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.

...and Division by Zero, by Dave Richeson.

I just found another interesting pedagogical weblog: Casting Out Nines, by Robert Talbert.

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.

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.
   

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.

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.

Here are two new math weblogs I have found: Secret Blogging Seminar and God Plays Dice.

I have a new favorite math weblog: Patrick Bahls's Change of Basis. I have had the fortune of meeting Patrick on a couple of occasions, and he struck me as being a very thoughtful professor. The weblog is as thoughtful as the person.

I use computational algebra systems in my research. I previously used Magma, but I now use GAP. I find that the documentation for Magma is nicer. I find that GAP is cheaper (free), and I like the fact that it is open source. The functionality is similar in both systems, although there are minor differences.

I use these systems as a lab of sorts. I study objects called "groups." GAP and Magma give me an environment where I can study the properties of groups. I liken this to how a scientist works: she observes something in a lab, thinks of a question, creates a hypothesis, tests the hypothesis, and then either has a result or creates a new hypothesis.

I find that GAP and Magma are useful in formulating questions and testing hypotheses—they are my lab. While a scientist could think of a question while observing chimps, this is tougher to do with abstract objects. GAP and Magma make this possible, and can further test the hypothesis by checking many examples.

Of course, once this is done, I need to leave the computer and figure out exactly why the hypothesis is true—I need to prove the result. But computational algebra systems help me know what I should try to prove.

This post is a response to Erica's post on the question what did your students get out of today's class that they did not have before?

I think that she is completely right to challenge the premise of his question. There is much too much emphasis on "facts" in our educational system, and not enough emphasis on "habits." "What did students take away from your class?" The only way to answer that simply is to answer with something akin to a "fact." "They now know that Dostoyevsky is Russian." "They now know that not everyone can have everything they want." These are "facts," and it really does not take very long to learn them.

Is this an important component of education? To a point, yet, but it should be secondary to other issues (below). Here are several things that I was taught in high school:

1. The Teapot Dome Scandal
   
2. The Krebs cycle
   
3. How to conjugate the Spanish verb "pagar."
   
4. The Hawley-Smoot Tariff Act
   
5. Hero's formula
   
6. The name of the male lead character from Wuthering Heights.
   
7. Descartes Rule of Signs
   
8. The maximum number of electrons in the third shell.
   

I remember some of these things, but not others. I am guessing that any reader of this list might know a few, but not all. I have a Ph.D., and I think it is clear that I, along with most of the readers of this weblog, am among the best educated people in the history of the world.

So how can I be extremely well-educated, but not remember many of these basic facts from high school (let alone college, when the courses were much more specialized)? I think that this is largely because "facts" are not terribly important in the scheme of things. (So you can probably guess that I am not a big fan of Hirsch, although I do not want to eliminate the content completely).

My opinion is main role of education is to nurture different thinking skills. Among the thinking habits I would like my students to have are:

1. To value of clear, concise communication.
   
2. To value, and use, evidence.
   
3. To employ supposition ("what if...")
   
4. To employ empathy (think of things from other people's points of view--this is different from "sympathy").
   
5. To value and look for connections among different ideas
   

(as you might guess, I am a fan of Deborah Meier).

Analogies are only somewhat useful (I am trying to get my education students to become aware of this), but here is one: I run almost every day. If you were to ask me, "What did you get out of running today?", I might not have a good answer. "I got cold," "My leg muscles might be 0.00001% stronger," "I had some time to think or listen to a podcast." Other days I might answer "Absolutely nothing."

So why run when I get, at best, only tiny bit stronger each time? Well...that's the only way to get stronger. I might not be able to measure how much stronger I got from that day's worth of running, but the effects are noticeable over time.

We might not be able to measure how much our students got out of one class, but one semester of a class might make them more likely to provide evidence for an argument. To spend an extra 15 minutes revising a paper so that it is clearer. To be more likely to think about how Afghan's think of the United States forces in their country (liberators or occupiers?).

After four years of this, we hopefully have a student who has become a lot "stronger" in these areas.

So the fact that I may or may not be able to say, "after today's lecture, the students should know _____" is not evidence that I am a good teacher, and it is not evidence that I am a bad teacher. I goals like that for the semester: "After the semester, the students will know ____," but it seems like it would be unwise to interrupt a classes struggle with larger ideas just to inject the mandatory dose of daily facts.

As for "assessment mentality," I find much of it troubling. I will start by saying that I strongly value assessment, but that I think that it is hard for someone without formal training to do well. What ends up happening is people tend to measure what is easy rather than what is valuable (guess which is easier to assess: facts or habits?). In the process of measuring the easy, less important things, we slowly erode our values. Over time, what is measurable becomes what is important, and we end up ceding control of education to the test administrators.

The excellent NPR science show Radiolab recently did a show on mathematics. They discuss topics such as an infant's understanding of quantity, Erdös numbers, and the surprising Benford's Law. You can listen or download here.

I taught 9th grade algebra for a couple of years, and the students always wonder why algebra is important to know. Mano Singham, a physicist with a science weblog out of Case Western University, posted a great example of how it might be useful. The video he posted is below, and I will be describing just one of the reasons why she has no business talking about science.

In the video, a homeopath describes how homeopathy works. She manages to mangle whole branches of physics, but someone having survived 9th grade algebra should be able to tell that she speaks nonsense. She claims that there is almost no mass in the universe (I think she is trying to say that atoms are mostly empty space), and ties this to Einstein's famous "E=mc²" equation. Since there is "almost no mass in the universe," the "m" on the right side of the equation essentially disappears. Since it disappears, she claims that Einstein's equation is essentially "E=c^2," and goes on to say something about how our eyes are really important.

Even if it were true that there is almost no mass in the universe (it isn't), the variable "m" would not "disappear" in the sense that we simply erase it; she would have to mean that it "goes to zero." In this case, the equation would be "E=0," not "E=c^2." She wanted "m" to be approximately "0," but she essentially said that it is approximately "1." Anyone surviving high school algebra should know that 0x=0. This should be a red flag to any educated person.


From now on, whenever anyone asks you why they need to learn math, you can answer: "so you do not get suckered by homeopaths."