**MED 383w: Learning and Teaching Problem
Solving **

**Fall, 2006**

**Call numbers**:

2218 TR 9:25 - 10:40 BA
202 CEP Dec. 14, 7:40 am

2219 TR 1:40 - 2:55 BA 214A CEP Dec. 14, 1:40 pm

**Instructor**: **
Office
hours**:

Jane R. Cushman MW 1:00 pm - 3:00 pm

Email: cushmajr@buffalostate.edu or by appointment

Office phone: (716)878-6319 Bishop Hall 343

**Course Description**: An introduction to the basic
techniques of problem solving, creative problem solving, the methods of
Polya, Schoenfeld, and others, with applications to the areas of
algebra, geometry, analysis, and recreational mathematics. The course also considers
strategies of teaching problem solving, as well as research in this
area.

Experiences in mathematical problem solving, learning through problem solving, and a consideration of diverse perspectives and problem solving approaches. Strategies for teaching the use of a problem-based approach and the historical and current roles of problem solving in secondary mathematics are also emphasized.

**Supplementary Text**: *Principles and Standards
for School Mathematics*, National Council of Teachers of
Mathematics, 1989, 2000. Copies
should be available at local bookstores, or you can order online from http://www.nctm.org/standards/buyonline.htm,
or access the text online at http://standards.nctm.org/document/index.htm. We
will be reading most of the sections on problem solving, reasoning,
communication, representation, and connections.

**Prerequisite mathematics:** MAT 270 and at least
junior standing. Most
of the problems assigned in this course involve only pre-calculus
mathematical concepts. You will be given a list (What You May Assume)
of these. If you think of using something not on this list in solving a
problem, check with me first to see whether it is acceptable. In some
cases, I will say yes and add it to the list. In other cases, I may say
you can use it only if you include a proof of it in your solution.

If you are rusty on the prerequisite material, you will be responsible for doing any necessary review. You may find the handout "What You May Assume" to be adequate for review. If not, here are some suggestions on library sources: the Math Club library has several precalculus textbooks; the E.H. Butler Library is a good place to find geometry texts. Also bear in mind that you may gain a better understanding of concepts in the process of using them to solve problems.

**Course objectives**: This is a course in problem
solving in mathematics, geared primarily toward prospective math
teachers. The goal of the course is to **improve** problem
solving skills. Of course, the only way to accomplish that goal
is to solve lots of problems. You will be solving problems in
class and at home, in groups and individually. However, our focus
will not be so much on the solution of a given problem as on the
process of solving it. You will be asked to reflect on your own
approaches as well as those of your classmates. To accomplish this, you
will be asked to keep a journal, and make substantial entries each
week (More on this later). You will also be presenting
problems in oral and written form. You will get experience in giving
and receiving feedback on your mathematical communications (both oral
and written). In addition, we will explore the place of problem
solving in the mathematics curriculum.

In order for you to get as much as you can from this
class, it is essential that you attend class regularly. In
addition to being here, you must **participate actively** in class
by presenting exercises, being attentive to solutions presented by
other students, allowing others to speak freely, asking questions and
offering constructive feedback to fellow students, accepting feedback
and constructive criticism offered to you, and generally contributing
to a healthy learning environment.

You must give yourselves lots of time to work on the assigned problems. You should start assignments early enough so that you have time to think about a problem, put it aside, come back to it, revise your approach, if necessary, and write up a well-thought out solution. The problems will all use elementary tools, available to a high school pre-calculus student. This is not to say that the problems will be easy. There are many "elementary" problems that have stumped mathematicians for centuries. There is no shame in being unable to solve all the problems. You will be frustrated at times; if you are not then let me know and I will find some better problems for you. If you are too frustrated, then come in to office hours and we will talk about it. It is up to you to find a balance of challenge and support which will make this class optimal for you. Above all else, do not be ashamed to ask questions; asking questions is the best way to learn.

**Relationship to Teacher Education Program Conceptual Model**:
The preparation of reflective facilitators of learning at Buffalo State
College is anchored in a foundation of professional knowledge:
knowledge of the learner and their characteristics, knowledge of the
content to be taught, and knowledge of pedagogy. The course objectives
for MED 683 address all three components of the conceptual model.

- Knowledge of the learner is fostered through classroom discussions.
- Knowledge of the content is developed as teacher candidates deepen their understanding of mathematical concepts through problem solving.
- Knowledge of pedagogy is developed as teacher candidates present and critique solutions to various problems.

**Attendance Policy**: Regular attendance is expected
at all class meetings. There
will be a large amount of in-class work which will be done in the class
and cannot be made up. All
absences must be discussed with me.
You can receive up to 5 points per day: 2 for on time attendance
and 3 points for constructive participation. If you attend and
participate every class day, you can earn extra credit.

**Policy on Collaboration**: Since unauthorized
collaboration is considered academic dishonesty, it is important that
you know what kinds of collaboration are and are not authorized in this
class.

1. The following activities are not only *authorized*
but *encouraged*:

- Working on a problem with someone when neither of you has yet solved the problem
- Asking someone for a
*small*hint if you have given a problem a serious try and are stuck. - Giving a student who asks for help the smallest hint that you possibly can.
- Asking someone to listen to and critique your ideas on a problem.
- Listening to a student's ideas on a problem and critiquing them without giving away the solution.
- Asking another person to read and critique your write-up of a problem.
- Reading and critiquing another student's write-up of a problem, pointing out errors but not correcting major errors.

2. *Unauthorized* collaboration includes:

- Asking someone to show you the solution to a problem that hasn't been handed in or discussed in class yet.
- Showing a student in the class a solution to a problem they have not yet solved and that hasn't been handed in or discussed in class yet.
- Copying, either word for word or by rewording, a solution that you have not played a significant part in obtaining. This includes a solution found in a book, a solution obtained by a student or group of students in this class, a solution originating in this class in a previous year, or any other source.
- Writing up a solution together with someone else, whether or not you have worked out the solution together.

Clearly authorized collaboration provides a learning
experience for both parties. Unauthorized collaboration benefits
no one and, in fact, is educationally detrimental. Please do not
put your classmates in a difficult position by asking to copy their
work. See the *Student Handbook and Calendar* for the section on
Academic Misconduct on page 25 in the 2006 - 2007 edition.

**Exams**: There will be two mid-semester exams
(during regular class time) and a final exam. The dates for
the mid-semester exams are:

Exam 1: Thursday, October 5

Exam 2: Thursday, November 9

I will only give makeup exams with a valid excuse.

**Grading**: Course grades will be based on all
relevant information I have about you: performance in class discussion
and problem solving, homework turned in, exams, journal, and any other
information I have pertaining to your work in this class and its effect
outside this class. Homework, exams, journals, and quality of class
participation will be weighted as follows:

Homework: 20%

Semester Exams: 15% each

Final: 20%

Class participation 20%

Journal: 10%

**Journal**: As part of your coursework for this
class, you are required to keep a class-related journal. The journal
will serve several purposes, including: encouraging you to reflect on
your problem solving behavior and other topics related to mathematics
and teaching, giving you practice writing about mathematics, providing
feedback to me, and providing another means for me to give feedback to
you.

You are expected to make journal entries at least twice a
week, with each week's entries being at least one handwritten, standard
sized page (or the equivalent word processed). *Please date each entry
and keep them in chronological order*.

Occasionally I may ask you (either the whole class or individually) to write on a specific topic, but usually the choice will be up to you. Possibilities include:

- Your reactions (thoughts, and feelings if you wish) to topics in the readings or discussed in class.
- Analysis of how you go about solving problems (e.g., what strategies you most often use), and how you might do so better.
- Insights you have had into various mathematical concepts.
- Comparing and contrasting how you and other students go about solving problems.
- Comparing and contrasting different solutions to the same problem.
- How you have used ideas discussed in this class in other classes or other situations in your life, or how these relate to what we've discussed in class. (Students who have an extended field experience or are student teaching this semester may have lots of comments related to those experiences.)
- How you might incorporate ideas in this class in your own teaching.
- How you might use what you learned in solving one problem in solving another.
- Describing problems you have made up, and why, when, and how they might be good teaching problems.
- Asking questions about concepts you don't yet understand fully.
- Requests for specific kinds of feedback.
- Suggestions on how to improve this class.
- Discussion of what types of problems you like best, and why.
- Comments on your progress in any of the areas of the course objectives.
- Information that might help me evaluate your performance in this class.
- (Don't limit yourself to just one of these topics, however. Anything related to mathematics and teaching mathematics is appropriate.)

You should *not* use your journal to record what
went on in class (except brief accounts to introduce your own reactions
to this.) You are expected to write in your journal outside class. If
you wish to take class notes, you should keep these in a separate
notebook or folder.

I will collect, read, and make comments on your journal every two or three weeks. Your journal grade will not depend on the correctness of the mathematical content of your journal, but on the thoughtfulness that went into your writing.

Use a looseleaf notebook or folder for your journal, so
you can write journal entries when I have collected journals to read. *Please
hand in sections I have already read along with new entries, so that I
can see old comments.*

**Procedures regarding disruptive individuals**:
Disruptive behavior by students in my class will not be tolerated.
Whenever I deem a student to be acting in a disruptive or threatening
manner, I will exercise my right to ask that individual to leave the
classroom. If refused, I will exercise my right to notify University
Police. The responding officer will determine whether an arrest should
be made or whether a referral to medical or counseling staff is
appropriate. If a student is perceived as a danger to himself, herself,
or others, the dean of students may propose an interim suspension until
a hearing is held. Any student removed from class will have the right
to a hearing.

**Students with disabilities**: Any student who
requires accommodations to complete the requirements and expectations
of this course because of a disability is invited to make his or her
needs known to the instructor and to the director of the Disabilities
Services Office, 120 South Wing, 878-4500.

**Cell phone policy**: Please leave your cell phones
either off or on vibrate. Do not text message at anytime during class.
If there is a call you must take, then step out of the classroom to
answer it.