At the beginning of the class, I see my job as preparing the students for my teaching style and my class. The students do not necessarily believe that an advanced mathematics class has anything to tell them about teaching at the secondary level. Thus, one main goal in the early days of the class is to provide the students with evidence that this class can help them be better teachers. A second goal is to break students of the habit of thinking of mathematics problems as having unique answers that can always be arrived at in the same way. Finally, I want to introduce students to the way I would like to see them write up proofs. The first day of class, the students are assigned to send me an e-mail math autobiography and answer various questions so that I have an idea about their background. The second day of class then begins with me handing out the first real homework set (with the problems mostly coming from the book) and having them work on it in groups.
The first problem of the set (although not specifically handed in) is for the students to write a program for their graphing calculator that does long division. That is, it will give them as many digits of the decimal expansion of a fraction as you might desire. The first question they must answer is to find the 57th digit in the decimal expansion of 1/97. The other questions from the text ask them to find the periods of the decimal expansions of 1/n for all integers n between 1 and 60, prove that various square roots are irrational, and make conjectures about the relationship of the period of 1/n and the value of n. I then have the students respond to an algebra word problem that appears to require fractions, and after they complete it, they are to watch a non-math major work the problem and comment on how the problems of equivalence of fractions show up for the non-expert.
The second problem set builds off of the first. Like the first set, I let the students work on the set in groups the day I hand it out. The goal of the set was to further the students' knowledge about rational and irrational numbers, encourage them in making conjectures, give them an example of a higher-order proof skills in use, and finally to help them understand problems of fractions. Unlike the first set, there is less direct connection to teaching high school, although it still rests in the background.