Goals, Assessment and Course Materials.
Exams and supporting materials are provided below.
This is a senior level algebra course required for all pure math majors,
including those planning to teach high school. We do not offer a
second semester of abstract algebra at the undergraduate level, though
from time to time advanced undergraduates continue on to take our graduate
algebra course.
Content:
I planned to spend and spent the first third
of the semester discussing the integers and basic number theory and the
definition of a group, the next half semester on group theory, and the
final sixth on rings and fields. I covered Lagrange's Theorem and
cosets and the group structure on quotient groups. I did homomorphisms
and isomorphisms thoroughly and the First Isomorphism Theorem, focussing
on the bijection between the cosets of the kernel and the image of a homomorphism.
We covered the definition and various examples of rings (commutative and
noncommutative), integral domains, division rings and fields, as well as
ring homomorphisms.
The content of the earlier homework assignments (based in number theory) focussed on determining when an object is an element of a given set. Many problems in this course depend on the students ability to deal with sets; proving that various sets are subgroups or normal subgroups of a given group depend on it. Many of the problems involved proving that two sets are equal.
Reasoning and communication:
Grading: The assignments were graded subject to the following rules:
Next time: I used this grading scheme in a sophomore-level bridge course at Penn State (4 sections in 3 consecutive semesters) and will use some variation of it in all "proofs" courses, as long as I have time; there is a lot of grading time, but it is so effective, that I find it worthwhile.
Accommodating a variety of levels of student preparation: In many assignments, I assigned two sets of problems, one labeled Fundamental and the other labeled Challenging, letting the students choose which problems to complete (for example, they were asked to do 2 of 4 problems, with two labeled Fundamental and two labeled Challenging). The Fundamental problems were always straight forward applications of the definitions and theorems given, though they were not always easy. The students were told that they would be held responsible for the Fundamental problems on the exams. The Challenging problems required mathematical ideas beyond the definitions and theorems of the section, and often required a trick that I did not want to explain over and over in my office hours to students still struggling with the definitions. In messages on their homework, I encouraged approximately the top third of my students to do the Challenging problems, indicating that I expected them to do so; many of these students did not consider themselves strong students, and I hoped that they would rise to the challenge. Orally, I encouraged everyone to try all problems.
Upshot: About a fifth of the students always did the challenging problems (sometimes getting help from other professors). Maybe another fifth did the Fundamental problems, but came to ask for solutions to the Challenging problems because they had tried and failed to solve them. I believe it kept my advanced students engaged, and gave me a way to teach them a little deeper into algebra without losing the weak ones.
Next time: The first time I did this, I only did it on about 4 assignments. The second time I did it about every other assignment. I may do it even more frequently in the future, depending on how advanced my advanced students are. Both times I taught abstract algebra, I had quite a few very strong students who took advantage of the challenging problems; the frequency that I assign such problems depends on the skills of the students at hand.
Exams:
I provided a study guide one week before each
exam. On it I gave the students a list of theorems and definitions
followed by a break down of the exam. The breakdown was always essentially
the following.
Upshot: On the exams, I often asked for a definition followed by a problem using that definition. My thought was that I was forcing them to receive the partial credit for the problem by knowing the definitions. Students writing definitions out when fishing for partial credit clutters any attempt at a solution; this was my attempt to avoid that clutter. I gave a couple of quizzes over definitions (worth little) before the first exam, so that when I told them to memorize the definitions for the exam, they believed me. This was effective in motivating the students to learn the definitions as well as emphasize the importance of precise language in mathematics. Most students received all the credit for this part of the exams. The 10% of the exam attributed to a new definition served to communicate to the students what I mean by "learn to deal with a definition." Most of the goal of that part of the exam was to communicate the abstract approach that unifies the various fields taught in undergraduate mathematics, rather than to assess the students ability to actually deal with the definitions. I believe they began to get the point; on the final, as a group, they clearly approached this part of the exam as one of the easier problems. (One student remarked, "Well, it can't be that hard since it is totally new," a sentiment opposite to their original attitude.)
Next time: I plan to structure my exams in this way again next
semester, though I am considering changing the 40% to 35% and putting 5%
more weight on the students general knowledge (the section worth 50%).