My basic feeling with final exams in upper division mathematics courses is that timed exams do not give me the type of information I want, and they do not promote student learning. I find this because I have difficulty asking questions that are different enough from the assignments that they do not simply reward the students that memorize proofs, but can be done by most B level students in the time period. One can solve this somewhat by asking questions that are more like essay questions, but the benefit here is minimal in terms of requiring a timed exam, since a take home exam can allow for students to think longer (and ideally better) about their answers.
The benefit to a timed midterm examination in my view is that it forces the students to review everything we have done to date, and that allows me to build on that immediately following the midterm. That is, I can try and make them reflect on the early material, which is relatively fresh in their minds again. On the final exam, however, this benefit is lost since I will not be teaching them again soon. Also, during the midterm, my schedule often makes finding time for grading a longer take-home exam more difficult.
When I give take-home finals, however, I like to give an oral portion too, so that I can probe answers (and failures to answer) more deeply, and so I can see what the depth of the student's understanding is. Thus, I chose to have a take-home exam paired up with an oral examination for the final.
The grouping of the questions (one of the first two and four of the last five) is to allow students a choice of questions so that they don't have to be able to answer everything, but also to make sure that they choose certain types of questions. The first two questions were supposed to be questions requiring synthesis of a major portion of the class, where I wanted them to synthesize everything covered in that topic. The last five questions were meant to be more direct procedure questions, but requiring a certain level of problem solving. Unfortunately, however, there was not a question that was well-paired with question number 3, and consequently most students avoided number 3, which was the only question that ended up requiring deep problem solving on the part of the students.
Question 1 is a standard type of question I like to ask in this class. While developing the solution of the general cubic in the class, we give three different derivations of the quadratic equation, something all of the students recognize that they will teach. We don't, however, spend a lot of time discussing why you would teach each of these derivations, and when you might teach them. On the other hand, I do tell the students why I am choosing to do each derivation, and how all of them tie together with how I teach the cubic solution. Thus, I am asking them to synthesize the reasoning for how we present the cubic with the derivations of the quadratic equation.
Question 2 is a pure synthesis question. I want them to summarize what they learned about the topic, but then they need to pick and choose what about the topic is important.
Question 3 is based in problem solving, framed as a problem a teacher might encounter. This question asks the students to think about the period of the product of two fractions in terms of the period of each of the terms of the product. The best possible theorem that I know a proof for comes from advanced number theory and an analysis of the groups of multiplicative units in modular arithmetic. That material is beyond most of the students in this class. On the other hand, all of the students should be capable of making some good conjectures and proving some good results that do not require quite as high level mathematics. In any case, however, this is probably the hardest question on the final.
Question 4 asks them to turn around the work from class on solving cubic equations and create an equation that satisfies the property that the solution method does not appear to give a rational solution, even though there is such a solution. Creating the equation is quite easy if you know how to do it, but it is quite difficult if you don't. Thus, the question gives a pretty good test of the students baseline understanding of solving a cubic. It then asks the students to work through the solution without simply using the derived equations since actually remembering the formula for the roots of a cubic is difficult (and frankly silly). This question is important for teachers to consider because on a day to day basis, they will need to create examples of polynomials having certain properties if they are not to be tied to closely to a text. For example, teachers should be able to quickly come up with polynomials having a given number as a root. The first half of this problem asks them to complete just such a task, except that there are certain additional requirements on the polynomial, just as there might be in a classroom environment, when they may need to do it on the fly.
Question 5, on finding a polynomial with the square root of 2 plus the cube root of 3 as a root, is asking them to apply material from class, but then further asks them to analyze why you would expect to need a 6th degree polynomial. This first tested the skill of how you create polynomials with certain properties, something they will need to do on a day-to-day basis in algebra 1 and algebra 2 classes. This question was also supposed to force them to think about the definition of the algebraic numbers and to think about the algebraic numbers as a vector space over the rational numbers, but it didn't do that particularly well.
Question 6, which asks them to explain how to construct an “ugly” number was meant to be a basic question that everyone should be able to answer. The students were to explain how to construct the number clearly and to justify why the construction worked. Ideally, this problem would let the students show that they understood the basic construction techniques, and also allow them to show that they could write down descriptions of these in clear ways.
Question 7, on Dedekind cuts, was meant to force them to encounter the idea of set notation in Dedekind cuts and to prove that the set they had really was a Dedekind cut. Unfortunately, this was too close to something done in the book, and it did not really challenge students. I say unfortunately, because this was the question that was supposed to have balanced question 3.
The oral portion of the final had me ask the students about one or two questions on their final, and then ask for a synthesis of some topic (that I chose). For students that had very clear grades, I mostly tried to make the questions help them learn better about the topics, but for students like Neal that had borderline grades, I used the oral portion to make decisions by asking serious questions to press them, and then help me make the borderline decisions.