Unique numbers

- Friday, August 27, 11 - 12 am

Links related to the course will be added here from time to time.

We will be reading most of the sections on problem solving, reasoning, communication, representation, and connections. Some of you may have read some of these portions before in other classes. If this is the case, be sure to read them again, focusing on any points indicated for attention in the assignment, and looking for points you may have overlooked, misunderstood, or not fully understood in previous reading. I find that each time I reread part of the

Course objectives

1. To improve your mathematical problem solving ability. (By "problem solving," I mean "figuring out" rather than following procedures that someone else teaches you.) This should help you in other courses you take, especially proof courses, as well as in other ways.

2. To improve your abilities in mathematical communication (including writing, talking, and listening).

3. To deepen your understanding of some basic mathematical concepts.

4. To enrich your understanding of what mathematics is, especially as this is relevant to teaching mathematics.

5. To develop the habit of reflection on mathematics.

6. To practice giving and receiving feedback.

Please note

1. These are "You can lead a horse to water …" type objectives.

2. This is a course in mathematics for prospective teachers. It is

- As with any University course, you are expected to spend at least twice as much time outside of class as in class working on the course Work outside of class will include working on problems, writing up solutions, reading from the text and possibly from other sources, and keeping a journal. (See below for more details on the journal.)
- In-class activities will include some group activities, but primarily presentation of problem solutions; discussion of problems, readings, and exercises; and occasional short lectures.

- Attend class regularly.
- Participate in class activities by presenting exercises when asked, volunteering to present solutions to problems, listening carefully to other students' presentations of solutions, asking questions and offering constructive feedback on other students' presentations, responding constructively to other students' critiques of your presentations, contributing constructively to class discussion, participating constructively in group activities, and treating your classmates in a respectful manner that contributes to everyone's learning.
- Do assigned reading and related assignments carefully and reflectively rather than superficially.
- Manage your time working on problems so that you can work on problems, set them aside, come back to them, write up your solutions, and revise your write-ups.
- Practice the problem solving techniques we discuss in class, to incorporate them in your problem solving habits.
- Learn from other students' methods and from feedback offered by other students or me, whether orally or in writing.
- Keep your journal as outlined below, and use it in a way that best helps your individual learning needs and opportunities
- Take responsibility for maintaining a balance of challenge and support that optimizes your learning in this course. (More on this below.)

1. The following activities are not only

- 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 them.

2.

- 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.

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 QA 551 shelf in the PMA library has several precalculus textbooks; the Textbook Collection in the PCL stacks is probably the best place to look for geometry texts. Also bear in mind that you may gain a better understanding of concepts in the process of using them to solve problems; so don't think you need to understand everything completely before you start the class.

Portfolio

Good or better quality on all of homework, exams, journal, and class participation, plus excellent quality on at least one of homework, exams, journal: A

At least good quality on at least three of homework, exams, journal, and class participation: B.

At least adequate quality on at least three of homework, exams, journal, and class participation: C

(See the second bullet in the section "What I expect of you" above for what constitutes good class participation.)

- Encouraging you to reflect on your problem solving behavior and other topics related to mathematics and teaching mathematics for understanding
- Giving you practice writing about mathematics
- Providing feedback to me about your progress in the course
- Providing another means for me to give feedback to you.

I will collect, read, and make comments on your journal every two or three weeks. You will not receive a grade on your journal, but are required to maintain it. Your journal will be part of the information I consider in determining your course grade.

- Use a double-pocket folder for your journal.
- Put new (unread) entries in the right hand pocket.
- Put old entries (those already read) in the left hand pocket, so that I can refer to them as needed in reading.
- After the first time your journal has been handed in, be sure to include the sheet of comments I have added to it, so that I can refer to it as needed in reading new entries.
*D**ate*each entry and keep them in chronological order.- Do
*not*include anything else in your journal folder.

- 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 the ideas in this class might influence 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/learning problems.
- Asking questions about concepts you don't yet understand fully.
- Requests for specific kinds of feedback.
- Asking questions about things in this class whose purpose you don't understand.
- 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.

You should

Dear 360M/396C student,

Welcome to M360M/396C. I hope this class will be rewarding for you.

As a mathematics teacher, you will share in the responsibility of helping to prepare future generations to solve many complex problems facing society. Increasingly, these problems are at least partly technical ones, often involving mathematics, so your impact as a mathematics teacher will be especially important. Many problems that do not involve mathematics can also benefit from the types of "higher order" thinking skills that go into solving challenging, non-routine mathematical problems, so what you teach in math classes can benefit students and society in ways beyond just mathematics. However, many of you have been shortchanged in your own precollege math classes by not having many opportunities to work on these types of challenging, non-routine mathematical problems. This course will give you an opportunity to work on such problems, as well as to learn about (and practice) some of the thinking skills involved in them and many other problem solving.

I believe that learning best takes place when the learner has an appropriate balance of challenge and support. I have tried to design this course so that there are ample opportunities for both. However, since what is challenge and what is support varies from student to student, you will need to take responsibility for achieving a balance that is appropriate for you.

Most students find most of the problems in this course challenging. If you do not, I expect you to take responsibility for pursuing other aspects of the course in a manner that is challenging to you. Possibilities include seeing how many ways you can solve each problem, polishing your mathematical writing, focusing on observing other students' problem solving efforts and how this observation can help you become a better teacher, trying to make your problem solving more efficient, learning to apply problem solving techniques discussed in this course to a course that is more challenging for you, reading more of the NCTM

If, like many students, you find the problems very challenging, you will need to make an effort to develop and utilize the supports available. No single prescription works for everyone, since what is supportive for one individual may in fact be challenging for another (Example: Some students find feedback supportive, but others find it challenging), but possibilities include:

- Establishing a good relationship with one or more students in the class with whom you can work on problems (Two heads are better than one!) or from whom you can obtain constructive emotional support. (Of course, you should be willing to expect to give as well as take in any such relationship.)
- Managing your time carefully so that you don't find yourself too rushed to set a problem aside and come back to it later.
- Making an extra effort to learn to use the problem solving techniques discussed in class
- Using your journal to help work through your struggles.
- Practicing constructive self-talk. (Example: If you start thinking that you are "dumb" because you didn't see on your own something that seems obvious after someone else explained it, remind yourself that that happens to everyone, including very smart people.)
- Making maximum use of your own and others' mistakes as
learning experiences.

You will undoubtedly find the course frustrating at times; it is in the nature of problem solving to feel frustrated at least sometimes. Don't let the inevitable frustration stop you. Learning to deal with it is an important part of learning to solve problems.

I look forward to seeing you learn and grow mathematically in this class, and hope that it will help you help your own future students learn and grow mathematically.

Sincerely,

Martha K. Smith

Professor of Mathematics