Posts Tagged ‘collaboration’

Conferences are important

June 2, 2011

I am not a great mathematician. I have many deficiencies. The deficiency I am going to focus on today is that I am not very good at generating my own research questions. But I have some ideas on how to get better below.

(Note: I am not a great mathematician, but I am an okay mathematician. This is because I have some amount of the most important quality for a mathematician: tenacity. I am not work efficiently and I may not have the background I should, but I am willing to sit down most every day and work. This is really huge).

First, I went to the Zassenhaus Group Theory Conference last weekend (I am one of the giants in the back of the picture). I was reminded how important these conferences are. For one, I got a lot of ideas for research questions from listening to other presenters speak. Some of these ideas were given to me directly by the speaker, and other ideas were tangential to what the presenter was actually discussing. But I also was a presenter, and I presented on a problem that I am stuck on. I received several great suggestions on how to proceed.

Second, I have been working with a collaborator for the past semester on a problem. Something clicked today about one problem I have: I search for a solution a little too directly. But reading my collaborator’s ideas, I realize that he plays with the ideas much more, collecting a bunch of ideas that may or may not be useful to the problem at hand. This seems like it would be enormously useful, and I am shocked that I do not do it already.

My goal is to play more. This is what I ask my students to do, and they sometimes just do not get it. Apparently, neither do I. I am hoping that

  1. I can work past this and
  2. I can learn what it took for me to work past this, so that I can help me students to learn to play more.
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Midterm Evaluations

March 11, 2010

I am pleased to say that I have been in the habit of offering midterm evaluations to my students for the past couple of years. I have always meant to hand them out, but I sometimes got lazy. No longer.

I have found that there are two advantages to these evaluations. First, I learn more about the class. I can learned how effective things have been, and I get a sense of how the students feel about the class. Second, the students have said they feel better about the course by my offering a chance to evaluate it. This is not surprising—everyone likes being listened to, and few people are listened to less than a college student.

One nice thing is that I can customize my evaluations to my course (as opposed to the evaluations that many schools require, which usually involves a bubble sheet and generic comments like “Bret rocks!” or “Bret sucks!”).

Here are the questions I asked this time:

  1. How helpful was the introduction of our 10 “toy” groups (S_3, D_4, quaternions, etc) to your learning? Should we have spent more time on these, less time, or did we spend the correct amount of time getting familiar with these groups.
  2. How useful is it when we go over proofs that people submitted for individual homework? How much do you learn from comparing these proofs?
  3. Has Bret provided enough support on \LaTeX for you to use it effectively?
  4. What are the benefits and drawbacks of our in-class exam format of “no surprises?” Would it be better to add a problem that you have not yet seen? Would it be better to add more “cooperative” questions? Should we leave the format the same?
  5. How useful has the feedback on the individual homework been?
  6. How could the in-class lecture time be improved? Should we be spending our in-class time differently?
  7. How effective have the cooperative groups been in helping you learn the material? Would you guess that you have learned more, less, or the same amount that you would have if you did all of the homework on your own?
  8. I am planning on following the textbook (Gallian) more closely from now on. How likely would you be to pre-read if I told you which section of the text would be covered in the next lecture?
  9. Overall, how much do you feel like you are learning in this class?
  10. What other suggestions do you have?

Here is a brief summary of student responses for these questions:

  1. Somewhere between “helpful” and “very helpful.” We spent roughly the right amount of time on them.
  2. Somewhere between “useful” and “very useful.” One student suggested that I have the students read through the proofs at home to save on class time. This was a brilliant suggestion, and I am going to change my course accordingly.
  3. Yes. Google is also very helpful.
  4. Most people liked the exam format, although some wanted more “surprise” computational questions. We will discuss this before the next midterm.
  5. The feedback has been helpful.
  6. Sloooooooooow doooooooown. I apparently go through proofs quickly. This response played a large role in my decision to start using Beamer for my classes. So far, it has been working well—a straw poll of my students suggests that we are now moving at an appropriate pace.
  7. “Very helpful” to “extremely helpful,” with perhaps five exceptions, who said that they learned an equal amount to if they had been working individually. But of those five, three said that they really did not meet much with their cooperative team. It seems like those who work with teams almost always get a lot out of it.
  8. Some said they would read ahead, some said they would. This information is embedded in my Beamer slides, so it is there for the taking.
  9. “The usual amount” to “an unbelievable amount.” No one suggested that they are not learning much.
  10. Sloooooooooow doooooooown.

One thing that I did not do this time was to ask questions that make students reflect on how they are responsible. I usually ask a question like, “What can Bret do to help you learn more?”. I follow this with “What could you be doing to help you learn more?”. I omitted it this time due to space considerations; also, I ask them a similar question every cycle (roughly a week) on their homework (so I guess I have mini-evaluations once per cycle).

Finally, I feel like I have the responsibility to report back to the class what the students said in their evaluations. This took the form of a three minute class presentation.

Cooperative Learning

January 9, 2010

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?

Making Algebra Less “Abstract”

January 8, 2010

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.

Course Collaboration Project—Part 4 (Homework Policies)

January 2, 2010

With goals and content in mind, I can now focus on how to best get the students to learn the material. One aspect of this is homework.

This is a proof-based course. My theory is that there are three things that need to happen if you are going to learn how to successfully do proofs:

  1. You must read a lot of proofs.
  2. You must write a lot of proofs.
  3. You must analyze the proofs you read.

The third point will largely be done in class, since I do not think I can expect students to know how to analyze proofs. I have several ideas for formats that will allow the students to read and write a lot of proofs:

  1. I might have students evaluate their own homework. Students would give me a photocopy of their homework, but keep the original for themselves. I would create a solution key/rubric. They would use the rubric to evaluate the homework outside of class; perhaps students could comment on the “differences, omissions, and additions” of their proofs compared to mine, and comment on how important these differences/omissions/additions are. Students would email me their evaluation, noting the strengths and weakness of their proofs. I would spot-check their work by using the photocopied homework.
  2. I might allow students to resubmit unlimited attempts on homework problems to me. Problems would have two possible grades: “Near-perfect” and “Incomplete.” Students would resubmit until they received a grade of “Near-perfect.” I would provided detailed comments on their proofs to help them with the next draft.
  3. I might have students evaluate other students’ proofs as part of their homework. I would create a packet of 3-5 student attempts at proofs. Students would be expected to contribute to class discussions on the proofs.
  4. I might have “homework committees.” This idea comes from from Patrick Bahls. Here, a committee of 2-3 students would look 1-2 selected problems from the homework assignment. This committee would look at all of the student solutions that were submitted, categorize the different approaches that students used, and discuss the relative strengths, weaknesses, and validity of each approach. The committee would give a short summary of what students did in class.

I think a combination of these approaches would work well to get students to read, write, and analyze a variety of proofs. I am leaning toward a combination of the first three approaches. I am planning on giving 3-5 problems that the students will self-evaluate each “cycle” (6 school days=1 cycle). Students would additionally get 1-2 problems that students would be allowed to revise as many times as needed. I would use these revisable problems to create the packets for students to evaluate. On the fourth approach, I am in agreement with Patrick that the homework committees might create more overhead than I care to handle.

I am strongly considering following Patrick’s lead and teaching the class LaTeX. I would then require students to use LaTeX on the revisable homework, which would make their revisions easier.

The one point that have not settled on: I would like students to give presentations. I have not yet determine how this should relate to the homework. I welcome input on how I should organize the course—on the subject of presentations, or any other aspect of homework.

Course Collaboration Project—Part 3 (Content)

December 30, 2009

For mostly my benefit, I will discuss the content for my abstract algebra course. I will also determine what I will emphasize and de-emphasized. The unfortunate fact is that I only have one semester; 36 class periods; 2520 minutes.

The chapters that are typically covered are:

  1. Introduction to Groups
  2. Groups
  3. Finite Groups; Subgroups
  4. Cyclic Groups
  5. Permutation Groups
  6. Isomorphisms
  7. Cosets and Lagrange’s Theorem
  8. External Direct Products
  9. Normal Subgroups and Factor Groups
  10. Group Homomorphisms
  11. Fundamental Theorem of Abelian Groups
  12. Introduction to Rings
  13. Integral Domains
  14. Ideals and Factor Rings
  15. Ring Homomorphisms

If there is time left, I will cover Polynomial Rings, Factorization of Polynomials, and Divisibility of Integral Domains. There will not be time left. If things go well in the first part of the semester, maybe I would do some Sylow Theory. In fact, I might have a tough time keeping myself from doing Sylow Theory, regardless of the amount of time we have.

This is a full semester. If I lectured all semester long, I think that I would be able to finish with just a little bit of time left. Since I am not going to lecture, this means that decisions will have to be made. Here are my basic ideas:

  1. I am planning on starting the semester by introducing several hands-on examples of groups: several cyclic groups, several dihedral groups, a couple of symmetric groups, and the alternating group on 4 letters. I also hope to introduce the quaternions in an easy-to-understand way. This should make the first four chapters much easier to understand. By the end of the semester, I hope that my students are experts in 8-9 different groups.
  2. I will de-emphasize direct products and the proof of the Fundamental Theorem of Abelian Groups. This should save some time.
  3. I will try to tie the ring theory to high school ideas as much as possible to ground it.

Course Collaboration Project—Part 2 (Goals)

December 27, 2009

This is the second installment in a series of posts on collaboration between Patrick Bahls and me. Today’s topic is “goals.”

I am astounded how frequently professors plan courses without expliciting stating the goals for the course. I include myself in this group—I certainly did not do this for multivariable calculus last semester. Still, I rarely hear people discuss this aspect.

Below are a list of my goals for any course I teach. I hope to reference each of these when creating the course—any feature that goes into the course should support one of these goals, and (ideally) all of these goals will be supported. Note that these goals are a variation of Deborah Meier’s goals, although they are not identical. The first goal will have to do with facts, while the others will be habits. My experience is that, without making a concerted effort to think about goals, professors only concentrate on the first, “facty” goal.

  1. Students should learn about the content specific to the course. In my case, it would be “abstract algebra.” My next post will be on this goal.
  2. Students should learn good communication skills. Students should be able to write and speak clearly and concisely. They should also be able to read and listen to others. They should be in the habit of refining their communication regularly to improve communication (i.e. there should be at least two drafts of any sort of formal communication).
  3. Students should be in the habit of using and requiring evidence. Students should justify any assertion they make, and students should require that others do the same (this is my favorite goal).
  4. Students should be in the habit of considering perspectives. Students should consider how other people think. They do not necessarily need to agree with others’ perspectives, but they should recognize that and how other people may view things differently (this can be difficult to achieve in a mathematics class, but it is far from impossible).
  5. Students should be in the habit of looking for connections. Students should automatically attempt to find similarities among different ideas they have learned.
  6. Students should be in the habit of applying supposition. Rather than only considering what has been presented, students should regularly “tweak” ideas to see how things change. “Suppose not A but rather B—what happens then?”

These goals are my current conception of what is important in education. I will plan to incorporate all of these goals in my course, and I will plan to omit other aspects that are not important.

Course Collaboration Project—Part I

December 26, 2009

I am pleased to announce that I will be collaborating with Patrick Bahls on our spring courses. I will be teaching an abstract algebra course; Patrick will be teaching a topology course.

This will largely be a pedagogical venture. We hope to give each other ideas on the course set-ups, and we hope to critique each other’s ideas. This will be done by a series of postings at our respective weblogs, along with a minimal amount private emails through Facebook (note: I am “Cogswell” on his weblog). We would like to keep this process as transparent as possible.

A couple of my upcoming posts will be on “homework committees” and “writing a course textbook.” These are ideas that Patrick has previously done; I will post on them to help me understand how they might be implemented.

I believe that collaboration is the best way to innovate. Communicating ideas to other people forces me to clarify my thoughts, hear others’ perspectives, and have more fun. This is true of both teaching and research. Patrick and I have similar goals for our students, and I am grateful to have found someone with whom to share ideas.