In the end, I was disappointed in the written portion of the final exam for the course. As the three sets of papers show, the exam did differentiat between the students somewhat. However, on the finer scale, I felt the exam did not do a good job of differentiating among the students. In particular, because problem 3 was dramatically more difficult than the problem it was ideally meant to be paired with, problem 7, the exam penalized students that chose to do problem 3. The other problem with the take home final from my perspective, was that by avoiding problem 3, students could avoid showing their problem solving skills. On the positive side, the final exam did do a good job of addressing the issue of transforming knowledge to pedagogical knowledge in questions 1 and 2. These problems showed that not all students were able to make this transformation, but on average students did. Below we shall analyze each of these.
The other issue is what does my experience with thsi final tell me for the future. The next time through the course, I will include problems like the first two, although I may require both, but I will try and put two problem solving problems on the exam, or force all students to do one problem solving type question. As mentioned in the introduction to this section, I find final exams to be somewhat problematic in advanced mathematics courses, which is part of the reason why I plan on keeping the final exam worth around 20% of the grade.
Next we analyze how the final played in to each of the course objectives:
Process:
Questions 1 and 2 directly addresses this, and looking at the three finals we have chosen to analyze, we see that the students show some success at this. For example, in analyzing the derivation of the quadratic equation, Neal points out
This derivation uses a lot of the “tricks” that are necessary to solve any equation: grouping terms in a useful way ...
Here he is using the same ideas of analyzing higher level mathematics to analyze the derivation of the quadratic formula and its usefulness for students.
Tom similarly analyzes the work on Dedekind cuts and sees a method of bringing it to the high school when he says:
... but one could mention (to high school students) the fact that if you were to define a cut of the line this way, you would have to include 1 and .99... in the same “half” of the line.
Again, he is looking at how the extremely technical idea of Dedekind cuts can inform his high school teaching.
Ellen's exam shows this less clearly. She certainly links each proof of irrationality to a topic covered in high school, but more in the sense that the proof could be done in the high school explicitly as you see in the statement
... the last proof would be helpful to show in a high school because it deals with limits and the completeness properties of real numbers.
While these are the topics covered in the proof, it is not the case that this would necessarily be good to do this proof in the high school as such, but rather it might be worthwhile to discuss the ideas underlying this proof.
The only problem that clearly deals with this goal is problem 5 on the final, although problems 1 and 2 have the potential to address this issue depending on the chosen topic. For example, in Tom's answer to problem 5, he sees the problem as connecting basic algebraic manipulations in creating the polynomial, abstract ring theory and dimension theory in explaining why he would expect the degree of the polynomial to be 6, and matrix theory in computing the inverse of the number. On the other hand, the problem (and Tom's answer) do not necessarily show these to be linked. In fact, Neal's answer on problem 5 makes this synthesis less clear since he avoids meshing dimension theory with explaining why he would expect the polynomial to be of degree 6.
On the other hand, Neal's problem 1, emphasizes different ideas that tie into solving the quadratic equation. In particular, he ties geometry to the algebraic technique. Moreover, if Neal had been able to come up with the third derivation, he could have linked graphical representations into the same theme. That Neal sees this is shown by his statement:
In these ways, this derivation can complement the geometric derivation very nicely so as to add more of this algebraic realm.
This goal is not really covered in the final, although the emphasis on solving a cubic (problem 4) and constructing numbers (problem 6).
Not covered in the final.
This is not clearly covered in the final exam, except that when students link things to the high school curriculum, it is possible to see evidence of them seeing this value, or at least seeing it as something to tell me, although in none of the three finals that we have looked at did students suggest a question that they might ask their students.
Problem 2 had the potential to address this specific goal, and Tom very clearly did so in his response to the problem when he said:
We have looked at numbers a (sic) ratio, length, solution to polynomials, but they have all failed in some aspect. Dedekind cuts on the other hand have not failed us.
Tom lists many of the various understandings we have shown. Moreover, while he discusses Dedekind cuts at length in this response, he also suggests at the end that he would not teach them to high school students, so he recognizes their weakness too.
Problem 5 also gives some inkling as to an understanding of some of rational numbers as functions on vector spaces, although using this in the general case is not particularly helpful. On the other hand, all three students saw how it could be helpful in finding an "algebraic" expression for the inverse of the element a in that problem.
This outcome had been more directly addressed on the midterm, and for several of the students, I addressed it explicitly in their oral final. Other students chose to present this in problem 2, although none of these three students did so.
Problem 4 clearly addreses this question, although the only one of the sample papers that addresses the derivation directly is Ellen's.
This outcome is directly assessed by problem 1. Neal's entire problem shows his learning on this outcome.
Problem 7 also addressed working with Dedekind cuts and set notations, and problem 2 in Tom's case also addressed the problem of the necessity of using Dedekind cuts. Of the sample exams, however, Tom was the only to provide any answer on problem 7. Moreover, he clearly showed an ability to work with the set theory notation in this problem. This was typical of the students that chose to do problem 7, although 2 students were not able to do so well.
No questoin on the final addressed this outcome.
No question addressed this outcome.
All in all, most outcomes were addressed in some way by the final, although
no student was required to respond on any particular problem. In the future,
I may require a different final exam set up to be sure that certain outcomes
can be assessed for all students, independent of how they choose problems.