Posts Tagged ‘Teaching’

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.

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.

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:

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 wordpress.com 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)

Ultra-Learning and the Drill Down Method

October 30, 2012

Scott Young completed every course from MIT’s computer science online offerings in one year. There are 33 total courses. Since a typical college student takes 8 courses per year, this amounts to roughly doing a bachelor’s degree in one year.

Of course, he did not receive a diploma for it. But he outlines how he did it in Cal Newport’s blog. He describes how he was able to learn so much so quickly in this post, and he describes this process as the “Drill Down Method.” It is a three step process, which I will summarize below:

1. Coverage. “You can’t plan an attack if you don’t have a map of the terrain. Therefore the first step in learning anything deeply, is to get a general sense of what you need to learn. . .A mistake students often make is believing this stage is the most important.”
2. Practice. Do problems.
3. Insight. “The goal of coverage and practice questions is to get you to a point where you know what you don’t understand. This isn’t as easy as it sounds. . .Often when you can identify precisely what you don’t understand, that gives you the tools to fill the gap. It’s the large gaps in understanding which are hardest to fill.”

I think that this matches up very well with the way I have been structuring my courses lately. The first half of my semester features students reading the textbook and answering clicker questions—coverage. Students are also taking quizzes during this time—practice.

The second half of the course features students working on problems and presenting them. This is designed to lead to insight, although I am not quite there yet. I need to figure out how to design the problems better.

Pseudointeraction

April 26, 2012

Dan Meyer (or maybe Jo Boaler) brought you psuedocontext. John Burke and Frank Noschese brought you pseudoteaching. Grace Chen brought you pseudoquestioning.

I propose a new addition to this group: pseudointeraction, which is a classroom where it appears that students are interacting, but they are interacting trivially or only very few are doing the interacting. A good example is to check “USA Lesson 2, Part 2” on this webpage by Oliver Knill.

I thought about this because I have heard a lot of mathematics professors say, “My classes are interactive lectures” (I have said the same thing myself). What this typically means is that it is a lecture, but students are encouraged to ask questions at any time and the professor will frequently ask students during class. Having observed many college mathematics classrooms by many different teachers (it used to be my job to view other classes), I think that I can safely say this:

1. Giving the students opportunity to interact is not the same as students interacting. Worse yet, if only a couple of students interact but interact frequently, it can give the illusion that the whole class is participating. In fact, I would estimate that a very good “interactive lecture” would have at most 25% of the students interacting at any point during class. So three times as many students never say anything as those who do, and those who do say something will often just say 1-2 sentences during the course of a lecture. So it looks like an interactive class from the professor’s point of view, but the average student basically does not participate.
2. The interactions in an “interactive lecture” are not the kind that we really want. The questions tend to be fact recall (What is 2+8? What is the derivative of $3x^6$?) or speculation as to what the professor is thinking (What is the next step in the proof?). These may have value, but the students are not really interacting with the mathematics—at best, “what is the next step in the proof?”-type questions are an attempt to see which students are already interacting with the mathematics on their own. Of course, since the professor moves on once he/she hears one student answer, the only information gained is “one student was following the proof.”

Finally, note that I am not talking about people who lecture, but then break for a think-pair-share-type exercise (I am sure there are a lot of some people who say that they do an “interactive lecture” and really do get. And I am not trying to say that “interactive lectures” lack value, and that we should stop doing them (I have not stopped completely). But I think that we should stop calling them “interactive,” since the level of interaction per student is ridiculously small, and I don’t think that this is the right type of interaction. Perhaps we should just call them “lecture” and not delude ourselves into thinking that the average student is interacting with the material in any serious way.

Library of Virtual Manipulatives

February 5, 2011

I am teaching mathematics for elementary education majors this semester, and we are currently discussing number systems. In discussing our base ten number system, I found the Library of Virtual Manipulative’s section on base blocks to be extremely useful. It turns out that it takes much less time to click on a button to create a virtual 10-by-10-by-10 cube than it does to build an actual 10-by-10-by-10 cube. This allowed me to run through many examples in class.

Semester Summary

December 17, 2010

I outlined a plan at the beginning of the semester here, here, and here. Now, we reflect. (This is really a first stab at what I will present at the JMM).

1. The Cooperative Learning was a tentative success. Anecdotally, the students this semester probably did slightly better as a whole than my classes have done in previous semesters. Slightly, but not a lot. I think, though, that there is great potential for improvement as I become a better wielder of CL. In many ways, I was a new teacher this semester. I made a lot of mistakes, and figured a lot of things out.

Importantly, my students seemed to be much happier this semester. Will just a few exceptions, my students really liked working together. However, most of my students would have preferred a little more lecture from me. I speculate that part of this is habit, although I think that I should include slightly more lecture—perhaps 5-10 minutes at the beginning and end of each class.

2. Students loved the Khan Academy videos.
3. The new grading system was a smashing success…by which I mean that it was an improvement over the old system. Students enjoyed it, and it really seemed to create a less adversarial relationship with the students. Moreover, there were a handful of students who made remarkable turnarounds. One student went from an very low F to a C, one went from a low D to a C, and one went from a low D to an AB. In my decade or so of teaching, I may have seen people make such a turnaround once or twice; it happened three times this semester. I believe that the grading system helped the students understand what they needed to do AND was forgiving enough that it was worth the time to make up the work.

That being said, it still has many of the same drawbacks as the traditional way of grading. Students were a little too motivated to “fill in a box” rather than to learn the material. I offered team quizzes during many of the classes: when a team felt like they all learned the material, a randomly-chosen team-member could take a quiz. If he/she got the question correct, the whole team got credit for the topic. It turned out that students would sacrifice learning time to take the quiz. Next time, I will either skip these or make it into a wager—students will get credit if they succeed on the quiz, but will have to eventually make up an extra question if they get one wrong. I am not sure what I will do.

So I will definitely keep this system, at least until I can figure out how to get closer to eliminating grades.

4. I ran to work four days each week (with some exceptions) until it got too dark in mid-November. I miss running to work already.
5. GTD is still working wonders. I am on the hiring committee for my department, and it is helping me keep everything organized.
6. My Seinfeld-method of keeping track of my research time seemed to work in non-November months. I spent at least an hour on research for 59 of 78 days this semester. Of the 19 that I missed, 15 were in November.
7. I now wake up at 4:30 most mornings. This was a monumental help in keeping up with my life. I scheduled all make-up quizzes, looked at job applicant files, and did research from 4:30 to 6:45 each morning.

It was a great semester with great students. Now if only my 59+ hours of research had been productive.