coloring proofs, working backwards, geometry, mathematical games and algorithms, non-routine calculus problems, problems from math olympiads and journals. Other topics may also be discussed. Students will be asked to present their solutions on the board. Working in groups is encouraged. The purpose of this class is to develop/improve problem solving skills, communicate your mathematical ideas, and to practice writing proofs in a correct and complete way.

Catalog description: A study of formulation of problems into mathematical form; analysis of methods of attack such as specialization, generalization, analogy, induction, recursion, etc. applied to a variety of non-routine problems. Topics will be handled through student presentation.

Course goals
Introduce students to a variety of problem solving techniques as well as challenging ("non-routine") problems accessible to 7-12 graders; practice communicating mathematical ideas verbally and symbolically in the problem solving setting.

Learning objectives
Upon completion of this course, students should know/understand:

• Importance of a formal (rigorous) proof
• Meaning of logical operations (and, or, negation, implication) and quantifiers (universality and existence)
• Difference between different types of proof (direct, by contradiction, by contrapositive, by example/counterexample, etc.)
• Principle of Mathematical Induction
• Dirichlet's Box Principle
• Divisibility properties of integer numbers; congruences
• The concept of invariant
• Basic graph theory terminology (vertex, edge, degree of a vertex, etc.)

Primary learning outcomes
Upon completion of this course, students will be able to do:

• Translate word problems into mathematical equations and relations
• Recognize problems which can be solved using one of the following techniques:
  • Dirichlet's Box Principle
  • Mathematical Induction
  • Divisibility and/or Congruences
  • Case Analysis
  • Patterns
  • Invariants
  • Coloring
  • Symmetry
  • Graphs
  • Working backwards
  • Equations

Grading policy: Your grade for the course will be based mostly on your performance on exams and homework (more information on these below). You will also receive points for submitting solutions of problems to jounals, class presentations, and reporting new mistakes in the book. The number of points awarded for these are shown below.

Exams: There will be three 50-minutes tests and one final comprehensive exam. Tests will be given in class. The final exam will be given during the final exam week. If for any reason you can not take a test at the scheduled time, please let me know as soon as possible, and certainly before the test. See schedule for exam dates. No books, notes, or calculators are allowed on any of the tests.

Class attendance is required. One "problem of the day" will be assigned for each class period. Students will be randomly called to present their ideas/solutions. These presentations are part of your grade. If you are called and you are absent, you will receive zero points for your presentation. (If you are sick or have another serious reason to miss a class, notify your instructor before the class to be missed; if you are unable to notify the instructor before class, please provide doctor's note, etc.) In addition to these presentations and new material/problems discussions, important course information will be given in class, sometimes handouts will be distributed.

There will be weekly homework due Wednesday at 2 pm, usually on the topic(s) of the preceding week. There will be 12 written homework assignments. Each homework will consist of 3-5 problems and will be worth 15 points total. Two lowest homework scores will be dropped. Homework problems will be new problems, sometimes somewhat similar to the problems discussed in class and sometimes quite different. You may have to spend a few hours on each homework (remember, the main purpose of this class is to learn to solve unfamiliar, non-routine problems). If you have any questions, or need someone to listen to and possibly comment on your ideas, please do not hesitate to ask your instructor (this is what her office hours are for!) Also, you are encouraged to work with your classmates. Please bring your homework to class. We will often discuss homework in class, so no late papers will be accepted unless you are sick or have another serious reason to miss the class (notify your instructor in advance, or provide doctor's note, etc.)

Participation in departmental problem solving activities is expected and counts toward your grade in the course. Below are the activities supported by the Mathematics Department. You are welcome to participate in any of these. Choose anything that interests you and fits your schedule.

• Journal Problems. During the semester, the instructor will bring a few journals with problems for you to try to solve. When you have a solution, write it down and show to the instructor. Depending on correctness/completeness of your solution and difficulty of the problem, you will get at least 10 points (possibly more) for your solution. If it is correct and complete, you should send your solution to the journal (and your name and/or solution may get published!) You can also present your solution in class. You will get 5 points for sending your solution to the journal, and 5 additional points for your presentation in class.
• Problem Solving Playground. Department faculty hold problem solving sessions, tentatively on Fridays from 1-2, starting September 7. Update: tentatively every other Friday starting September 14 from 1-2. Also PSP will meet every other Friday starting September 21 from 2-3, however, that conflicts with our class schedule. One of the goals of the PSP is to practice for the Putnam Exam (see below) as well as the Fresno State problem solving contest (also see below). You will get 5 points for attending each session.
• Fresno Math Circle. This is a math enrichment program for children in grades 3-12. We are focusing on problem solving and deep understanding of mathematics concepts. You are invited to participate/help run the sessions, which are held every other Saturday starting September 8 from 10-11:30 AM for grades 3-8 and every other Thursday starting September 6 for grades 9-12. No prior experience is necessary. Contact your instructor if you would like to participate/help. Your choices are:
  • Atttend and participate (as a student) in Thursday sessions for high school grades. You will get 10 points for each session.
  • Help with Saturday sessions for grades 5-6 (lead by your instructor). To prepare for each session, we will meet for about an hour earlier in the week. You will get 15 points for each session.
• Problem Solving Contest. The contest is usually held the week of the DMD (in November), and is open to all Fresno State students. One good way to practice is to attend the Problem Solving Playground meetings (see below). You will get 10 points for participating in this contest, and additional points if you place in the top 10 (10pts for 1st place, 9pts for 2nd place, and so on).
• Putnam Exam. This is actually not an exam, but an annual nation-wide competition in problem solving for undergraduate students. All undergraduate students enrolled at Fresno State are welcome to participate. It is held on the first Saturday in December, however, advance registration is required. The registration date is usually in October (the exact date will be announced as soon as it becomes available). One good way to practice is to attend the Problem Solving Playground meetings (see below). You will get 20 points for participating in this competition.
• Putnam Prep Sessions. In the past, department faculty have held putnam prep (problem solving) sessions. Problems from previous Putnam Exams, as well as possibly other problems, may be discussed. If such sessions are held again this year, they will be announced as information becomes available. You will get at least 5 points for attending each session (if they are 1 hour long; 10 pts if 2 hours long).
• Department of Mathematics Day. This annual department event is held on November 16 this year, the last Friday before Thanksgiving. You may choose to give a talk, e.g. about your solution to a journal problem, or participate in the Math Jeopardy game. The schedule is given in the flyer. You will get points for your participation in the above activities: at least 10 points for the talk if it is related to problem solving (possibly more depending on the topic and time spent), 10 points for participation in Math Jeopardy.
• If you earn more than 40 points participating in the above activities, you will receive extra credit (up to another 40 points).

Extra Help: It is essential not to fall behind, because most classes will use the material studied previously. If you have trouble with some material, seek help in the following ways:

• Ask me, either in class or privately. My contact info is given on the course web page. Don't be shy to ask questions. If you don't understand something, chances are very high that somebody else doesn't understand that either. In particular, you are more than welcome to ask questions about solutions in the book. But I can not tell you how to do the homework problems. Those are for you to solve. Remember, the purpose of this class is for you to learn to solve non-routine problems. However, if you'd like to know a solution to a homework problem after it's due, let me know, and I'll be very glad to discuss the problem.
• Work with your classmates. Note: working on your homework together is allowed and even encouraged, however, every student should write down their solutions on their own. The best thing to do, after you have worked with other people, is to put aside all the notes you made, and write a complete solution from scratch, in your own words. Also, if you have worked with someone, please indicate that on your paper.

Computers: At California State University, Fresno, computers and communications links to remote resources are recognized as being integral to the education and research experience. Every student is required to have his/her own computer or have other personal access to a workstation (including a modem and a printer) with all the recommended software. The minimum and recommended standards for the workstations and software, which may vary by academic major, are updated periodically and are available from Information Technology Services or the Universit Bookstore. In the curriculum and class assignments, students are presumed to have 24-hour access to a computer workstation and the necessary communication links to the University's information resources.

Students with disabilities: upon identifying themselves to the University, students with disabilities will receive necessary accommodation for learning and evaluation. For more information, see http://www.fresnostate.edu/studentaffairs/ssd/

Academic honesty: cheating and plagiarism will not be tolerated in this class. For information on the University's policy, see the University Catalog (section Policies and Regulations).

Honor Code: Members of the CSU Fresno academic community adhere to principles of academic integrity and mutual respect while engaged in university work and related activities. You should:

• understand or seek clarification about expectations for academic integrity in this course (including no cheating, plagiarism and inappropriate collaboration)
• neither give nor receive unauthorized aid on examinations or other course work that is used by the instructor as the basis of grading.
• take responsibility to monitor academic dishonesty in any form and to report it to the instructor or other appropriate official for action.

Disruptive Classroom Behavior: student conduct which disrupts the learning process will not be tolerated and may lead to disciplinary action and/or removal from class.

Copyright policy: Copyright laws and fair use policies protect the rights of those who have produced the material. The copy in this course has been provided for private study, scholarship, or research. Other uses may require permission from the copyright holder. The user of this work is responsible for adhering to copyright law of the U.S. (Title 17, U.S. Code). Digital Campus course web sites contains material protected by copyrights held by the instructor, other individuals or institutions. Such material is used for educational purposes in accord with copyright law and/or with permission given by the owners of the original material. You may download one copy of the materials on any single computer for non-commercial, personal, or educational purposes only, provided that you (1) do not modify it, (2) use it only for the duration of this course, and (3) include both this notice and any copyright notice originally included with the material. Beyond this use, no material from the course web site may be copied, reproduced, re-published, uploaded, posted, transmitted, or distributed in any way without the permission of the original copyright holder. The instructor assumes no responsibility for individuals who improperly use copyrighted material placed on the web site.

Subject to Change: This syllabus and schedule are subject to change in the event of extenuating circumstances. If you are absent from class, it is your responsibility to check on announcements made while you were absent.