SCI 360: An Advanced Perspective on School Algebra

Spring 2005
Unique# 11285

Instructor: Jennifer Christian Smith
Office: SZB 462B
Phone: 232-9682
Email: jenn.smith@mail.utexas.edu
Office Hours: Tuesdays 11:00-12:00 , Wednesdays 4:00-5:00 , and by appointment
AIM: drjennsmith

Course Description:

What do courses such as abstract algebra, number theory, and analysis have to do with the algebra you will be teaching in middle and high school? What are the connections between university level and secondary school mathematics? Why is it important for future high school mathematics teachers to take such courses that seem on the surface to have little in common with secondary math? In this course, we will address these questions by carefully examining important concepts in the school algebra curriculum, such as real and complex numbers, functions, polynomials, and equations. We will solve and extend problems, discuss the history and applications of algebra, analyze problems from high school algebra from an advanced perspective, and explicitly make connections between the concepts you will teach and their more abstract analogues which you have encountered in your undergraduate mathematics courses. We will utilize appropriate technology whenever possible, and will also investigate the research literature on students' difficulties with algebra.

Course Information:

•  Meets Wednesdays, 5:00-8:00 , in SZB 344
•  Prerequisite: At least one of M328K, M343K, M341, M361K

Required Materials:

•  Usiskin, Z., Perssini, A., Marchisotto, E., & Stanley, D. (2003). Mathematics for High School Teachers: An Advanced Perspective . New Jersey : Pearson Education, Inc. (Available at the Co-op bookstore.)
•  Packet of readings, available from IT Copy
•  A graphing calculator, such as a TI-83 plus, or a TI-89.

Course Evaluation:

•  Participation: 40%. This includes attendance, regular presentation of problems, preparation for class (reading and working problems in advance), and degree of participation in small group and whole class discussions. (See rubric below.)
•  Written assignments: 40%. You will be required to turn in written explanations for 2-3 problems each week. These problems will be chosen at the end of each class meeting and will be due at the beginning of class the following week.
•  Final project: 20%. Projects will be chosen from the book. You may work alone, but I prefer that you work with a partner. You will present the results of your project in class and turn in a paper. (See Project information below.)

Course Policies:

Attendance is critical in a course that meets weekly! Because of the structure of this course, it is virtually impossible for a student to make up a missed class. More than one absence will lower your participation grade by a letter .

1. This course will be problem-based, and so it is critical that you have read the chapters and worked through the problems prior to class. Your participation (and hence your grade) will be hindered if you are not prepared for class!
2. I will utilize BlackBoard for course announcements and assignments. It is your responsibility to check the course page regularly.
3. Homework is due at the beginning of class. Late work will receive a grade that has been reduced by one "letter". That is, late work that would have received a B will be given a grade of C.

Reading Assignments:

There will be a reading assignment from the text and from the packet for every class meeting. Keep in mind that reading a math text is different from reading any other kind of text. Mathematical writing is compact, non-repetitive, and uses lots of symbols. As such, it takes more time to read a mathematical text. Please allow yourself that time. Take notes in the margin, re-read parts you didn't understand. Work through examples on a separate piece of paper. No, really . Trust me on this!

Class Structure:

We will typically follow this format:

5:00-5:30 Discussion of packet reading
5:30-6:00 Small group discussion of problems
6:00-7:45 Presentation and large group discussion of selected problems

There will be a 15 minute break at an appropriate time.

Goals of the course:

•  Examine the structure of secondary algebra from an advanced perspective.
•  Make connections between topics taught in high school algebra and topics studied in undergraduate mathematics courses.
•  Deepen ability to analyze and generalize problems and their solutions.
•  Deepen understanding of previously learned concepts.

The vehicle by which we accomplish these goals is the discussion of problems we've solved.

NOT Goals of the Course:

•  To find and present answers to as many problems as possible.
•  To learn specific content; i.e., to “cover” the text.
•  To learn new mathematics topics.

When examining a solution presented by a classmate, consider the following:

•  Is the solution correct?
•  Do you understand the solution?
•  Do you understand the problem?
•  Why is this a correct solution to the problem?
•  Can the solution be generalized?
•  Can the solution method be generalized or applied in other problem situations?
•  What mathematical concepts or theorems can be applied to this problem?
•  Is the problem an example of a particularly important concept?
•  Does the problem highlight a common misconception students have?
•  Are there connections we can make between different areas of mathematics based on this solution?
•  What alternate solutions methods (geometric, algebraic, estimation, numerical reasoning, modeling, etc.) could be used? What insight would we gain about the problem from using them?

Snacks:

If everyone in the class agrees, we will have a sign-up sheet for snacks. One or two people will bring snacks for the group to each class meeting.

Study Groups:

I strongly encourage you to meet with your classmates outside of class to discuss the readings and work on problems. Start building those collegial relationships now!

Participation Grade Rubric

If your participation does not meet the standards for a C above, you will receive a participation grade of D. This is bad.

Reading List

Carlson, M. (1998). A cross-section investigation of the development of the function concept. CBMS Issues in Mathematics Education, 7, 114-162.

Chazan, D. (2000). Chapter 3: Towards a "conceptual understanding" of school algebra. Beyond Formulas in Mathematics and Teaching (59-111). New York : Teachers College Press.

Chazan, D. & Yerushalmy, M. (2003). On appreciating the cognitive complexity of school algebra: Research on algebra learning and directions of curricular change. In J. Kilpatrick, W. Martin, & D. Schifter (Eds.), A Research Companion to Principles and Standards for School Mathematics (123-135). Reston , VA : National Council of Teachers of Mathematics.

Driscoll, M. (1999). Chapter 5: Expressions generalizations about functional relations. In Fostering Algebraic Thinking: A Guide for Teachers Grades 6-10 (90-114). Portsmouth , NH : Heinemann.

Herscovics, N. & Kieran, C. (1999). Constructing meaning for the concept of equation. In B. Moses (Ed.), Algebraic Thinking (181-188). Reston , VA : National Council of Teachers of Mathematics.

Kieran, C. (1989). The early learning of algebra: A structural perspective. In S. Wagner & C Kieran (Eds.), Research Issues in the learning and Teaching of Algebra (33-56). Reston , VA : National Council of Teachers of Mathematics.

Laughbaum, E. (2000). Testing students with hand-held technology. In E. Laughbaum (Ed.), Hand-Held Technology in Mathematics and Science Education: A Collection of Papers (184-191). Ohio: Teachers Teaching with Technology Short Course Program.

Mulligan, C. (1988). Using polynomials to amaze. In A. Coxford & A. Shulte (Eds.), The Ideas of Algebra, K-12 (206-211). Reston, VA: National Council of Teachers of Mathematics.

National Council of Teachers of Mathematics (2000). Algebra standard for grades 9-12. In Principles and Standards for School Mathematics (296-306). Reston , VA : National Council of Teachers of Mathematics.

Thompson, P. (1994). Students, functions, and the undergraduate curriculum. In E. Dubinsky, A. Schoenfeld, and J. Kaput (Eds.), Research in Collegiate Mathematics Education I (21-44). Providence, RI: Conference Board of the Mathematical Sciences.

Usiskin, Z. (1988). Conceptions of school algebra and uses of variables. In A. Coxford & A. Shulte (Eds.), The Ideas of Algebra, K-12 (8-19). Reston , VA : National Council of Teachers of Mathematics.

Van Dyke, F. & Craine, T. (1999). Equivalent representations in the learning of algebra. In B. Moses (Ed.), Algebraic Thinking (215-219). Reston , VA : National Council of Teachers of Mathematics.

Waits, B. & Demana, F. (2000). Calculators in mathematics teaching and learning:Past, present, and future. In E. Laughbaum (Ed.), Hand-Held Technology in Mathematics and Science Education: A Collection of Papers (2-11). Ohio: Teachers Teaching with Technology Short Course Program.

Zazkis, R. and Campbell, S. (1996). Divisibility and multiplicative structure of natural numbers: Pre-service teachers' understanding. Journal for Research in Math­ematics Education, 27(5), 540-563.

Final Project

Projects will be chosen from those at the end of each chapter, in consultation with the instructor. I prefer that you work with a partner on your project (since more learning occurs when people communicate about mathematics than when a person works alone), but you may choose to work alone if you wish. You and your partner should choose a project by March 23 . After your project has been completed, you will write a paper and make a presentation in class. Both of these should be joint efforts. And yes, I do know that collaborative writing is difficult! It's a great professional skill, though: you'll likely write papers for teaching journals and grant proposal with colleagues.

Paper

For your project, you will turn in a 5-7 page paper. The paper will include the following sections:

1. Introduction: Explain the problem in terms understandable by a peer. Why is this problem interesting to you? Why is it important?
2. Background: What did you know about this problem before you began trying to solve it? What mathematical topics are involved in understanding the statement of the problem? Give a brief history of the problem, if applicable. Introduce any definitions or terms here, if necessary. Please refer to our course readings, if possible.
3. Methods: Describe the methods you used to solve the problem. What strategies did you try that didn't work, but that gave you some insight into the problem?
4. Results: Describe your solution to the problem, giving as much detail as possible. This should be an expository paper, not merely a list of equations. Emphasize why your solution makes sense.
5. Discussion/Conclusions: How could you extend your solution, or adapt your methods to solving other problems? What new concepts or strategies did you learn or discover while solving this problem? What are the applications of this problem, and how does your solution deepen your understanding of those applications?
6. Implications for teaching: What have you, as a future high school teacher, learned from this problem that will be useful to you in the high school classroom? Could you adapt this problem for use in your classroom? Explain. How has your solution of this problem deepened your understanding of the related mathematical concepts?

Class Presentation

You will have 15 minutes to present the results of your project in class on May 4 . This presentation should summarize and present your paper in the form of a PowerPoint presentation. It should be interesting and informative! Rather than spending a lot of time on the mechanics of how you solved the problem, highlight what is important about the solution.

Course Schedule
SCI 360, Spring 2005

We will endeavor to stick to this schedule! Reading assignments should be completed prior to class on the date indicated. The problems listed should have been attempted and worked through prior to class on the date indicated. At the beginning of each class meeting, there will be time allotted to discuss these problems in groups before whole-class discussion begins.