525 Algebraic Structures for Teachers

Spring 2006     W 5:30-8:15pm

Matt Jones                   email: mjones@csudh.edu        

Website: http://www.csudh.edu/math/mjones You will find the syllabus and course assignments on the website.  

Office:  NSM A-120    phone:  (310) 243-2410

Office Hours:    M 5:15-6:00pm

                        W 1:00-3:45pm (1- 2:30 only on February 8, 22, March 8, 22, April 12, 26, and May 10)

W 8:15-9:00pm

                        And by appointment

Text and Materials:  Course reader and a graphing calculator or a laptop with calculator.

Suggested references:  A Research Companion to Principles and Standards for School Mathematics; Contemporary Abstract Algebra, Third Edition, by Joseph A. Gallian

Course Description:   This course is designed to provide a deeper examination of the connections between undergraduate abstract algebra and high school algebra, with an emphasis on concrete examples, and to examine research on effective teaching and learning.

Goals:  Students will understand:

·         The integers under modular arithmetic as an example of a set with algebraic structure

·         Applications of modular arithmetic to cryptography and check digit schemes

·         Group as an algebraic structure and examples of groups:  modular arithmetic, geometric transformations, symmetry, and matrices

·         Ring as an algebraic structure and its application to modular arithmetic, integers, and polynomials

·         Field and examples of fields:  finite fields, rational numbers, real numbers, complex numbers, and rational functions

·         Research on effective teaching and learning in mathematics, especially as related to algebra, and its classroom implications

Expected Outcomes:  Students will be able to: 

·         Create or work with sets which are examples of groups, fields, or rings, and work out operations on elements in the set

·         Prove results about cryptography and check digit schemes

·         Prove and apply theorems about groups, rings, and fields

·         Create or identify activities consistent with research on successful teaching and learning of mathematics, and implement them in a classroom setting

Assessment:  This course will be graded out of a possible 600 points.  Points are distributed into categories as follows.

  • Homework (100 points): There are 5 homework assignments, each worth 20 points, due during the semester.  If you cannot be in class, have someone turn your homework in for you or turn it in to my office on the day it is due.  Late homework is not accepted.  Rubrics for the grading of each assignment will be provided.
  • Reading Prompts (100 points):  In most weeks, research reading will be assigned.  Many of these readings will be sent to you by email (see schedule page Note).  For research readings, you must type up (single-spaced, 12 point font) your responses to the following questions:
    • Summarize the thesis of the reading in 50 words or less.
    • Highlight three to five of the most significant points of the reading.
    • For each highlight above, cite at least 2 implications for your own teaching.
    • In one sentence, describe the “take-away” message of the reading, that is, what you would like to remember in a year or tell a colleague in the hallway about how this research should impact your teaching.
    • You may (but are not required to) type your responses into a graphic organizer which I will email to you
  • Article Project (100 points):  Details will be provided separately.  Presentations will be given April 5 and April 12.
  • Exam (100 points):  An in-class exam covering both content and research will be given March 8.
  • Reflection Project (100 points):  A final reflection highlighting important learning about both research and algebra will be due on May 10.  Details and a scoring rubric will be available later in the semester.
  • Final (100 points):  The final will be held Wednesday, May 17, from 5:30-7:30pm, and will be cumulative.

Grading Scale:  A:  92% or better, A-:  88-91%, B+:  85-87%, B:  81-84%, B-:  78-80%,

C+:  75-77%, C:  71-74%, C-:  68-70%, D:  65-67%, F 64% or below

Creating Conditions for Successful Learning:  Research shows success in math class depends very much on two factors:  the amount of time spent working on the material, and the student’s beliefs about mathematics and what it means to understand and do mathematics.  With this in mind, here are some suggestions: 

  • Be in class, every class, and be on time.
  • Be prepared to participate in group work and discussions every day so that class time is not wasted, and
  • Spend at least 1 hour every day, not including class time, working on homework assignments and readings, and studying.
  • Realize that mathematics is not just a set of procedures, and that mathematical concepts involve a lot of thinking and reasoning.  Consequently, being able to execute procedures accurately is only one part of doing well in this class.
  • Realize that success in mathematics is less about “ability” and more about willingness to think and to work hard to make sense of things.

In addition, you need to have:

  • your assignments with you and ready to turn in on the day they are due
  • the numbers and emails of at least 2 classmates so that you can be informed if you miss a class.

Make-up Policy:  I do not accept late or make-up work.  If you experience a major emergency, special arrangements may be made at my discretion.  Please make every effort to contact me as soon as possible when you know you will miss a class due to an emergency; do not wait until the next class to ask about being excused from an assignment.

Classroom Norms:  As we will spend a lot of time working in partnerships, in groups, and in class discussions, here are some rules to help you navigate what may be an unfamiliar experience in math class. 

  • Never call out an answer until the person leading the classroom has given permission.  Raise your hand.
  • This is a safe environment.  That means that you should feel free to ask a question or offer an opinion or an answer, and no one will make fun of you for what you say.  We will discuss how to disagree with or question fellow students when they are sharing their work.
  • If you are working with classmates, work with them.  Do not wait and hope that others will do your work for you, and do not move on to other assignments while your classmates are struggling to understand the current one.
  • Be considerate of others.  In addition to the ways to be considerate listed above, do not dominate group or class discussions.  Remember that everyone needs an opportunity to share his/her ideas.
  • Do not expect me to validate your answers or those of anyone else.  You are responsible for making sense of answers and solution methods, and you should always look for ways to verify your work.
  • Cell phones should be off or set to “vibrate.”  Do not place a call during class, and do not answer a phone call without first leaving the room.

These rules are meant to benefit the entire class, and to ensure that everyone has the opportunity to contribute and to learn.

 

Academic integrity is expected.  I enforce university policies on academic integrity.  In particular, cheating, fraud, plagiarism or other academic dishonesty is unacceptable and will be cause for disciplinary action.

 


 

Week

Topic

Assignment or Project due

Reading discussed

Author

1/25

Modular Arithmetic Introduction

 

Facts and Algorithms as Products of Students' Own Mathematical Activity*

Gravemeijer & van Galen

2/1

Cryptography

 

Moving Beyond Teachers’ Intuitive Beliefs about Algebra Learning

Nathan

2/8

Modular Arithmetic and Check Digits

Mod Assignment

 

 

2/15

Mod and Check Digits II

 

Conceptual and Procedural Knowledge of Mathematics:  Does One Lead to the Other?

Rittle-Johnson & Alibali

2/22

Modular Arithmetic Theorems

 

Interference of Instrumental Instruction in Subsequent Relational Learning

Pesek & Kirshner

3/1

Dihedral Groups and Multiplication Tables

Groups I

 

 

3/8

Symmetry Groups

 

The Core-Plus Mathematics Project:  Perspectives and Student Achievement

Schoen & Hirsch

3/15

Matrices and (Z/nZ)*

 

The Effects of Curriculum on Achievement in Second-Year Algebra: The Example of the University of Chicago School Mathematics Project

Thompson & Senk

3/22

Groups

Groups II

 

 

4/5

Group Theorems and Fermat’s Little Theorem

Article Project

On Appreciating the Cognitive Complexity of School Algebra*

Chazan & Yerushalmy

4/12

Rings and Z/nZ

Article Project

Stasis and Change*

Smith

4/19

Polynomial Rings and Factorization

 

Representation in School Mathematics:  Learning to Graph and Graphing to Learn*

Monk

4/26

Fields and Z/nZ

Rings and Fields

Knowing What to Believe:  The Relevance of Students’ Mathematical Beliefs for Mathematics Education

De Corte, Op’t Eynde, & Verschaffel

5/3

Frieze and Crystallographic Groups

 

Patterns of Misunderstanding:  An Integrative Model for Science, Math, and Programming

Perkins & Simmons

5/10

 

Reflection Project; Groups and Fields

 

 

5/17

Final Exam

 

 

 

Note:  Articles marked with a (*) are from the volume A Research Companion to Principles and Standards for School Mathematics.  The book is available in the CSUDH library, or can be purchased from NCTM or any other bookseller, such as Amazon.com.  Copies of these articles (other than the first one) will not be provided to you—you are responsible for obtaining them.  All other articles will be provided to you either in hard copy form (highlighted articles) or as .pdf files sent by email (all other articles).