Number Theory for Teachers

This summer I'm taking a course in Number Theory designed for secondary school math teachers, specifically those of us who went through the Traders-to-Teachers program at Montclair State University. As such, it will cover a subset of the topics usually found in introductory number theory courses, with, I suspect, less rigor. From the syllabus: "Our class discussions will relate number theory topics to middle and high school level curriculum."

At the beginning of the first class the instructor asked what we hoped to get out the course. I kept silent, as my situation - unemployed - and interests differ from the rest of the class. However, I continued to ponder the question and hope to answer it here. Aside from the obvious and trivial response - to satisfy a requirement for full teacher certification - here's what I want to get out of the course:

• I want to get over my "abstractophobia": fear of math that can't be easily visualized. In college I aced three semesters of calculus, but stumbled when I tried abstract algebra and topology. I didn't pursue advanced math after that (advanced math being any subject whose textbook didn't have a lot of illustrations). I did well in probability and mathematical statistics because I could "see" their applications. This preference has persisted: last year in the Traders-to-Teachers program I was more comfortable with the geometry section that with the algebra section. 
• A somewhat lesser goal is to accommodate my visual preferences by coming up with illustrations for number theory concepts. This means extra work with Geometers Sketchpad, Geogebra and other drawing software. Along with this effort, I hope to explore different ways of writing mathematics, meaning putting math into digital form for storage, presentation, online instruction, and communication. By doing this, I can use my downtime due to unemployment to add various software packages to my skillset.
• I also want to put this available time to use by researching the historical development of topics that are covered in class. I think that some of the difficulties individuals have in learning math, such as the acceptance of negatives, irrationals and √-1 as numbers, are reflected in the historical development of number. Certainly it will be help me to teach math if I can provide a rich context that includes history and applications.
• Finally, I want to pursue number theory with more rigor and depth than what is required in this course. This means, for example, attempting the starred problems in the text, and supplementing the text with others, such as Naive Set Theory  by Paul Halmos, which I've borrowed from the library. Again, I have the time and interest; I believe that as a teacher I need to reach a level of understanding of the subject that is measurably greater that I expect from my students.