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.
I always lectured the entire class period, which
lasted one hour and fifteen minutes and met twice weekly. I had 34
students. Of these, 2 students dropped, there were 1 F, 3 D's, 10 C's,
7 B's and 11 A's (1 of which was a B students that received a high A on
the final exam). My colleges informed me that 6 of those 11
students are the strongest this department has seen in years, that I was
lucky to have them, and not to expect so many A's in general.
Content:
I planned to spend and spent the first third
of the semester discussing the integers and basic number theory, the next
half semester on group theory, and the final sixth on rings and fields.
I covered Lagrange's Theorem and cosets, though not to the group structure
on quotient groups. I did homomorphisms and isomorphisms thoroughly and
the First Isomorphism Theorem, though since we did not have a group structure
on the quotient, we just showed 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.
Next time: I will cut the number theory part by half and begin group theory seriously by the second sixth of the semester. I used the number theory because I wanted to get them off to a good start with easier problems (and I wanted to use the number theory freely later on), but I would like to try to give the same writing and reasoning instruction through the group theory in order to get through more content. I hope to add the group structure on quotient groups and the rest of the isomorphism theorem, without losing any of the topics listed above.
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: From time to time, 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: I think I did this only on about 4 assignments. Next time, I plan to try to do it much more frequently, maybe once per week, depending on the students. Last time I started this about a month into the course, after the wide range of preparation that the students had became apparent. Next time, I may start it earlier so that I can push my strongest students a little farther.
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 and 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 had much the desired effect. On the first exam most students did not receive even half credit on this part. A few students with lower grades did complete this section, and I made an effort to help those students individually to understand how to use their apparent reasoning skills to solve other problems. In one rewarding case, a student who received a low B but solved that part of the exam responded to my prodding and shot to the top of the class, getting a nearly perfect score on the final. On the second exam about two thirds of the students got at least 7 out of the 10% on that problem, and on the final, as a group they clearly approached it as one of the easier problems. (One student remarked, "Well, it can't be that hard since it is totally new.")
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%).