What Was Accomplished

In looking at the class as a whole, we first must ask what was accomplished: did students reach the desired outcomes, did the outcomes change for the class, and what unexpected outcomes (both good and bad) came about. We begin by looking at the original outcomes.

**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.
Again, the information from the final exam and the analysis on Neal seem to confirm that students did get a better understanding. Looking at one of Alan's responses to an interview question of what he got out of the course:

*(Now) when I do a homework problem I try to get an understanding of how does that relate to the real numbers, does that make sense, and how does it relate to the things I know.*We see that he is looking for connections. Brad had a similar comment regarding the projects

*Well just in general, we had to use, as I am sure everyone did, many different aspects of math from a bunch of different classes and I have never had that in any other math class before, and that's what we need as future math teachers because we need to show to our students, you know, this relates to everything, you're gonna have to remember this, use what you know, … yeah, use what you know to discover what you don't know, and I really found that in ours.*Brad went further in his statement to point out that he saw this as necessary as a future teacher also. Based on these, I feel that all students gained a deeper understanding here.

- 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
This is probably the hardest to measure. If one restricts to asking whether students concept of what makes a good mathematics problem changed, the preference survey results suggest that the definition of a good problem is richer after the class. Whether this carries over to them learning the value of asking richer questions is not as clear.

Based on the information from both the final exam section and the analysis of Neal, I think it is clear that the first outcome was addressed by the class, and that it was reached to some extent by all students in the class.

Certainly by covering the geometric impossibility theorems, the solvability of equations, the existence of transcendental numbers, and issues concerning p and e, this outcome was satisfied.

Four of the five groups had successful experiences, and even the fifth had some success in that they did an investigation and came to some conclusions. Thus I felt it was satisfied. For more evidence, refer to the project artifact.

**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.
Only about half of the students chose to do this problem on the final, but most responses showed some understanding of how they might help in teaching (the answers on this were the best of any class I have taught previously).

- 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.
As commented on in the final exam artifact, 10 of 12 students were able to do the problem on the final relating to this outcome.

- 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.
Definitely satisfied by the research projects.

I believe the students did learn this, but it is hard to support using the available evidence in this portfolio. Tom certainly appears to have learned this from his final, but he was among the top two students in the class. From the group interview, Neal said:

I think in a sense what I came away with the class is a better number sense. Construction of the real number line and what things fit whether, where certain numbers come from.Several of John's comments also hint that he might have been gaining an understanding of the various ways of understanding numbers, but again the linkage is tentative.

This goal appears to have been satisfied based on the oral portion of the final.

The final exam showed that almost all students were able to solve this problem.

The students did well on the homework addressing these questions (not included in the portfolio). Also, several of the projects show good student work on understandings of e and of p. In particular, the ten cards problem discusses the number e at some length, and does a good job with it from the Taylor's series perspective.

The positive outcomes that I added in the class were two regarding student use of technology for investigation, which the entire class did. Most project groups needed to use technology to get started, and the first homework set certainly added on to this. Finally, if one looks at the spirals project, one can explicitly see technology playing a major role. The other outcome dealt with understanding how math is done. Again, this became a central tenet of the projects, and Alan upon being asked what relationship he saw between the class and the projects, quickly responded:

They were both about how mathematics is done.

Other students didn't recognize this, but allowed that they did learn that from the class.

In addition to evaluating the outcomes that I was after, it is useful to look at other outcomes, some negative, some positive. One negative outcome, was that two students left feeling quite negative towards group work. Jill, felt this way previous to the class, but Mary did not. Other negative effects of the class, include leaving three students (Ron, Ellen, and Teri) apparently convinced that they could not really do mathematics. In all three cases, I saw capability at doing a research project, but in each of these cases, the stronger students in the groups left these students feeling slightly out of place.

On the other hand, there were other interesting positive outcomes. Ron stated near the end of the interview

That's it! That is what I finally learned from the class that I did differently, I learned to talk more. Cause after 496, I am in 425 right now, I don't have anyone except the professor, which is limited hours to talk with. With Neal I did that in 496 and I was really good, help me learn, I realized verbal discourse helps me retain more of the knowledge.

Brad's comment was:

That's what I meant, that made me feel good, like we could actually come up with something, be creative, with something like that and you know gonna be a teacher trained at getting the kids to see all these different ways you can do it, why things happen, why you do it this way, and stuff like that is really important, I got that from this course.

Namely, he saw the importance of teaching that allows for different ways of doing things. In some ways, this outcome while not specific to the goals of the course is the kind of thing I hope for a teacher to do when they are in the classroom.