Connecting advanced mathematics to high school mathematics

Lecturer | Professor Amy Cohen |

acc@math.rutgers.edu | |

Office | 530 Hill Center, Busch |

Tentative Office Hours... | Th 2:30-3:30pm and by appointment |

Course Meetings... 4:30-7:30p.m. in Murray 115 on CAC

No textbook. Readings will be distributed

Registration requires a special permission number.

Students in the fifth year of the Five-Year Certification program will receive a special permission number by e-mail. Other students desiring to take this course should consult with Professor Cohen before attempting to pre-register for this course.

This course attempts to answer some very sensible questions: What can a future teacher get out of advanced math courses that makes a teacher more effective? How do the ideas, methods, and procedures of advanced math provide the kinds of insight into high school content that allows a teacher to answer questions, correct misunderstandings, motivate future work, etc? How does mathematical content knowledge strengthen pedagogical content knowledge?

Completing advanced mathematics courses, including advanced calculus, abstract algebra, and college geometry, is an important component of students' preparation to become teachers of high school mathematics. In these advanced mathematics courses, students are exposed to the importance of mathematical precision and rigor as well as new modes of mathematical reasoning. By studying topics from high school mathematics from an advanced mathematical viewpoint, students' understanding of these concepts can be enriched so that they should be better able to teach these concepts in their classrooms. For a variety of reasons, these potential benefits are not always realized and future teachers frequently question the value of completing advanced mathematics courses.

The purpose of this course is to help you build connections between your experiences in your advanced mathematics courses and the high school mathematics that you will teach. The first nine weeks of this course will focus on mathematical content with frequent mention of pedagogical issues. The rest of the course will focus on mathematical pedagogy with frequent reference to mathematical issues. In some cases the content we discuss will overlap with that content you will teach. One example is the reason "spurious roots" arise in solving certain algebraic equations. In other cases the content will be an advanced concept that you will not teach, but which provides insight into material you will teach. An example here is the interactions among algebraic ideas like "group homomorphism", linear algebra topics like matrix multiplication, geometry topics like rigid motions, and high school topics like trigonometry.

Your responsibility will be to deepen your understanding of the mathematics and to develop insights into the ways that advanced mathematical issues can inform the way you teach and the way you react to your students' developing understandings and misunderstandings.

** Goals of this course**

- to learn to see high school mathematics in terms of the abstract definitions and structures studied in advanced mathematics classes.
- to use links between high school and advanced mathematics to better understand both.
- to consider ways in which this understanding might affect the way your teach.
- to work on using mathematical terms precisely in speech and writing.
- to recognize some epistemological and didactical obstacles to the learning of mathematics.

**Expectations of this course**

- to attend all classes. If you cannot attend for some unavoidable reason, you should let the instructors know by e-mail and catch up on the work you missed.
- to be engaged and attentive in class. We do not assume that you have a strong proficiency in all the topics that we discuss. We do expect you to ask and answer questions, to work hard inside and outside class to make new connections in your knowledge, and to maintain a positive attitude toward learning mathematics.
- to be aware that this is a
math class . A significant portion of your grade will depend on how much mathematics you learn and how clearly you can communicate about mathematics. The course is designed to facilitate your learning. However, you must take responsibility for your own learning. Please seek help from classmates and instructors when you feel you can use help.

**Grading and assignments**

- Each week you will write a short
summary of the mathematics that you worked with in that week's class. It should be approximately one page long, double-spaced. It is due by the end of the Thursday following the class. Send it to the instructor in the body of an e-mail message. This summary should be written with the standards you would use for any written homework assignment submitted for grading. It should be thoughtful, grammatically correct, well-organized, and free of typo's. - You are also invited to share your reactions to the course weekly. This reaction should be separate from the summary. It may be more informal in presentation.
- There will be one midterm exam. This will occur around the ninth week of term, as the course shifts its focus from mathematics to pedagogy. This exam will assess your mathematical understanding of the material we have discussed. Details will be given closer to the time of the exam.
- There will be a final group project. Each group will pick a topic from high school mathematics and discuss how advanced mathematics relates to it. A list of potential topics will be provided, but you may also pick other topics with the approval of the course instructors. Your group will produce a paper discussing how the advanced mathematics relates to the topic, discuss how insights from advanced mathematics affect your understanding of the concept, and describe how your teaching would be changed by the insights your describe. Each group will make a presentation of its project to the class. More detail will be provided later in the term.

** Very Tentative Course Schedule**

Week 1... | Recall content from advanced math courses |

Week 2... | Groups and homomorphisms |

Week 3... | Rotations, equivalence relations, matrices |

Week 4... | Rotations, equivalence relations, matrices |

Week 5... | Modular arithmetic and divisibility |

Week 6... | Equations, inequalities, properties of number systems |

Week 7... | Solving equations, implicit implications |

Week 8... | Exponential functions, the number |

Week 9... | Limits, real arithmetic |

Week 10.. | Generating examples |

Week 11.. | Definitions |

Week 12.. | Didactic obstacles |

Week 13.. | Reading mathematics,turning exposition into dialog |

Week 14.. | Presentations |

**Textbooks and readings**

There is no formal textbook. There will be assigned readings. Short readings will be distributed in class. Longer materials will be put on reserve in the library.