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.
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.
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.
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.
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.
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.
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).
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.
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.
Definitely satisfied by the research projects.
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.