I am sad to say I am leaving Livejournal. I have enjoyed my time here, but I have decided to move to wordpress.com. Please go to my new website, Symmetric Blog for details.

A student came to me today to discuss a conjecture that we made in class. First, I am impressed that my students are coming up with conjectures spontaneously. Second, I am impressed that some of the students are trying to prove them.

The conjecture was that for all elements g in a group G, the order of g divides the order of the group G. This student proved this by independently developing the notion of a coset, which I think is a difficult idea for students. We are going to learn about them next week in class, so I will find out then how difficult students find this idea.

Unfortunately, I think that this student had these skills before I ever had him as a student—I wish that I could take credit, but I cannot.

I typically write my own homework packets, rather than just selecting problems out of the book. I have several reasons for this:

1. Students are forced to use the space that I provide. Since there is a bigger problem students not using enough white space, I can make most students' assignments neater by forcing enough white space.
   
2. Students cannot simply look up the solution in the solution manual; they need to figure it out or speak with someone about it (note: I still have the flexibility to assign these problems if I want them to be able to see the solutions, and I usually have some problems of this sort).
   
3. I am not limited to the problems of a particular textbook: I can borrow/steal problems from other sources, and I can write my own.
   

Today I am going to focus on writing homework problems. Many of my problems are of the "Prove or disprove" variety, as opposed to the "Prove" variety. I think that this is important, since it forces the student to decide if the proposition is true. In a larger context, this forces students to think critically, since they are not told to blindly believe that the proposition is true. This is a habit I hope students build in my classes—to evaluate whether a statement is believable.

I have found a challenge in this, though: it is somewhat difficult for me to create "Prove or disprove" statements that the students are supposed to disprove. I create some based on common student misconceptions. For example, I would include a "Prove or disprove: (a+b)²=a²+b² for all real numbers a and b" if I were teaching algebra. However, it is generally difficult to create statements that "look" true, but are actually false. With luck, I will be proficient at this by the end of the semester.

Steven Strogatz just stated a series for the New York Times that explains mathematics to the lay-person. His first post, called From Fish to Infinity">From Fish to Infinity, discusses natural numbers and addition.

My college is now accepting applications from its students for summer research. I met with my first student today, and I fortunately have a handful of research questions that would work for him. However, I find this to be a very difficult task; heck, I find it difficult to come up with my own research questions.

Last semester, I started trying to fix my lack of questions by creating one question every other day. The question does not need to be good, but it has to be something that could theoretically be a research question. I was not as successful as I would have liked, although I did come up with one third of the questions I had hoped for. I used a couple of them with my student.

This is one of my weaknesses—I am not very good at creating questions. This makes me think that I should nurture this quality in my students, lest they end up like me. I think I will re-read The Art of Problem Posing, which was recommended to me by Juliana Belding.

I finished grading my first assignment for my abstract algebra class, and all is well. I was particularly concerned about how the cooperative homework component would be perceived by the students—I worry about student reaction whenever I change the usual school procedure. In particular, I was concerned that students would not like having other students responsible for their grade.

It turns out that my students enjoyed it. I took a straw poll in one of my classes when I first introduced this policy, and only one student (out of 13) said he was nervous about the system. Everyone else felt either positively or neutral about the policy.

I used some class time to give students an anonymous evaluation of how the first cooperative homework assignment went. I asked some open-ended questions ("What went well?," "What could your team improve upon?," and "What could Bret do to help?"). I also asked the students to rank their experience on a scale of 1 (Bad Experience) to 10 (Great Experience). The high score was 10, the low was 5 (only 1 occurrence), and I think that the mode was 9. There seems to be good support for this policy.

Derek Bruff tweeted about a post on teachingcollegemath.com. The post is about how a student's concept of what mathematics is is correlated to his/her study habits and success. In it, she suggests pre- and post-testing students on their conceptions to help evaluate the worth of a teaching method.

The first day of abstract algebra went reasonably well. The students seem great (as always), and I did roughly what I expected to do. Here is what I accomplished:

1. We started with a bingo-type icebreaker activity. At the very least, this avoids the creepy pre-semester silence that happens immediately before some classes begin.
   
2. The students cut out all of their models for groups.
   
3. Everyone introduced themselves.
   
4. I gave an introduction to abstract algebra, including a list of questions that abstract algebra helps answer.
   
5. We went through the "rules" to all of the models.
   

So far, so good. I wish I could have got to some of the syllabus, since that is where I discuss the homework policy. However, I explained this a bit when I emailed out the first homework assignment. Also, I prefer to concentrate on mathematics more than policies on the first day of class.

The first homework assignment was to determine the number of elements that are in each group for each model.

I attempt to add a new, proven feature to my teaching each year. This semester, I am concentrating on adding true cooperative learning to my classes.

Any sort of learning can be categorized into one of three categories: "Individual learning," "competitive learning," and "cooperative learning." An individual learning environment is where one student's learning is not affected by any other student's learning; every school where I have worked has had predominantly (solely?) a focus on individual learning (my courses included). A competitive learning environment occurs where one student succeeds at the expense of the other. An example of a policy that encourages competitive learning is the true grading curve, where only 10% of the class could earn an A. A cooperative learning environment occurs when students succeed or fail together.

Cooperative learning is more than simply using group work. Two aspects of cooperative learning that I have usually not included with run-of-the-mill group work are positive interdependence and individual accountability. Positive interdependence means that the group succeeds and fails together—there is no room for some of the group members to succeed while others fail. Individual accountability means that I have developed policies so that students cannot just let others do all of the work.

I am implementing cooperative learning policies in my course because the psychology research overwhelmingly shows that students learn more in cooperative environments than individual and competitive environments (individual environments tend to improve learning more than competitive). This is really the only reason I need, but the research also shows that students who have experienced true cooperative environments strongly prefer cooperative learning environments to individual or competitive environments.

I am going to introduce cooperation into my classroom through three policies:

1. Students will work cooperatively on homework. I will assign them to groups of 3-4, collect all assignments from the group, randomly select one of the papers, and give the grade of that one randomly selected paper to the entire group. Of course, the students will be instructed to meet to make sure that all of their papers are correct.
   
   This policy promotes a positive interdependence by giving everyone in the group the same grade. This encourages students to teach each other to make sure that they all understand the material. There is individual accountability because any one of the group members' papers could be selected for grading; a slacker will cause the entire group to do poorly.
   
   (Note: There will also be individual, rather than cooperative, homework. There is definitely a place for individualism).
   
   
2. Students will have a similar experience for each midterm. I will again assign groups (likely the same groups from the previous homework assignment), give them an exam problem in advance, and then ask the students that question on the in-class portion of the midterm. Each group will get a grade based on how the entire group does. I have not yet decided on the method for determine which one grade all group members receive (feedback would be appreciated), but options are: randomly selected a question to grade, averaging the group members' scores, using the lowest grade, or using the second lowest grade.
   
3. Students will be creating a textbook for the class. This idea is from Patrick Bahls. This will be a lower stakes cooperative task, since I will not be giving the entire class a grade depending on how the students do. Rather, it will be a (hopefully) enjoyable task that promotes learning.
   

I welcome comments, particularly on the following two issues:

1. How should I grade the cooperative homework? I strongly favor de-emphasizing grades, and I have previously been give an "All or nothing" grade with re-writes. However, I am afraid that I will not be able to grade everything if this happens (I am allowing unlimited re-writes on the individual homework assignments). I have considered a 0-3 scale for each problem, but that does not give them the feedback I would like. I really have not thought of a solution that I am happy with—please help.
   
2. How should I score the cooperative question on the midterms? Average? Randomly selected? Lowest score?
   

I am teaching abstract algebra next semester, and I have decided to focus on 9 10 finite groups at the beginning of the semester. These groups are:

1. Cyclic group of order 3
   
2. Cyclic group of order 6
   
3. Cyclic group of order 7
   
4. Dihedral group of order 6
   
5. Dihedral group of order 8
   
6. Symmetric group of order 6
   
7. Symmetric group of order 24
   
8. Alternating group of order 12
   
9. The quaternions
   
10. (edit): The direct product of two cyclic groups of order 2 (thanks to Jill for recognizing my omission)
    

I have chosen these particular groups because:

1. They have relatively small orders, so are relatively easy to understand.
   
2. They represent a variety of different "types" of groups (note: I understand that there are no non-solvable groups).
   
3. They will make the ideas of "subgroup," "normal subgroup," "quotient group," "isomorphism," and "homomorphism" easier to teach.
   
4. Except for the quaternions, they all have physical representations for the students to study.
   

I have chosen to spend time concentrating on a handful of groups because:

1. They will make abstract algebra less abstract. The physical representations will (hopefully) give the students a way of accessing the group structure. I have created physical representations for the students to use; see my website for details (note that I have borrowed—stolen, really—liberally from Patrick Bahls for the LaTeX section of this page. Also, my syllabus is only a draft in two senses: first, I might revise it more before classes start. Second, I intentionally let the students decide on many of the course policies, so the final draft will not be ready for another couple of weeks).
   
2. Being very familiar with a handful of groups is the best way of producing counterexamples to conjectures; in particular, it seems like Alt(4) or the quaternions is the smallest counterexample for about 90% of false conjectures.
   

Of course, it is all just a hypothesis that having an intimate understanding of 9 finite groups will help students learn. I look forward to determining if the hypothesis is true.