Math 265 – Mathematical Problem Solving
Fall 2005: 4 credits, MTWF, 8:00-8:50, MC 419
Instructor: Rich Maresh, Associate Professor of Mathematics, Dept. Chair
Office: MC 521, 796-3655, Hours: MWR 12-1, R 1-2 (Home: 526-4988)
Final Exam: Tuesday, 13 Dec 2005, 7:40-9:40 am
COURSE DESCRIPTION: This course will focus on a variety of techniques for solving mathematical problems. It will also take a look at pedagogical issues involved in teaching problem-solving. A number of “classic” mathematical problems will be considered. This course is intended for education majors with a minor in, or particular interest in, mathematics. Prerequisite: grade of C or higher in MATH 110 or equivalent.
TEXTS: (1) Problem Solving Through Recreational Mathematics (Averbach & Chein, Dover Publications, 2000; originally published in 1980, Freeman and Company)
(2) Problem-Solving Strategies for Efficient and Elegant Solutions (Posamentier & Krulik, Corwin Press, 1998)
(3) Problem Solving (Smith, Brooks/Cole, 1991)
Student Goals and Outcomes:
1. Students will become familiar with the literature on the use of problem solving in the mathematics classroom.
(a) Students will read the NCTM (National Council of Mathematics) documents regarding the use of problem solving in mathematics education.
(b) Students will locate and read two articles on the use of problem solving in mathematics teaching.
(c) Students will write and present to the class a “position paper” that makes evident their own philosophy of the role of problem solving in mathematics teaching.
2. Students will develop their own problem solving skills.
(a) Students will explore in a methodical way (the Posamentier text) a wide variety of problem solving strategies.
(b) Students will solve a variety of problems and will engage in class discussion regarding the process.
(c) Students will improve their ability to orally present their solutions.
(d) Students will improve their ability to present their solutions in written form.
3. Students will explore ways to integrate problem solving into their own mathematics teaching.
(a) Students will practice integrating problem solving into their lessons.
(b) Students will demonstrate that they understand the difference between problem solving and rote learning (or drill and practice) in mathematics teaching.
(c) Students will learn to create a grading rubric and demonstrate that they can use it.
This course is aimed at elementary education majors who have declared a minor in mathematics. The Wisconsin DPI requires Elementary Education Mathematics minors to take a course in Problem Solving, so in that sense we are satisfying a specific requirement, but let us hope there is also worthwhile work to be done here. In one sense, it matters little what specific types of problems we solve – it is the practice which is important.
Journals: I don’t want to make written journals a burden, but I do think they give me a good way to keep in touch with what’s happening to each of you as you interact with the material. As a compromise, I’d like to ask you to turn in (email is fine, maybe better) a self-reflective journal entry around the middle Monday of each month: Sep 12, Oct 17, Nov 14, and Dec 5. These should be discussions of how things are going for you in the course, what breakthroughs you may have achieved, and about any concerns you might have. They are worth 5 points each.
Blackboard: I will be using “Blackboard” to at least some extent. It’s not entirely necessary, since we will be meeting 4 times per week and thus have ample opportunity to communicate in person, but Blackboard does give me a nice tool for giving you Internet links or for you to turn in written assignments, such as your journal entries.
Homework: It is always important that you do the assigned homework, but in particular when the focus is on solving problems it is absolutely necessary that you try problems so that we will have something to discuss in class. Working with classmates on assignments is a delicate matter – sometimes it’s appropriate and helpful and sometimes it’s more or less “cheating”. Probably the main point is that you need to make sure you are learning, that you are understanding the material yourself.
Problem Sets: You will be frequently turning in assignments – it is important that I see your work on a regular basis and that you get feedback on your progress in the course. There will also be one problem each week, a “Problem of the Week”, that you should take seriously enough to carefully write up your solution, along with some discussion of the process you went through to solve the problem.Level of Difficulty: You are all elementary education majors and so you see yourself teaching mathematics to children in the future, clearly a laudable goal. But you yourself are not in the fourth grade, so the problems you need to work on have to be at an appropriate level for YOU, not for your future students. It is not always easy to pick problems at the right level of difficulty, and I’m sure we will have an occasional bump on the road, but I hope you can understand that for you to improve your own problem solving skills, the problems need to be challenging at your level.
Problem solving can be fun, but if a problem is either too easy or too hard it isn’t fun and nothing is gained from doing it. I enjoy doing crossword puzzles, but the one in our local paper is almost too easy – it usually takes me only about 5 minutes – while others can be so difficult (all about ancient Persian coins!) that there is no way to do it. As a teacher you yourself will have to deal with this issue, the appropriate level of challenge. Of course, it is even more complicated than that, since your students can vary so much in their skill level.
Exams: There will be two exams during the course, one basically a mid-term, given just before our “break”, and the second the Final Exam. These will include some take-home problems, but will also include an in-class exam – I want to see what you can do on your own, with you feet held to the fire, as it were.
Portfolio: I am going to ask you to compile a portfolio of your work. It will contain two things: (1) your solutions to FIVE problems which you think best demonstrate how you met the challenges of the course, and (2) a collection of at least 5 websites which contain problem resources for teachers, and at least one specific problems you found on each site. I want this portfolio to both show your work as a problem-solver and also to specifically indicate how you will use the internet as a resource when you are in your own classroom. This portfolio will come due at the final exam, but I will ask you to report to the class on your findings regarding the internet resources during the last couple days of class.
Americans with Disabilities Act: If you are a person with a disability and require any auxiliary aids, services, or other accommodations for this class, please see me and/or Wayne Wojciechowski, the campus ADA coordinator (MC 335, 796-3085), within ten days to discuss your needs.
If problem solving is treated as “apply the procedure”, then the students try to follow the rules in subsequent problems. If you teach problem solving as an approach, where you must think and can apply anything that works, then students are likely to be less rigid. (Suydam, 1987)
All too often we focus on a narrow collection of well-defined tasks and train students to execute those tasks in a routine, if not algorithmic fashion. Then we test the students on tasks that are very close to the ones they have been taught. If they succeed on those problems, we and they congratulate each other on the fact that they have learned some powerful mathematical techniques. In fact, they may be able to use such techniques mechanically while lacking some rudimentary thinking skills. To allow them, and ourselves, to believe that they “understand” the mathematics is deceptive and fraudulent. (Schoenfeld, 1988)
A teacher of mathematics has a great opportunity. If she fills her allotted time with drilling her students in routine operations she kills their interest, hampers their intellectual development, and misuses her opportunity. But if she challenges the curiosity of her students by setting them problems proportionate to their knowledge, and helps them to solve their problems with stimulating questions, she may give them a taste for, and some means of, independent thinking. (Polya, 1973)
The primary goal of most students in mathematics classes is to see an algorithm that will give them the answer quickly. Students and parents struggle with the idea that math class can and should involve exploration, conjecturing, and thinking. When students struggle with a problem, parents often accuse them of not paying attention in class; “surely the teacher showed you how to work the problem!” How can parents, students, colleagues, and the public become more informed regarding genuine problem solving? How can I as a mathematics teacher help students and their parents understand what real mathematics learning is all about? (Hadaway)
As a teacher, what I hate hearing from a student is, “We never did anything like this problem before!” As these previous citations state, real learning comes from doing problems that challenge a student to think, not from just doing drill and practice. But sometimes it feels like I am swimming upstream – students and parents so often see learning mathematics as practicing the mechanics, when I want to see it as learning to think and solve problems. (Maresh, 2005)
MATH 265 Fall 2005 Schedule
Aug 29-Sep 2 Introduction, NCTM Documents: the “Standards” – review, reaction paper
Sep 7-9 (Mon = Labor Day) [Posamentier/Krulik] Chap 1, 2: Introduction, “Working Backwards”
Sep 12-16 [P/K] Chapter 2: “Working Backwards”
Sep 19-23 [P/K] Chapters 3-4: “Finding a Pattern”, “Adopting a Different Point of View”
Sep 26-30 [P/K] Chapters 4-5: “Adopting a Different Point of View”, “Solving a Simpler Problem”
Oct 3–7 [P/K] Chapters 6-7: “Considering Extreme Cases”, “Visual Representations”
Oct 10-14 [P/K] Chapters 7-8: “Visual Representations”, “Intelligent Guessing and Testing”
Oct 17-21 [P/K] Chapters 9-10, Exam #1, (Fri = Semester Break)
Oct 24-28 [P/K] Chapters 10-11: “Organizing Data”, “Logical Reasoning”
Oct 31- Nov 4 [Averback/Chein] Chapter 1: Following the Clues
Nov 7-11 [A/C] Chapter 2: Solve it with Logic
Nov 14-18 [A/C] Chapter 3: Algebraic Problems
Nov 21-25 [A/C] Chapter 4: Topics from Number Theory
Nov 28 – Dec 2 Developing a Grading Rubric (Wed-Fri = Thanksgiving Break)
Dec 5-9 Problem Resources: Student reports on resources found
Dec 13 (Tue) Final Exam 7:40-9:40 am