MAT 240: Methods of Problem SolvingFall 2008 Syllabus

Professor: Robert Talbert, Ph.D.Office hours: Old Main 128 MF 11:00-12:00, MTRF 1:30-2:30 and by open-door drop-in, appointment, or instant messenger.Voice: 317.738.8268Email: rtalbert@franklincollege.eduAOL instant messenger: rtalbert235Google Talk instant messenger: robert.talbert

Course Materials• Textbook: T. Sundstrom, Mathematical Reasoning: Writing and Proof, 2nd ed., Prentice-Hall. • Course website: http://mat240.wikispaces.com (Note: This is different and separate from the course

Angel site.) • Additionally, students should have 24/7 access to a computer for class work involving Maple, web

access, and LaTeX typesetting.

Catalog Course Description An overview of various methods of problem solving to discover patterns, construct and modify conjectures, and develop proofs of those conjectures in topics including number theory, algebra of complex numbers and set theory. Proof by induction will be discussed. Incorporates use of computer algebra systems. Prerequisite: MAT 142 (Calculus II).

Informal Course Description and Course GoalsMethods of Problem Solving is a course about how mathematics is discovered, developed, and communicated. It is intended to help students make the transition from calculus to higher-level mathematics courses centering on difficult problems, often problems requiring proofs, as opposed to simple exercises as you saw in calculus. A particular emphasis is placed on constructing and writing proofs for mathematical conjectures.

The successful student in MAT 240 will be able to do the following: • Employ structured problem-solving strategies to understand a problem; plan out a reasonable solution

based on reconnaissance problems and experimentation; execute a solution plan with fluency in the necessary tools; and examine a solution critically to weight its correctness and soundness.

• Develop fluency in the mathematical content treated in the course (from the areas of calculus, logic, set theory, number theory, and complex number arithmetic).

• Construct examples, counterexamples, and reconnaissance solutions, employing appropriate technological tools when helpful.

• Construct complete, clear, and correct solutions to problems, including both problems to find and problems to prove.

• Refine oneʼs work based on feedback from the professor, from classmates, or from individual experimentation.

• Create a professional document displaying solutions to problems using the LaTeX typesetting platform. • Perform all of the above tasks both as an individual and as a member of a working group.

What to ExpectYou cannot learn problem solving except by doing problems. This class, accordingly, will be very active on a daily basis, and it will depend on each student to prepare and think well for each class and participate actively in each class in order to get the most out of the course. The professorʼs role in MAT 240 will not be that of a lecturer but rather a course manager (setting up assignments and grading work) and a discussion coordinator. The studentsʼ job is to give a diligent, good-faith effort to prepare well for each

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class and then bring their thoughts and ideas -- even partial ones -- to class every day in order to motivate a stimulating discussion period. This involves: • Reading the textbook carefully and working the Progress Checks as you go; • Giving a serious, significant effort to working each of the problems assigned for discussion; • Keeping well-organized records of your notes, questions, graded work, and grades; • Seeking help from the professor as needed; and• Staying on top of work that is due in the future.

Especially important is this rule: DO NOT FALL BEHIND, AND DO NOT PROCRASTINATE. If you do either, even occasionally, you will find that it is extremely hard to catch up. The kind of work we do in MAT 240 is not the kind that can be done in a single hour just before it is due. It takes time, repeated and persistent effort, and patience (and a healthy sense of humor). Falling behind or procrastinating will spell almost certain failure, even if done early in the course. On the other hand, putting forth a consistent effort to do well and stay up to speed in the class will pave the way for a successful and enjoyable future in your math courses, as all such courses from here on will use MAT 240 as a prerequisite and touchstone.

Making the transition from lower- to upper-level mathematics is difficult and often frustrating. You should take encouragement from the fact that most class discussions are formative in nature, meaning that we all expect mistakes to be made and incorrect ideas to be floated. This is part of the learning process, and even professional mathematicians usually start a new line of research with mostly bad or wrong ideas. The goal of class meetings is to hammer out correctness from the raw material you bring. The more raw material you bring, and the more honest the hammering-out process is, the more you will learn and the better you will do on summative assessments (e.g. midterm and final exam) where you are expected to do things correctly the first time.

It is appropriate, given the difficulty level of the class, that the professor should extend himself proportionately to the same degree you are being ask to extend yourselves. I hold regular office hours and maintain an open-door policy for unannounced drop-ins; I also make myself available through instant messaging. Please do not hesitate to call on me for help; itʼs my job.

Assessments and GradingYour grade in the course will be determined by the following items of work: • Preview Activities (10%). Each day when a new section is covered, students will be responsible for

completing the Preview Activities in the textbook for that section. The Preview Activities consist of 2-3 multistep exploratory problems that are designed to stimulate your thought about a new topic and prepare you for further work on the new topic. Students may do Preview Activities individually or in groups of 2 or 3.

• Participation (10%). Students will receive 4 points per class for showing up on time and providing thoughtful (not necessarily correct) responses to all questions posed to him or her. Students will also rotate through being the “scribe of the day”, which involves writing up course notes and solutions to problems and posting them to the course web site for a grade.

• Written Assignments (15%). There will be five written assignments given during the semester consisting of discussion problems we do not complete and other supplementary problems.

• Professional Development (5%). Each student will have the opportunity to participate in various outside-of-class activities having to do with mathematics or math-oriented professions. A list of Professional Development opportunities will be maintained on the course website. Most activities will be worth 5-10 points; students must accumulate at least 50 points throughout the semester.

• Midterm Exam (20%). This exam will be given in class on Thursday, October 9. • Problem Portfolio (20%). This major assignment will consist of individual students working through a

collection of 10 problems during the entire semester, writing them up professionally, and submitting them as a portfolio. More information on Problem Portfolios will be given in a separate handout.

• Final Exam (20%). A comprehensive final exam will be given on Wednesday, December 10 from 1:30--3:30.

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Your semester grade will be determined using the following table: Grade Percentage Range Grade Percentage Range

A 93-100 C 73-76A- 90-92 C- 70-72B+ 87-89 D+ 67-69B 83-86 D 63-66B- 80-82 D- 60-62C+ 77-79 F 0-59

Keep in mind that most students in MAT 240 need a C- or higher to “pass”.

Course Policies

Academic Honesty. All work that is submitted by a student must be the work of that student alone. Submission of work that properly belongs to someone else constitutes plagiarism and is heavily punished by Franklin College. Please see the notes on academic honesty which are appended to this syllabus for more details.

Attendance. The effectiveness of this course depends upon your preparation, attendance, and participation in the class meetings. Each student is expected to attend class every day and participate in an active, well-prepared discussion. You will not receive participation credit for any day that you miss. Absence on the day of the midterm exam (October 9) or final exam (December 10) must be accompanied by a documented excuse, signed in ink by an adult in charge of the situation (doctor, police officer, etc.) and submitted to the professor within 24 hours of the midterm in order to qualify for a makeup. Otherwise a grade of “0” will be given.

Late Work. Late submissions of preview activities and written assignments will not be accepted. This includes handing in preview activities late because a student was late to class. If a student knows in advance that he or she will be missing a class, all work that is due for that day must be submitted in advance (email is good for this) or by proxy (= giving it to another student to hand in).

Students with Learning Disabilities. Students with documented learning disabilities are eligible for alternate exam environments, including extended times and alternate locations. Please see the professor as soon as possible to arrange such accommodations if you are eligible.

Technology. It is assumed that students in MAT 240 have basic proficiency with the operation of a personal computer and with the resources on the campus network. Technological difficulties will not be considered valid excuses for late work. For example, failing to hand in a written assignment because “the printer wonʼt work” will result in a grade of 0; the student should instead email the assignment as an attachment to the professor or hand in the writeup on a flash drive. It is assumed as well that you will back up your work to multiple locations besides your personal computer (e.g. your G: drive, a flash drive, as an email attachment to yourself, using a web-based storage service such as box.net, etc.) in the event of a catastrophic computer failure such as a hard drive crash. Students will be responsible for checking their Franklin College email and the course web site at least once per day for announcements and assignments.

Writing. A key element of MAT 240 is effective communication, particularly technical communication and especially as regards mathematical proofs. Even if you never write a proof in your post-college career, you will be called upon to argue logically for or against an idea and to communicate your thoughts (or the thoughts of your employer) with clarity and precision. A large portion of your grades on all the

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assessments in MAT 240 will be based on the quality of your writing, which in mathematics also includes the correct use of mathematical notation and terminology. Therefore it is implicit in every exercise or problem you work that you must give a complete, correct, and clear explanation of your answer and not just give the answer itself. (For many problems, the “answer” is itself the explanation.) Students are expected to use English correctly, including correct spelling and grammar, and to format their mathematics in a professional way. There are hints in the textbook for how to write mathematics, and you will be expected as part of your nightly reading to absorb and implement these hints.

Academic Honesty in MAT 240 and at Franklin College

One of the primary, if informal, goals of MAT 240 is to get you to “think like a mathematician”. The overall goal of the course is to develop your problem-solving, proof-construction, and communication skills to the point where you are a confident problem-solver in any context and a fluent lifelong learner of mathematics, having followed your own path toward understanding and appreciating this amazing subject.

As such, all of the work that you complete as part of the requirements for this course must be your own work, or the result of an honest and equitable collaboration among the members of your working group. When I grade your work, I am looking to see your own personal development in the understanding of the material. I must be able to trust that the work that you are handing in reflects this development and understanding accurately, even -- especially -- if there are problems or errors in it. I have no interest in your merely emulating the work of one of your classmates, copying or even paraphrasing work from a web site or textbook, or in any way otherwise passing off someone elseʼs work as your own.

Plagiarism is the term usually given to define the act of handing in work as if it were your own, when in fact it is not. Academic dishonesty is a broaded term that encompasses plagiarism as well as other actions such as using unauthorized implements on a timed exam. Academic dishonesty is so named, and plagiarism is included under its heading, because academically dishonest behavior is intended to mislead the professor into thinking that your work is an accurate reflection of your learning.

To be clear: Academic dishonesty is not a “youthful indiscretion” or something that can be rationalized away because of the stresses of college life or because so many get away with it. It is a deliberate, conscious choice on the part of the student to mislead his or her professor, and it demolishes the mutual trust upon which all of education is predicated. If you plagiarize or commit academic dishonesty, it is not just the one instance that I cannot trust; your entire body of work (past, present, and future, and not just for MAT 240 but for all your college career) becomes untrustworthy. And it is supremely unfair to the students who are struggling but doing so honestly.

The penalties for academic dishonesty in any form are appropriately severe at Franklin College. If a professor suspects academic dishonesty on an assignment, the professor is required to investigate it. (Note: This is not a choice on the professorʼs part but a job-related obligation according to the Faculty Handbook of Franklin College.) If the professor, in his or her professional opinion, finds that academic dishonesty was committed, each student involved receives a grade of “0” on the assignment, and each studentʼs letter grade in the course is lowered by one full letter, after the “0” has been factored in. That is the penalty for the first offense in the studentʼs career at Franklin College. If it is the studentʼs second offense -- or if the student commits a second offense later -- the student is expelled.

While professors do have some leeway in recommending alternative punishments for academic dishonesty, it is my personal policy not to do so, but rather recommend the full force of the penalty in all situations -- whether the assignment in question was a final exam or a 5-point reading assignment.

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Given the severity of academic dishonesty and its punishment, it is appropriate to lay out precise guidelines for academic honesty in MAT 240 in various cases.

• On Written Assignments, every sentence that you write should be one that you have generated yourself and that you understand. You are permitted to collaborate with other classmates on overall strategies for solutions and on big ideas and hints. But you must be working alone when you write your solutions. Additionally, all collaboration with other students on Written Assignments must occur with students who are currently at the same stage of the solution as you. For example, if you are making no progress on a solution and find a classmate who had finished the problem, and then get help on how to do the problem, that is considered plagiarism (collaborating with someone not at the same level of progress as you). If you are making no progress on a problem and get together with 2-3 classmates who have also made no progress to brainstorm big ideas for the solution, then this is acceptable collaboration. However, if one of those students in your brainstorming group comes up with the correct idea for the solution, and you simply write down their work without working out the details for yourself and without real understanding, then thatʼs plagiarism (using someone elseʼs work as your own).

• Also on Written Assignments, the primary resource you should use is the course textbook and your notes (and the notes that are on the course web site via the daily scribes). However, you may find it helpful sometimes to look up additional reference material in other books (such as your calculus book). If you use such information in a significant way for your solution, you must attribute it properly using the title, author, and page numbers of the resource you used. However: It is plagiarism to use other books or other mateirals to get completed solutions or significant parts of completed solutions; this is using someone elseʼs work as your own.

• Finally for Written Assignments, no contact whatsoever is allowed with past students from MAT 240. Those students have attained a level of expertise that places them automatically at a higher level of performance than students who are already in the class.

• On Preview Activities, students may work individually or in groups of 2 or 3. Students working individually are to operate under the academic honesty guidelines for Written Assignments -- keeping in mind that all you need to do is give a significant, serious good-faith effort to get a perfect score! If students work in groups, then collaboration may happen freely within the group (also keeping in mind that individuals need to understand the reading for participation opportunities), but groups may not interact with other groups outside the guidelines above.

• The academic honesty requirements for the Problem Portfolio are significantly more restrictive: Here, you will not be allowed any collaboration with any person, book, web site, etc. other than the professor. More information on the Problem Portfolios in general will be given to you in a later document.

• For all other assessments -- participation, professional development, midterm, and final -- student work is such that plagiarism is not really an issue. However, obviously exams will be closely monitored so that only authorized implements are being used.

The easiest route to take in order to avoid issues with academic dishonesty is just simply to recognize and avoid the temptation to engage in it. It is much better to turn in work that has problems but honestly reflects your best efforts than to turn in something that, for all practical purposes, lies to the professor about you. You might lose points in the short term, but you will learn better, perform better, and enjoy your mathematical future better if you stay honest.

PS: In order to “walk the walk” here, I should mention that portions of this document were adapted from Ted Sundstromʼs syllabus for this course, which is available at his web site at Grand Valley State University.

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MAT 240 Course Calendar

M T R

8/26/2008Problems vs. exercises; four stages of problem solving.

8/28/2008Calculus review part 1: Problems to find. Preview activities: Calculus exercises pt. 1.

9/1/2008Labor Day; no class.

9/2/2008Calculus review part 2: Problems to prove.Preview activities: Calculus exercises pt 2.

9/4/2008Conditional statements (1.1). Preview activities: 1.1 (pp. 1--3).

9/8/2008Constructing direct proofs (1.2). Preview activities: 1.2 (pp. 13--15).

9/9/2008Statements and logical operators (2.1). Preview activities: 2.1 (pp. 29--30).

9/11/2008Logically equivalent statements (2.2).Preview activities: 2.2 (pp. 37--39).

9/15/2008Predicates, sets, and quantifiers (2.3). Preview activities: 2.3 (pp. 47--48)

9/16/2008Quantifiers and negations (2.4). Preview activities: 2.4 (pp. 58--60).

9/18/2008Finish 2.4. Preview activities: None.

9/22/2008Direct proofs (3.1).Preview activities: 3.1 (pp. 76--78).

9/23/2008Finish 3.1. Preview activities: None.

9/25/2008More methods of proof (3.2). Preview activities: 3.2 (pp. 93--95).

9/29/2008Finish 3.2; Proof by contradiction (3.3). Preview activities: 3.3 (pp. 107--109).

9/30/2008Finish 3.3. Preview activities: None.

10/2/2008Using cases in proofs (3.4). Preview activities: 3.4 (p. 120).

10/6/2008The Division Algorithm and congruence (3.5). Preview activities: 3.5 (pp. 128-129).

10/7/2008Review for Midterm.

10/9/2008Midterm Exam. (Problem solving; calculus; Sundstrom 1-3.)

10/13/2008Operations on sets (4.1). Preview activities: 4.1 (pp. 151--153).

10/14/2008Proving set relationships (4.2). Preview activities: 4.2 (pp. 168--169).

10/16/2008Fall Break; no class.

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M T R

10/20/2008Finish 4.2. Preview activities: None.

10/21/2008Properties of set operations (4.3). Preview activities: 4.3 (pp. 180--182).

10/23/2008Cartesian products (4.4). Preview activities: 4.4 (pp. 190--191).

10/27/2008The Principle of Mathematical Induction (5.1). Preview activities: 5.1 (pp. 218--220).

10/28/2008Finish 5.1. Preview activities: None.

10/30/2008Introduction to functions (6.1). Preview activities: 6.1 (pp. 264--266).

11/3/2008Finish 6.1; More about functions (6.2). Preview activities: 6.2 (pp. 278--280).

11/4/2008Finish 6.2. Preview activities: None.

11/6/2008Injections, surjections, and bijections (6.3). Preview activities: 6.3 (pp. 290--292).

11/10/2008Finish 6.3. Preview activities: None.

11/11/2008Composition of functions (6.4). Preview activities: 6.4 (pp. 306--308).

11/13/2008Inverse functions (6.5). Preview activities: 6.5 (317--319).

11/17/2008Relations (7.1). Preview activities: 7.1 (pp. 349--351).

11/18/2008Equivalence relations (7.2) Preview activities: 7.2 (pp. 359-362).

11/20/2008Equivalence classes (7.3). Preview activities: 7.3 (pp. 372--373).

11/24/2008Modular arithmetic (7.4). Preview activities: 7.4 (pp. 384--385).

11/25/2008Finish 7.4. Preview activities: None.

11/27/2008Thanksgiving Break; no class.

12/1/2008Arithmetic and representation of complex numbers (handout). Preview activities: Complex number arithmetic and plotting handout.

12/2/2008Polar form of complex numbers and exponentiation (handout). Preview activities: Polar coordinates and exponentiation handout.

12/4/2008Review for final exam.

Final Exam: Wednesday, December 10, 1:30--3:30 PM

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