At the beginning of the course, I listed for myself several desired outcomes. However, as happens in most courses that I teach, I modified the list as the course went on. Moreover, I would argue that some of the desired outcomes for the course, were never explicitly stated, although in reflection they can be recognized. There are myriad reasons for changing outcomes, the main one being a recognition of knowledge that the students lack. My initial list, was made early on in the course from my course diary. As a habit, I do not include all of my desired outcomes on the course syllabus, as I believe that some outcomes are better enacted if the students come to see the importance of these outcomes on their own.

**Initial List: **(compiled from course notes). My initial list of desired
outcomes broke into two pieces: what I have termed process goals and content
goals, although these words should be looked at pretty loosely. (I dislike
the dichotomy of syntactic knowledge and substantive knowledge found in Schwab.)
In the course syllabus, only the process goals
are listed, although the sequence of topics in the course are listed. Clicking
on the assessment link at the end of each, will lead you to my assessment
of whether this goal was achieved and the reasons for this assessment.

**Process:**- For students to be able to look at advanced mathematics topics and see how they are reflected in the high school curriculum and how these advanced topics might inform their teaching of high school mathematics.
- For students to gain a deeper understanding of themes of mathematics, and how various subjects in mathematics can relate to a common theme.
- For students to see some of the questions that drove the development of mathematics answered in the context of the course.
- For students to successfully experience mathematical research on their own.
- For students to learn the value of asking richer mathematical questions.

**Content**- Students will learn various ways of understanding numbers and the strengths and weaknesses of each.
- Students will have familiarity of the proofs of the geometric impossibility theorems and be able to show a rudimentary understanding of the flow of the proofs.
- Students will learn some theory of equations:
- To solve a cubic equation and derive its solution (modulo some complex number theory), and
- Learn multiple ways to derive the quadratic equation and how each might help in teaching.

- Students will learn the fundamental reason for the necessity of defining
the real numbers via Dedekind cuts (or some other analytic property) so
as to learn about:
- infinite decimals, and
- working with set theory notation.

- Students will learn proofs of some of the fundamental properties of the numbers pi and e.
- Each student shall participate in solving a mini-research project that is related to the high school curriculum.

In truth, I would say that all of these outcomes were outcomes that I desired by the end of the course, however, I would say that there were two additional outcomes.

- For students to gain a deeper understanding of how and why to use technology in mathematical problem solving and developing understanding.
- For students to gain a deeper understanding of how mathematics is done.

The first of these was added after the second day of class (see the first snapshot of the course), when I discovered that some of the students had never used technology to solve problems.

The second was there from the beginning of the course (and in fact from about the 2nd time the course ran), but had never been explicitly codified. In fact, this last outcome is probably the central process theme of the class.