The first day of class is an important one for setting the tone of the class, and in particular for setting the stage for the student projects. As in any class, I begin by introducing myself and discussing the syllabus. While discussing the syllabus, I talk at length about how this class will be different than many of the mathematics courses they have taken, and how it will be similar. Because the students know that it is a capstone specifically intended for prospective teachers, they may hope it will be a course that spends a lot of time discussing basic mathematics and what makes it difficult. While the course does this from time to time, that is not the purpose of the course, and I need to say this to them multiple times, so they gain the expectation that this is a mathematics course they are taking.
After telling them some about how the course will work, I next discuss the research projects. I tell them that they will be doing such a project so they can get a feeling for what it is like to do mathematical research more similar to what a mathematician does. I explicitly point out that most of the problems they have ever been assigned, they have been given about a week (or maybe two) to do. I then mention that I was given my first research question in mathematics 15 years ago, and that I still haven't solved it - but that I have written 3 papers on it. This allows me to talk about the nature of mathematical research, how you don't know if you will solve your research problem, but that if you attack the problem mathematically, you should get some worthwhile mathematical understanding out of it.
After a little more time spent on student questions, I then enter into the first class topic, fractions and rational numbers. I start this day with a discussion of what we mean by 1/2. The key point to be made here is that we mean many different things. If this discussion goes well, I will be able to get the students to suggest, one of two, the midpoint of a line of length 1 (or something of this sort), the decimal .5, and a pie with half of it shaded in. Usually, I have to push the students to suggest that we also mean 2/4, 3/6, etc. At this point, I can begin the discussion of what do we mean by a number, and in particular, set the course up since from these meanings we have the rational numbers, number as length, and number as decimal, all of which are sections of the course.
Once this is completed, we then end the day by talking about rational numbers as equivalence classes of fractions. This lecture is mostly intended to set the tone in the class. I want the students to start by feeling comfortable in throwing out ideas, and I want them to see that something as simple as 1/2 can lead to something complicated. (Thus, I usually discuss the problem of proving that the relation on ordered pairs of non-zero integers (a,b)~(c,d) if and only if ad=bc usually is a tough one for students to show is an equivalence relation).