University of Arizona

Math 407, Synthesis of Mathematical Concepts, Fall 2008

William McCallum, MW 2:00—3:15 p.m., Gould-Simpson 849

This course is about high school mathematics—algebra, functions, geometry, probability, and statistics—viewed from an advanced perspective. We will see how concepts that arise early in the study of mathematics evolve at higher levels, and how simple mathematics problems can lead to sophisticated insights. We will learn to make connections between different areas of mathematics by identifying underlying structures and principles.


Course Information


Assignments

There will be periodic assignments. You may discuss them with me in office hours and on the course listserv. However, your work must be your own. They should be legible, with pages stapled and your name at the top of each page. I will not accept late assignments. It is your responsibility to remember to hand in your work. The first assignment is due Wednesday, September 3.


Participation in the class listserv

A portion of your final grade will be determined by your participation in email discussions on the course listserv. Your grade will be determined not so much by the correctness of your postings as by your willingness to respond to questions and participate in discussions about the mathematics.


Midterm Exam and Final Project

There will be a one-hour midterm exam in class on October 15 and a final project with a written component and a class presentation component. The midterm exam will be a test of basic knowledge of high school mathematics, whereas the final project will be a test of your understanding of the deeper mathematics underlying high school mathematics.


Grade

You can earn up to 600 points: 250 points from assignments, 50 from participation in the course listserv, 100 from the midterm exam, and 200 from the final project. Your grade will depend on your score out of 600: 90% or higher for an A, 80% for a B, 70% for a C, 60% for a D. A score below 60% will give you an E.


Policies

I will follow the standard university policies on missed exams, withdrawals, incompletes and academic integrity.


Final presentations

  • Monday December 1: Nadia, Sarah, Robin, Andrew
  • Wednesday December 3: Liz, Danielle, Heather, Jaymie, Jessica
  • Monday December 8: Eric, Monica, Ellie, Sean, Trish
  • Wednesday December 10: Shannon, Christina, Geraldine, Sam, Carlie
  • Choose an area of the high school curriculum that presents some interesting difficulties either mathematically or pedagogically, and write a paper with about 2–3 pages of text (not including diagrams) about (a) why you chose it (b) the underlying mathematical ideas, going deeper or further than what you would necessarily present to students (b) how you would teach it, including a discussion of why students might find it difficult, possible student errors, and tasks, examples or activities you might use. In class you should plan a 10 minute presentation of the key points. You will be graded on both (a) and (b) in the essay, and also on the clarity and quality of your in-class presentation. Your paper is to be handed in at the beginning of the class where you give your presentation.

Syllabus


Algebra (4 weeks)


Functions (3 weeks)


Geometry (3 weeks)


Linear Algebra (3 weeks)


Probability and Statistics (3 weeks)


Lecture notes

Assignments

  • Due September 3
  • Due September 18: Write a task that assesses and develops students' ability to interpret expressions and equations, or to recognize and describe the purpose of the particular form of an expression or equation, or to see and understand the structure of an expression or equation.
  • Due October 1. Complete parts 5 and 6 of this document describing the triangle problem. We essentially covered part 6 in class, but I would like to see a clear statement and write-up.
  • Due October 29. Give a geometric proof of one of the angle sum formulas: sin(a + b) = sin(a)cos(b) + cos(a)sin(b) or cos(a + b) = cos(a)cos(b) - sin(a)sin(b). You may consult references, but the proof you write must be in your own words and show your own understanding.
  • Due November 19: Compare and contrast the Wald and Agresti-Coull formulas for p and SE[p], given on pages 6–8 of the slides from Walt Piegorsch's talk. Use the algebraic form of the expressions to explain what effect extra terms in the Agresti-Coull formula have on the 95% confidence interval. For what situations are the two formulas in close agreement, and for what situations are they different?

Preparation for midterm

Here is a practise test from Massachusetts that you can use to start preparing for the October 15 midterm on teacher knowledge.

William McCallum

Course listserv

math407@listserv.arizona.edu

Office hours

Room 839, Gould-Simpson Building, Monday 11–12, Wednesday 1–2, or by appointment.

Phone and email

Cinderella license

Download the latest version of Cinderella from http://cinderella.de/download. Then download license.cdy, open the file with Cinderella, and then quit Cinderella.