## Posts Tagged ‘abstract algebra’

### Inquiry-Oriented Instruction

February 15, 2017

I was part of a grant last semester to implement a set of teaching materials that has been refined over the last decade. The materials use a teaching method called inquiry-oriented instruction, which I would say is a subset of inquiry-based learning (IBL). I used these materials in my abstract algebra class, although there are materials for both linear algebra and differential equations, too.

A very brief description is “intuition comes before definitions.” The materials introduce quotient groups by discussing Even and Odd integers, which students could easily see is a group at that point (using rules like “Even + Odd = Odd”). Once they got familiar with the idea that we could have sets of elements make up a group, we slowly backed our way into the definition of coset. It was pretty impressive to see students very naturally come up with definitions—having the right prompts helped a lot.

As part of the grant, I went to training at North Carolina State to use the materials. I also had funds to have student video my class, which will be used to analyze how well instructors who were not involved with the development of these materials can implement them.

We also used the class video as part of a weekly online working group. The purpose of this group was to prepare us, both in terms of pedagogy and course materials (not everyone was an algebraist), to teach the class. We discussed the purposes of the prompts, talked about what was going well and poorly, and watched video of each others’ classes. I found this immensely helpful.

I would use these materials again (in fact, I am planning on using the linear algebra materials next year). My sense is that my students had an abnormally good grasp of the definitions; previous students have struggled to understand what a coset means, for instance. My focus for the next time I use the abstract algebra materials is to work harder on the technical proofs—I think that my students did better on writing proofs than the previous time I taught the course, but not by a lot. Still, I think that the gains in intuition were worth it.

Links to the abstract algebra, linear algebra, and differential equation materials can be found here in the middle of the page.

### New IBL-Peer Instruction Hybrid Model

September 18, 2013

Here is my plan for my abstract algebra class in the spring semetser. This is probably a little early to post this, but it ties in with Stan’s post on coverage in IBL classes.

My plan for the spring is to run an IBL course. I wrote my own notes this summer (although they are based heavily off of Margaret Morrow’s notes). One problem that I have with most of the IBL notes for abstract algebra is that they do not do much with ring field and field theory. In creating my notes, I included just about everything that I would want to include in a first abstract algebra course (including a section on group actions). This, of course, is too much content to cover in a semester in an IBL class (I suspect, anyway).

Here are the details: I figure that I can expect the students to discuss 5 problems per class, I can assign 1 other problem as a special type of homework, so I have accounted for 6 problems per day. Since there are about 30 days of class, this means that I can expect them to do 180 problems on their own. But I created a set of notes with 234 problems, and I expect to add more throughout the semester. This is too many problems.

But my solution is similar to Stan’s: I have roughly 50 extra problems for 30 classes. I can simply do three of the problems for students via screencast for them each class period (then I get some extra days for exams, review, and snow days). This has a couple of advantages. First, it allows me to cover all of the material I want to cover over the course of the semester. Second, it gives students model proofs to help them learn how to write proofs.

A second feature that this course will have is a better integration of IBL and Peer Instruction. I am a fan of both pedagogies because of the learning gains reported in the research. I am a fan of IBL because of the level of independence it promotes; Peer Instruction does not do this (at least, the way I do it). I am a fan of Peer Instruction because of the way it stamps out misconceptions and helps students make sense of mathematics; IBL does not do this (at least, not the way I do it). So I am continually looking for ways to combine these pedagogies.

Peer Instruction (for me) works best when the students have already been exposed to the content. I have previously tried to merge the two pedagogies by splitting the semester into halves. This has its advantages, although I am trying something new out next semester: I am going to have IBL classes on Mondays and Fridays (30 classes), and I will have Peer Instruction classes on Wednesdays based on the material that was covered on the previous Monday and Friday.

The basic idea is this: students are introduced to an idea the first time in preparing for an IBL class. They see the material a second time in class. They see the material a third time on the next Wednesday’s Peer Instruction class. They see the material a fourth time on homework/tests/whatever I end up planning.

I am really looking forward to this. Please let me know of any potential problems or improvements that you can think of.

### The Many Ways of IBL Conference

June 26, 2013

I attended the University of Chicago’s “Many Ways of IBL” conference last week. Here is a brief list of my thoughts for the week, in no particular order.

1. It was utterly great to see a couple old friends. I have been blessed to have had good colleagues everywhere I have been, and I wish that I could have taken many of them with me to my current position.
2. It was great to meet a bunch of new friends. I hope to stay in touch with many of them.
3. Part of the conference was to watch John Boller teach an IBL class on real analysis to a bunch of super-motivated high school students. Both John and the students did a fantastic job. I told John that it was so enjoyable that he could charge admission.
4. One big thing I was failing at with IBL last year: I did not discuss the statements and meanings of the theorems before students presented. Boller did this, and it must help students understand everything about the course better.
5. Paul Sally continues to be amazing. He is also hilarious.
6. In many classes, I have students read the textbook rather than lecture. I have no idea how to mesh this with IBL, but it is something I value. I realized from the conference that the reason why I value this is that it helps students learn how to learn on their own.
7. Even though I have been calling my recent hybrid classes “a mix of Peer Instruction (PI) and IBL,” I no longer think that I have been doing IBL. At best, it is IBL-Lite, although it is probably just “students presenting problems.”
8. This will lead me to alter a paper that I recently wrote on a PI/”IBL” calculus class; I will now qualify that my IBL is pretty weak.
9. I am now fairly certain that my courses for pre-service elementary education majors are IBL.
10. I might do IBL in my abstract algebra course this spring. If so, I might interweave IBL and PI differently: I might mainly do IBL, but then have some PI days to make sure students understand the ideas that have already been presented.
11. In abstract algebra, I might also create a class journal, where students can submit homework problems to an editorial board (of students) for peer review.
12. In IBL classes, have students take pictures of the board work. They can then upload the pictures to the course website as a record of what happened.
13. Matthew Leingang gave me a nice way of communicating course rules. He has “The Vegas Rule” for his class: “What happens in Vegas, stays in Vegas” where “Vegas” is defined as “the world outside of this classroom.” This is a nice concise way of reminding students to not use previous knowledge and outside sources.
14. Leingang also got me excited about paperless grading. Now I just need to find \$1200 for an iPad and scanner.
15. Ken Gross uses an “adjective-noun” metaphor for fractions, where the adjective is the number and the noun is the whole. That is, you can explain common denominators by doing something like: $2/3$ “units” $+ 1/2$ “units” equals $4/6$ “units” $+ 3/6$ “units,” which is equivalent to $4$ “sixths of a unit” $+ 3$ “sixths of a units” $= 7$ “sixths of a unit” $= 7/6$ “units.” Most of the work then is just changing the “noun” and finding the appropriate “adjective” for each of the new nouns.

### 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.
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.

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.

### Beamer

March 9, 2010

A confluence of events has conspired me to switch how I use my in-class time. I have switched

1. I gave my students a midterm evaluation. They are mostly happy, but thought that I was moving really quickly through the proofs.
2. I taught with Matthew Leingang, who, for years, has been doing what I am about to describe.
3. I attended a presentation about MOBIs, which are similar to SMART Boards.
4. One of my classrooms has poor lighting, making it difficult for some of the students to see the board.

These events led me to switch from primarily using the chalkboard to primarily using Beamer presentations. Beamer is like a $\LaTeX$ version of Power Point, but at least 500% more awesome. The biggest push came from the presentation about the MOBIs. I got really excited about the presentation, but then realized that 90% of the benefits could be gotten by using Beamer presentations. Here are the advantages; this is all speculation so far, since I have taught only one day with Beamer.

1. Beamer will allow me to be a more efficient teacher. I will spend less time writing, which means that I can spend more time on in-class projects for the students.
2. Beamer will allow the students to be more efficient students. They print off handouts of the presentations before coming to class, and use these as their notes. They do not need to write, but rather just concentrate on understanding the ideas.
3. Beamer allows me to write up pristine proofs. It is important to see a lot of good proofs when learning to write, and my Beamer proofs are much better than my blackboard proofs. This is because I take short cuts on the blackboard to reduce the writing time; there is no such problem with Beamer.
4. Beamer is more visible than the blackboards in one of my rooms.
5. Beamer will allow me to do more complicated examples. For instance, my students are familiar with $S_4$, but I rarely do examples of subgroups of $S_4$ because it takes so long to describe the subgroup. This is a snap with Beamer.
6. Beamer will ultimately reduce my prep time. The next time I teach this course, I will have an outline that I can modify; I will not need to start planning from scratch.
7. Beamer will help me plan better lectures. When I teach this class again in a couple of years, I will have a concrete reminder of what I did. I will write notes in my file that tell me how to improve. I do not have this type of “memory” with blackboards.
8. Students can look over the notes before class to be more prepared for class.
9. Students who miss a class can more easily see what they missed.

Here are some potential (and already actualized) drawbacks:

1. The prep time is concentrated up front. Producing slides for 3-4 days of class takes an entire 8 hour workday.
2. Some students may feel like they do not need to come to class when the class slides are available online.
3. There is a risk of going too quickly through slides.

There are likely others, and I will be happy to point them out as the experiment continues. The first drawback is very real, but will hopefully be worth the long-term investment (I am thinking about installing a geothermal heat pump in my home, which has similar advantages and disadvantages: a bit up-front price, but you ultimately save over the long run). I do not think that the second drawback will be a problem for my students, who are quite good about attending class. I could see this being a problem at other schools, though. Finally, I have asked the students for help in slowing me down. In particular, I have given them all a red index card. When they want me to slow down, they are supposed to hold it up so that I can see I am going too fast. They were not used today, but they could be helpful later on.

Finally, my students seemed to think that today’s Beamer lecture went well. Also, here is a link to my webpage, where the handouts for the presentations are posted.

### How much homework is enough?

March 2, 2010

As a tenure-track professor, my colleagues visit my class every semester to better get to know me. Aside from the fact that this is an evaluation with professional consequences for me, I find this to be a great practice when done correctly—in fact, I wish that tenured professors would make a habit of visiting each other’s classes. It is easy to become too familiar with one’s own teaching, and it is good to periodically have an outsider question your choices.

I recently had a visitor, and he was concerned about my homework assignments. It was a blessing that we had this conversation, because it reminded me about why I make some of the decisions that I do; I have internalized these, but it is good to make them explicit from time to time.

1. I believe that any sort of upper-level mathematics that is worth doing is worth taking your time on. Upper-level mathematics is about ideas and problem-solving, not computations and memorization (for the most part). I believe that a well-written homework assignment should require that the student take time to complete it; it should require stops and starts, dead-ends, and sometimes flashes of inspiration. These problems cannot be churned out, factory-like, on a schedule. Because of this, I eschew the practice of giving proofs after every class period; rather, I prefer to give students a “cycle” (6 days—recall that my college/s is/are weird) so they can dwell, fail, try again, and ultimately succeed.
2. I believe that the teacher’s job is to make him/herself expendable. I used to revel in the fact that my students would come to my office hours on a daily basis for help. However, I no longer think this is in the best interest of the student. While it does make me feel good to feel needed, I have (in the past) had a tendency to set up co-dependent relationships with my students, where the students never feel like they can do the mathematics outside of my presence. This, in my opinion, is the opposite of education, and I have been improving on this by leaps and bounds over the past eight years. My latest tool is cooperative learning, which has been considered a great success by my students. A second success is giving my students more flexibility in when the homework is done; again, I have found that giving students homework every cycle creates less dependence than making homework due every lecture—even if the amount of homework is the same in both cases.
3. I believe in quality over quantity, if I must choose. I prefer to give fewer homework problems while expecting a higher quality (and giving a higher quality of feedback) than more homework problems of lower quality.
4. I believe that mathematics homework is not just about mathematics; rather, this is an opportunity to improve writing skills, which will ultimately be used more than the mathematic skills for 99% of my students.

One of my main points is that I do not think that having homework due every lecture meshes with my values, but this does not mean that I do assign homework that is due the next lecture period. Rather, I reserve these homework questions for simple computational problems or problems that apply a definition.

These are ideas that I have come to in my years of teaching, although I make no claim that these are optimal. I prefer weekly/cycle homework to daily homework for proofs, but I feel like I could easily change my thinking on this if I hear a good argument. I would appreciate suggestions and other people’s rationales in the comments.

### Useless Mathematical Applications

February 22, 2010

Someone (not me) recently put two posters for “Mathematics Awareness Month” outside of my door. Here are the two posters:

I think that the first poster is good, but I think that the second poster is awful. The difference between the two is that, while both are applications of mathematics to sports, the first one (baseball) is reasonably authentic, and the second (basketball) is not authentic at all.

I could imagine the Colorado Rockies doing a mathematical analysis of the effect of humidity, and then making a decision about whether to humidify or not based on that analysis (I am assuming that the poster is about humidifying the baseballs, which they can carefully control, but the poster does not make this clear). On the other hand, I have shot as many jump shots as anyone reading this post, and I have never wondered about the chances of making a jump if the angle were 30 degrees. Moreover, I cannot pretend to come up with a reasonable estimate for my initial angle on a free throw—let alone a three-pointer or an 8-footer with a 6’4″ defender guarding me. Even if I did the analysis to determine the optimal angle, the angle would change according to my position on the court, the various talents and deficiencies of my defender, and my level of fatigue. For any given shot, I almost certainly would not have all of the data needed to describe the conditions of the shot—let alone do a detailed analysis of that particular shot—to help me increase my chances of making the jump shot.

This is another example of the scourge of “fake applications” in mathematics education. There are too many questions like, “the population of a certain type of fish is t^2, where t is time measured in years. Find the rate at which the population is changing when t=2.” Some amount of these fake applications may be useful in conveying how the mathematics would theoretically be used, but I think there is a greater harm: these types of problems (fish and jump shots) are pretty clearly not useful to anyone in the real world, yet they are clearly presented as an attempt to convince students that mathematics matters in real life. If I were a student, my reaction would be, “Really? Is that all you got? Jump shots?” I would leave feeling that the examples are contrived (which they are), and so there may not be real world applications.

As much as possible, I am for including authentic applications (my problem is that I do not know of many, since I am a pure mathematician who has been too lazy to find the authentic applications); I am going to include some coding theory and cryptography in my abstract algebra class this semester. I think that authentic applications have great value; contrived applications likely do more damage than good.

### Optimally Challenging

February 21, 2010

@republicofmath tweeted a link to this article on Advanced Placement courses. This article nicely summarizes my feelings on AP calculus—very few students learn much calculus beyond the algorithms (I am not citing anything here because this is little more than my impression). Combining this with the push to make algebra as a required course for students as young as 7th grade, and we begin to see a pattern of the maxim “earlier is better.”

One particularly dangerous way this manifests itself is the in the corollary “harder is better.” In fact, I find myself doing that a bit this semester in my abstract algebra class—I find myself introducing graduate-level ideas to my students at times. This has mostly been fine, since my students have been really good. However, what is the cost of this? For every graduate-level idea I introduce, an undergraduate idea is left unsaid (roughly). But somehow I want to introduce these topics.

I do not think that I am alone in this habit of making courses more challenging than they need to be; in fact, I think that I far from the worst offender, since I am aware of this tendency and work hard to keep the material at the level of my student. I know of other professors who brag about making very difficult exams and homework assignments. I fall into this trap, too, when I am not consciously thinking about this issue. I cannot speculate about everyone else who prides themselves on being a “tough” teacher, but here are my best guesses as to why I am this way:

1. I find the material more interesting. Since the course is for the students, this is not a terribly good reason. “Research” should be the outlet for the material I find most interesting.
2. I find that my ego is stroked when I teach harder material. It is rarely a good idea to do things just for ego, though.
3. I think that I am being a good teacher by challenging my students. However, this is not true. It is very easy to make a course that is too easy or too hard. Unfortunately, students do not learn much in these classes. It is relatively hard to create a course that is optimally challenging for the students, which is where they learn the most. Instead of aiming for “hard,” I should be aiming for “just hard enough.”
4. I find that my colleagues respect me more. It seems like the fastest way for a professor to lose the respect of his/her peers is to gain the reputation of being an “easy teacher.” This is easily done, since many of my colleagues in this country think that anything short of “students killing themselves to make it through a math class” is too easy.

I do not want to make it sound like I have done a bad job this semester—I have actually been very pleased. I can merely point to occasions when I have introduced ideas that are too hard. If anything, I think that I might be developing the reputation as being “too easy” on my students (this will likely be the topic of my next post).

### My students are impressive.

February 10, 2010

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.

### On creating lies

February 3, 2010

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.