Posts Tagged ‘Peer Instruction’

Linear Algebra Class Structure

February 21, 2014

I was originally scheduled to teach abstract algebra this semester, but my section was cancelled due to low enrollment. Instead, I am teaching linear algebra, as we had higher-than-expected enrollment there.

The good news is that I can use the same basic course structure for linear algebra that I was planning to use for abstract algebra. The model is this:

  1. The semester is divided into two parts. The first part, from January 15th until March 31st, is where we learn the content. The second part is all review and assessment.
  2. For the first part, we do IBL-type presentations on Mondays and Fridays. Each day, we can do 4–6 presentations in 55 minutes. On Wednesdays, we review what we learned on the previous Monday and Friday. The reason why I chose Wednesday as the review day was so that students could have at least three nights to prepare for each presentation day.
  3. For the second half of the semester, we will alternate between assessment days and review days. Students will be able to choose what they want to review based on what they found most confusing from the first half of the semester AND from the recent assessments.

One advantage of having the Wednesdays saved for review is that I can use it for an emergency presentation day if a Monday or Friday class is cancelled; this has happened twice so far this semester, due to cold and snow (including today).

One problem that I have is that the course notes I wrote for linear algebra have 314 problems in them. Since I am compressing the presentation part into the first part of the semester AND only using Mondays and Fridays for presentations, I only have 20 presentations days for the 314 problems. This means that we need to average 16 problems per presentation day. I accomplish this by designating 6 problems as “Presentation Problems” (which will be presented, naturally), creating video solutions for another (roughly) 6 problems, and then leaving the remaining four-ish problems without solutions (these are mostly computational problems for which the students were given a video “template” on how to do the process).

It took a while to create the videos, but they are pretty much necessary for our course. This course serves as a very gentle “Introduction to Proofs” course, but the level of proof that is expected is of the “figure out how the proof follows directly from the definition” type. Since there are more complicated proofs that need to be done in the course, I would either need to lecture in class, have the students read the proofs from a textbook (which we don’t have), or create video lectures.

Also, given that we only have six Presentation Problems each day, I have developed a method of having the students volunteer for the problems that cuts down on the amount of work that I have assigning students to problems. My usual way of doing this is putting one essay quiz on Moodle that asks “Which problems would you like to present?” I still do this for my capstone course, in which we present 15 problems per day. For linear algebra, though, I put one quiz consisting of one multiple choice question for each problem that is to be presented. The students are given three choices: “I want to present this problem,” “I really want to present this problem,” and “I changed my mind—I no longer wish to present this problem” (a student who does not want to present does not need to complete the quiz for that particular question). I assign each question 10 points, 5 points, and 0 points, respectively. These points do not affect a student’s grade, but a there simply so I can look at the quiz summary to see each student’s preference quickly without much clicking. The drawback to this is that there is a lot more to do on Moodle (6 quizzes per day instead just one). However, I created all of the quizzes at the very beginning of the semester, and it didn’t actually take that long to do once I learned about the “duplicate” feature on Moodle.

We are just over halfway through the presentation days, and the class is going really well. I think that I have a remarkably good class, so I cannot really say how this class structure is working; I think that any class structure would work with this particular group of students. On the other hand, this shows that this class structure can work, given the right set of students.

Again, a new IBL-Peer Instruction Hybrid Model

December 24, 2013

I am continuing to try to figure out a way to effectively use both IBL and Peer Instruction (“clickers”) in my classes.

First, my main constraint: my favorite grading scheme requires students to be given many chances to get questions correct. Ideally, this means that we would finish with new content for the course 1/2 to 2/3 of the way through the semester.

Here is the approach I have been using up until now:

  1. First part of the semester: Students get the content from reading the textbook.
  2. First part of the semester: Students assimilate the content through Peer Instruction.
  3. Second part of the semester: Students do something that resembles (but isn’t actually) IBL.
  4. Second part of semester: Assess the students a lot.

Below is the same model I discussed last summer for my abstract algebra class. That abstract algebra class was closed due to low enrollment, and I was assigned linear algebra instead. I am keeping the same model, although I have a lot more exercises/theorems/conjectures in my linear algebra notes than I do for my abstract algebra notes.

Here is the new approach:

  1. Mondays and Fridays during first part of the semester: Use IBL and student presentations to introduce the content.
  2. Wednesdays during first part of the semester: use Peer Instruction to review and solidify ideas learned on the previous Friday and Monday.
  3. Second part of the semester: We review the most difficult material through Peer Instruction and in-class practice.
  4. Second part of semester: Assess the students a lot.

Here is the main problem that I am facing: I have 312 exercises in my IBL notes; I basically wrote the notes that I wanted—including many examples to build intuition—and I am now trying to figure out how to shoehorn all of the content into 1/2 to 2/3 of a semester. This works out to an average of about 7 exercises per day if we did IBL work every day of the entire semester, 10 exercises per day if we did IBL work on Mondays and Fridays (and review on Wednesdays) every day of the semester, and 20 exercises per day if we did IBL work on Mondays and Fridays (and review on Wednesdays) every day for half the semester. So I want to see if I can do between 10 to 20 exercises per class IBL class period, which is too much to do without some modifications. Here are the options I can think of to make this happen:

  1. Cut some of the content. I don’t want to do this.
  2. Provide screencasts of some of the exercises. I want to do this anyway, since part of the goal of our linear algebra class is to introduce students to proofs, and I believe that it is very useful for students to see worked examples. But I don’t want to have to provide 10–15 screencasts each class period.
  3. Simply do not cover many of the intuition-building exercises in class; Dana Ernst suggested this to me yesterday, and I think that it is brilliant. There is not reason why I have to do everything in class. Perhaps I could just take questions on any intuition-building exercises after we do the main theorems; I could provide screencasts for some of these if we run out of time.
  4. Other ideas?

Right now, my plan is to have students present and thoroughly discuss roughly 5 problems per IBL day, I would do screencasts for roughly 5 problems per day, leaving roughly 10 intuition problems to leave for the students to do.

Do any of you have ideas about how to improve this?

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.

So you suddenly have 68 students enrolled?

August 11, 2013

Suppose you are an instructor who uses Inquiry-Based Learning. You are used to running a particular course with 30–35 students, and you are about to start teaching that course in two weeks. But then you find out that you have 68 students registered for your class. What do you do?

Peer Instruction, hands down. Here is how you do it.

Since I am assuming that you only have two weeks to prepare, this is the most basic way of implementing Peer Instruction. Robert Talbert’s Guided Practice idea would be better to include if you are able.

If it is too late to get “clickers,” use Poll Everywhere, Socrative, or Learning Catalytics. I would tend toward Poll Everywhere, since it is pretty cheap ($65 per month for 68 students—get someone else to pay for it), and students only need a texting plan to use it. But Learning Catalytics seems pretty awesome; I just don’t trust all of the students to have a tablet or smart phone.

Do you have a textbook for the course? If so, here is the recipe:

  1. On the first day of class, assign students to fixed teams of 2 or 3. This will help every student feel like they are part of a community in your class. Students should sit together with their team. You may want to change up teams later in the semester.
  2. Students read a section of the text the night before class. You prepare 5–10 multiple choice questions based on the section. These questions should cover the main points of the section. Some questions will only be to help students understand a definition/concept, other questions will force students to confront misconceptions. Peer Instruction is awesome for confronting misconceptions. Just make sure that you have good distractors for each question.
  3. Everyone comes to class.
  4. If you need to pass back papers, make administrative announcements, etc, you can do that at the beginning of class. But do not, under any circumstance, give an overview of the section; this will teach them that they do not need to read the section, and the result will be that your class will eventually morph into a standard lecture. Instead, simply start the first clicker question.
  5. Display the question on the screen. Have students silently think about the question themselves and “click” their favorite answer when they are ready. You may want to give them a fixed time limit here, although I usually do not; I can usually tell how much students need by the number of students who have already responded. But I usually do not have 68 students.
  6. Look at the results, but do not let them see the results (mute the projector if you need to). If the students overwhelmingly get the correct answer, display the results and give a very brief explanation about why the correct answer is correct AND why the other answers are incorrect. (Note: there is a high bar for “overwhelmingly correct.” For instance, on a True/False question, if half of the students know that the correct answer is True, say, and the other half guess blindly, then 75% of the students will answer correctly. This is bad, since half of the class does not understand. So you might want 90% correct answers on a True/False question, slightly lower for a question with three options, etc. This is an art and not a science, though).
  7. On the other hand, if the students do not overwhelmingly answer correctly, tell the students to discuss their answers with their team. The students should try to convince the other team members of their answer, but the students should be open to changing their mind. Once the team agrees on a single answer, have them re-vote. You should wander around the class as much as you can here, eavesdropping. Once most students have responded (or your time limit is up), display the results to the class.
  8. Now, explain why the correct answer(s) is (are) correct AND why the incorrect answers are incorrect. You can tell how long you should spend talking about this by how the teams did in the most recent round of voting. If they did well, do not talk for long. If they did not do well, give them a more thorough lecture (although you probably will not need to talk for more than 10 minutes).
  9. Repeat with the remainder of your questions until class ends.

This will get every single student involved, and my students have overwhelmingly loved the experience. There is also evidence that Peer Instruction will help students learn enough to increase grades by half of a grade.

If you do NOT have a textbook, you should do your best to find some sort of a free online text for them, write your own notes, create your own lecture videos for students to view before class, and/or use existing videos (e.g. Khan Academy) to use to “transfer” knowledge to the students before class. Then you can use class time to have the students make sense of the new knowledge.

Failing this, lecture. But build some number of clicker questions into your lecture. The process is the same as outlined above, but you will just have fewer questions.

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.

An example of why lecturing does not work very well

March 2, 2013

We just started discussing confidence intervals in probability and statistics. As expected, students had a difficult time with it.

As usual, they read the section, answered some questions online, and came to class. In class, we worked on clicker questions. The first was basically:

Q: The 95% confidence interval for the population mean \mu is [x,y]. Based on this interval:

  1. There is a 95% chance that \mu is in this interval.
  2. 95% of the observations are in this interval.
  3. This method of creating intervals works 95% of the time.

This is a tricky idea, but the third choice is the best answer of the three. In my second class, only 2 out 26 students got it correct. This was to be expected, though, since it is a tricky subject.

So I basically gave a 15-20 minute lecture as to why the third one was correct and the first two were wrong. Actually, it is more accurate to say that I repeated a six minute lecture three times about how to think about this.

We had two more clicker questions related to confidence intervals, and then I gave them the following question (perhaps you recognize it):

Q: The 95% confidence interval for the population mean \mu is [x,y]. Based on this interval:

  1. There is a 95% chance that \mu is in this interval.
  2. 95% of the observations are in this interval.
  3. This method of creating intervals works 95% of the time.

The class was completely split into thirds as to which of the three answers was correct (to be fair, the question was only isomorphic to the first question, not equal).

I re-gave my two more variations of my six minute lecture explaining how to think of each of the three choices.

Then I re-gave the question, only with the following choices:

  1. There is a 95% chance that \mu is in the interval.
  2. The probability that \mu is in the interval is 0.95.
  3. 95% of the observations are in this interval.
  4. Exactly two of these answers are correct.
  5. Each of the first three answers are correct.
  6. None of the above answers are correct.

The correct answer is “None of the above,” of course. Three of the 26 students got it correct, even though I had literally just told them why the first three choices were wrong two minutes prior to voting.

This means one of two things. Either

  1. Either learning is incredibly complex, and lecturing is not a good tool to help people understand, or
  2. I suck at lecturing.

To be fair, Peer Instruction was not working, either. But it is surprising to me that Peer Instruction works as well as it does, and it is surprising to me that lectures fails as miserably as it does. The confidence interval lesson is a good reminder of the latter.

The point is not that my students are dumb—they are not. Nor is it that they are bad students—they are not. The point is that learning is difficult (especially with tricky ideas like “confidence intervals”), and one must be sensitive to this fact.

When to start flipping

February 1, 2013


Joshua Bowman tweeted the following question:

The underlying question is: should your first flipped class be a class you have taught before, or should it be a new class?

The argument for the former seems clear to me: it is smart to reduce the number of moving parts. If you have the content and assessments down, you can focus more on the pedagogy.

But I probably lean the other way: I think that it might be better to first flip a class you have not taught before. The reason: you don’t have the safety net of a pre-prepared lecture to fall back on, so you are forced to solely think about the class from a flipped perspective.

Of course, this might just be because of my personal experience. My first attempt at flipping a class was in linear algebra, which I had taught twice before. I had the students watch some Khan Academy videos and do problems out of the textbook before class, and we worked on problems during class.

The problem was that students would ask me questions in class, and I could immediately turn to all of my pet examples (which I had not reviewed beforehand) that I developed the two previous semesters. So the first third of the semester was as much a straight lecture as a flipped classroom. Once I realized this was happening, I rebooted the class to be a better version of a flipped classroom (but you never want to be forced to reboot anything).

Other people may not have this trouble, but I did. But it worked out: I taught real analysis—which I had not taught before—and the flipped classroom went well. There are a variety of reasons for this, but it helped that I did not already have a lecture-mindset for that class.

Anyway, here is my advice for anyone considering flipping a classroom:

  1. Start with a class you haven’t taught before.
  2. Use Peer Instruction (PI). Not only will it provide you with a great framework for your in-class work, but many people do it so you can borrow/steal a lot material. Best yet: even if you completely screw up the class, you will still be no worse off than a brilliantly-done lecture.
  3. Choose a class textbook that is readable for the students. Have the students read it before each class.
  4. Have some sort of mechanism for collecting the students’ questions prior to each class. Classroom management systems like Moodle/Blackboard/etc work, you could set up a class blog on and have them use the “comments” for their questions, or you could just use email.
  5. Get someone else’s PI “clicker questions” to use a foundation for your course.
  6. To prepare for a class, read through the section and create several clicker questions of your own before reading the clicker questions you stole from someone else (this is to get practice, but also to focus on what you think is important about the section). After you have written some of your own, merge them with the reference questions you got from someone else. This can be done well before the class actually meets.
  7. The morning before the class, look through your students’ questions. Pick the appropriate clicker questions from your reserve that will best answer their questions, writing new ones if needed (this is optional, especially if you have an 8 am class). Be sure to keep some questions on the most important topics, though, since students sometimes do not ask questions on this.
  8. Go to class, ask the questions, and have fun.

Notice that I did NOT recommend “creating videos.” I think that this is a nice thing to do for the students, but it is a lot of work. Students can definitely learn from a reasonable textbook.

As for “clickers,” I use TurningPoint, but only because that is what my campus decided on. Several people use iClicker, and Learning Catalytics is supposed to be awesome if you are sure that everyone has a device (and you have some money to spend). But do not discount low-tech solutions, either: I believe Andy Rundquist prefers colored notecards to electronic clickers (students raise a red notecard for option a, green for b, etc).

I am a big fan of the flipped classroom for most college-level classrooms. Please contact me if you are interested in getting started.

As always, please feel free to critique anything that I have said in the comments.

(photo “Flip” by flickr user SierraBlair, Creative Commons License)

Students demand: “More clicker questions!”

May 3, 2012

In my IBL-flipped hybrid complex analysis class, we finished the “flipped” portion (and the entire textbook) in half of the semester. We spent the next quarter of the semester finishing IBL presentations.

I asked what they wanted to do for the remainder of the semester. They said “more clicker questions” (they also requested time to work on problems in class, including application problems). This is what we have been doing.

After starting class working through some of their homework problems, I asked how the class wanted to spend the last 30 minutes of class—I had both clicker questions and some new problems to work on. We found that 79% of the students wanted clicker questions (clickers are nice for a variety of reasons). We did a couple, and the remaining questions start a new theme, so I decided to stop with clicker questions for the day so we could work on some of the problems.

That was a mistake. They were mad. Okay, not mad, but they really wanted more questions. So we did them for the remainder of class.

I asked why they had such a preference for clicker questions, they gave two answers:

  1. It is too easy to get stuck on the problems that I have been giving (I have been giving them trickier proofs, since this is their third time through the material).
  2. The clicker questions really help them learn.

I was really happy to hear that.

Semester Reflection, Part I

December 12, 2011

I am back to blogging after a semester of figuring out how to be the parent of two kids.  We are slowly figuring it out.


Anyway, below is a summary of what I did for the semester followed by how I would change in future semesters.  Recall that I am teaching real analysis.


  1. Students read a section of the text and watch some screencasts before class.  Students had to answer some questions online before class; if students did not answer the questions, they got a nagging email asking why.  This led to a very high completion rate.

  2. Students could request screencasts, thereby giving them a customized lecture (of sorts).

  3. For the first 60% of the semester, students spent about 75% of the time answering clicker questions (individually and in teams of three).  The remaining 25% of the time was spent starting homework problems.

  4. For the last 40% of the semester, we reviewed.  Students had to re-read a chapter before class.  In class, I gave the students four proofs to do in teams of two on a whiteboard.  Two proofs were very basic, and two were more complex.  I went around and gave feedback to each of the teams individually.  The idea was to run through the proofs of these four problems by the end of class (I put the proofs on slides), but we rarely got to all four questions.  I would also present the proof of a major theorem from the chapter about halfway through class.

  5.   Students were graded according to a midterm, a final, a portfolio, and a “practice portfolio.”  The exams are fairly standard.  The portfolio is a collection of each student’s best proofs throughout the semester, and the student has to provide evidence that he/she understands each of the course topics.  These are yet to be graded.  The practice portfolio was the same idea mid-way through the semester; this was graded on completion only, since the purpose of the practice portfolio was to get the students used to this different way of grading.

  6. Students who wanted to get an A for the semester had to do a project.  This means that they had to create screencasts on a section of the textbook that we had not covered during the semester (I used Abbott’s textbook, and he has them designated as “project sections”).


What went well:


  1. The clickers/peer instruction.  Analysis is full of ideas that are difficult to understand; if you do not understand them, it is even more difficult to prove anything about them.  The clickers really gave everyone—with virtually no exception—a solid idea of what was going on.

  2. The last 40% of the class was terrific.  We essentially went through the textbook twice, and the students made huuuuuuuge improvements the second time.


What I would improve next time:


  1. During the “clicker” portion of the semester, the class time spent starting the homework was not effective (in part because I did not give it enough time, but I don’t think that it would have been great with ample time, either).  I would recommend giving them the “basic” proofs that I did in the review portion of the semester each class period instead.  Perhaps do 50% clickers each class and 50% “basic proofs” (two would probably suffice, and most teams would probably only get to one).

  2. Do the practice portfolio much earlier.  I did it right after midterm, and that did not give students enough time to digest it.  Also, I recommended whether someone should do a project based on this, and students would have more time for projects if the practice portfolio went earlier.

  3. I also did two Calibrated Peer Review assignments. These failed due to errors on my part. First, I had students put their proofs on Moodle, which they linked to on the CPR site. This was a problem because students do not actually have access to the files on Moodle (it worked when I tested it because I have more permissions). Second, I told the students the wrong deadline for the second assignment. I think that this tool has a ton of potential, but I need to eliminate the user error first.
  4. I screwed up the standards a bit. For example, I was missing “Cauchy sequences” and “Limits” (Limits!). I was able to come up with fair workarounds for the students, but I think that I will only release standards to the students as we reach them in later semesters. This should force me to think through the standards an nth time, and I likely won’t miss anything major by doing this.


The jury is still out on portfolios.  We will see.