Teach Like a Freak, Part One

Inspired by the latest installment from Steven Dubner and Steve Levitt, Think Like a Freak, in this two-part series I consider what it might mean to Teach Like a Freak. In part 1, I take up the idea of experimentation and how I am currently experimenting with my teaching.

One of the central premises of Think Like a Freak is that one should be willing to experiment, and to make decisions based on the data gathered. For quite a long time, I have been teaching using inquiry-based learning (IBL), a mode of instruction in which students are the focus of classroom activity, deeply engaged in collaboratively making sense of the content. Evidence has been mounting that IBL specifically (http://www.nctm.org/publications/article.aspx?id=42527) and active learning more generally (http://www.pnas.org/content/111/23/8410.abstract) are more effective than lecture across multiple outcomes. At the same time, I have been reading research about Interactive Engagement (http://www.ams.org/notices/201308/rnoti-p1018.pdf), and have been experimenting with trying to blend IE with IBL. The question for me is how to structure my class to take maximum advantage of these approaches. To force myself to take this question seriously, I promised to speak about what I learned at the JMM 2015 in San Antonio. 

Last year, when I taught Transition to Proof, I had made a handful of IE questions, but class time was spent mostly on students presenting their work, and our discussions of those presentations, and a little bit of pair work. What I am doing this semester is using IE questions every week, which means about 15-25 minutes out of 150 minutes are spent on these short questions, with the rest of the time being spent the same way as last year. So, my goal is to compare the two classes in their understanding and skill in writing proof. The next step is to decide how to assess the impact of blending IBL with IE. This involves deciding what to measure. Another issue is that, once I decide on appropriate measures, it can be difficult to get good comparative data. For purposes of assessing impact, I do not have two classes running simultaneously with which to carefully set up a comparison. The best I can do is to use the data I still have from last year’s class. More specifically, in the previous year's course, I had tried a handful of IE questions, and so I have the results from those as well as exam scores for that class. The key component of my assessment, then, is to compare student exam performance last year to the performance this year, when I am using IE questions on a weekly basis. I do not claim that this will give me a definitive answer, but at least it will be a start. Another thing I have been doing is keeping track of participation in whole class discussions, so that I can compare the number of participants in discussions on IE days with discussions on non-IE days. Although counting the number of participants in whole-class discussion is a somewhat superficial measure, it gives me some quantification of how things run differently with the IE questions. If it seems that students are benefiting from more IE questions, I will keep making time for them in class.

Backing away from the specifics of this question, one thing that I have decided to do with courses that I teach regularly is to keep results of exams broken down by question. The reason for this is that I often modify exams from year to year, and so exam scores from year to year are not directly comparable, but there will be questions that are directly comparable. Another thing that I am learning to do is to keep a log of each class day’s activity. This way, in addition to evidence of student learning, I have a record of the kinds of interactions that occurred in class meetings. Together, these provide two kinds of data that help me to know whether what I am doing is working. Although I have always modified my teaching over time, by Teaching Like a Freak, I can hope to have evidence of whether the changes are making a positive impact.

In part 2, I will take up two other ideas from Think Like a Freak, targeting small problems and thinking like a child. 

Keep Tinkering

I am always making adjustments, tinkering with my courses, both during but especially between iterations of the courses. My teaching is never a finished product. It is in the nature of teaching that what worked in one year for one course may not work for another course or in a subsequent year of the same course. I want to share one change that I have made over the past year, the effect it had, and what I am doing as a result.

In my Transition to Proof course last fall, I began building concept questions to supplement the regular proofs, and using them to target specific misconceptions or difficulties that students are having (or that I expect based on past experience). By concept questions, I mean short questions, usually multiple choice or true/false, that are designed to draw out students’ thinking and generate productive disagreement. Every time we had one of those discussions, I was exhilarated by the amount of discourse in the room. This practice evolved because I promised myself that I would focus on getting more discussion out of students in that class, since, in the past, I felt that there were too few students able to comment or question the proofs presented by their peers at the board. With the concept questions, I felt like I was seeing what the students were getting or missing from those proof presentations. In particular, the questions really helped to draw out the main points of proofs, points that I thought they would have gotten from a direct discussion of the proof, but which may have been less apparent than I had assumed. I almost feel like students in previous iterations of the course were shortchanged because they did not get this added layer of discussion to push their thinking forward. That’s when tinkering pays off.

As a result, I have planned some form of concept questions into both of my courses for this fall. Accompanying this change, I have also included the concept questions into the course grade. In addition to using the concept questions as a teaching tool, I am curious as to: (a) whether simply participating in the concept questions correlates with performance in the course, and (b) whether answering questions correctly on the first try correlates with performance in the course. Most of all, I would like to know whether using the concept questions as a tool in class improves the class’ understanding of the key concepts, but this will be hard to measure. I am thinking that I may have some items on some exams in one of the classes that I will reuse from prior years, so that I can compare performance. That’s not as good as an experiment, but at least I will have a basis for comparison.

The larger message is that it is healthy to revisit one’s goals for a course, to think about personal goals for improving one’s teaching, and to be willing to try new ideas that show promise of bringing students closer to the learning goals, and to measure the impact of the changes so that what works remains in place, and tactics that don't work are revised or edited out of the course. 

Building an Effective In-class Learning Environment, Part 3: Balancing Group Work And Student Presentations

In this series, I explore the questions: What are some advantages and disadvantages of group work and student presentations? How can students be held accountable for learning in groups and from student presenters? What defines a good balance of group time with whole class presentations? In Part 1, I focused exclusively on group work.

In Part 2, I focused mainly on student presentations. Finally, in Part 3, I discuss considerations involved in balancing time allocated to each of these modes of classroom organization, and balancing the strengths and weaknesses of the two modes against each other. 

Striking a balance between group work and student presentations:

In Part 1, we learned that group work is an effective way to organize classroom learning, but that there are sometimes issues that need to be resolved or discussed by the entire class, or problems that many groups are unable to resolve on their own. In Part 2, we learned that student presentations are good for putting the focus of the class on a particular solution, but that there is the potential for students’ preparation for class and participation in discussion to suffer. Therefore, I find that using both groups and individual presentations helps to keep students engaged in class, and that a good mixture will encourage students to prepare for class on their own time.

Given the challenges and opportunities associated with group work and student presentations, what mix of these two forms of classroom organization is best?

For me, there is no one right answer. Even for a particular semester with a particular section of a class, I am sure that different blends of class organization would be valuable. Still, I have found that I tend to favor more groups or more presentations in different sorts of classes. I use two basic models in my classes. In what follows, I will describe the models and why I feel that each one is valuable in the particular courses where I use it.

Group-centered model:

In this model, roughly 60-70% of each class period is spent in groups. Usually, I have classes that meet twice each week. Typically, on one of these two days, class begins with me introducing the topic of the day, in some courses explaining the manipulatives we will be using, or the calculator functions that they may need to do the day’s mathematics, and then sending the groups to work on problems. I may tell the groups to give a report when they have a solution to a particular problem. As groups work, I monitor their progress, check to ensure that everyone is participating in the work of the group, and ask questions as needed to help groups make progress in their thinking. If a group is ready to report, then I ensure that all group members contribute to the report, and if they answer all my questions satisfactorily, then I approve the report; otherwise, they are told to work more and call me back when they are ready. If I did not request group reports, then I am usually making note of which groups have done work that I think should be discussed in front of the class, and which problems are sticking points for groups. If all groups become stuck, then we transition to a student presentation or a whole-class discussion of how to proceed. Otherwise, groups continue to work until I feel that most of the class is ready to discuss the key ideas and the work that I have identified for presentation. Presentations then serve as a way to codify the important concepts, as a way to compare different solution ideas, and for groups to ask questions regarding issues they had while working. Everyone is sent home to work on problems, and to come back ready to discuss solutions.

When students return for the next class, they begin in groups right away. Sometimes, I will announce a jigsaw, so that particular groups are assigned to focus on solutions to a single problem and prepare the explanation they will give later. Other times, I make sure that everyone has a colored pen, and I quickly identify which problems will be presented and who will present them, so that we move into student presentations rather quickly. Because students are using colored pens, I can tell what they have done on their own time, and yet taking good notes on the presentations can boost their homework score. It may happen that after a particular presentation, students have the tools they need to solve other problems on which they were stuck, in which case they get time to work in groups again. Or, I may have follow-up problems that build on what was presented, and again the groups are charged to apply what they have learned from the presentations. Depending on time, we may begin a new cycle of looking at a new topic while working in groups.

I have used and refined this model since I first had my own classes. I find that this is a good model for lower-division mathematics, including courses like Mathematics for Elementary Teachers, where a number of the problems involve computations and generally involve more familiar or concrete concepts. The problems lend themselves to groups being steadily engaged. It is more difficult to use this model when the problems are longer and more abstract. One reason for this is that the average time to solve a problem is longer. This makes it more difficult to launch into a topic during class time and have sufficient progress made by all groups within 30 minutes or so. Therefore, in classes like Transition to Proof, Abstract Algebra, or Modern Geometry, I use a different model.

Presentation-centered model:

In the presentation-centered model, roughly 70-80% of the time is spent on student presentations (this includes the think-pair-share time in which partners are discussing presentations). Class begins in one of three ways. Either, a) students are encouraged to discuss their solutions while I ensure that everyone has a colored pen and I sign up the presenters for the day; b) the class begins with a set of prompts, in which I put up a short set of questions, often true/false or multiple choice, and students are asked to think-vote-discuss-revote, similar to Interactive Engagement in physics and elsewhere; or, c) I announce a jigsaw, and partners are assigned to one of two problems that they will shortly have to explain to another person. At the conclusion of any of these events, we launch into student presentations. Each presentation is discussed in detail, until the class is satisfied with the mathematics, and I am satisfied that the class has identified the important ideas. Occasionally, in between presentations, partners may be asked to look at a related problem that either applies the ideas from the most recent presentation, or anticipates the ideas that may come up in the next presentation. After the conclusion of all the day’s presentations, usually four to six of them, then I may point students to the next topic or assignment, and partners will often be asked to do some preliminary work with definitions or examples that may help them.

When I first began teaching proof-oriented courses, I used presentations and accompanied them with think-pair-share, as I do now, but I did not use the jigsaw and prompts. I find that beginning the class with the partner work gets the class into a discussion-oriented mindset, which helps to make the presentation discussions more lively. Using the prompts makes for a nice formative assessment where I learn where the whole class stands with key concepts, and I can see and react immediately to what the class thinks. I also find that students are very highly engaged during jigsaws, so that I often structure the problem sets so that there are two closely related, more accessible problems that lend themselves to a jigsaw. But because a jigsaw depends on a large portion of the students being able to solve the assigned problems, not everything can be handled this way.

Final comments:

Stepping outside of my own classroom, I know that different instructors have preferences for whole class or small group mode. Each mode demands slightly different skills from the instructor. In small groups, the instructor has to travel from group to group, listening and occasionally contributing questions, and making mental or written notes about the discussions for later summative activities (whole-class presentations or sharing, or instructor summary). The noise and activity level tend to be high. With whole class presentations, the challenge is to ensure that all students are engaging with the content of the presentation, and to do as much as possible to have broad participation. Ultimately, the goal is to have as many students as possible engaged in creating mathematics and making sense of the core ideas of the course for themselves, so that students develop the mathematical thinking skills that will serve them long after the course is over. One of the benefits of inquiry-based learning and the active modes of instruction described here is that there are many opportunities to gain evidence of students’ thinking—to conduct formative assessment, so that adjustments to instruction can be made before an exam reveals critical gaps or misconceptions among the students. And, as the recent Proceedings of the National Academy of Sciences paper indicates, evidence favors active learning in STEM courses over lecture. So, whether an instructor prefers groups or presentations, if students are engaged, chances are good that they are learning. 

Readers, what classroom organization works for you? 

Dealing with Misconceptions, Part 2

In my last post, I dealt with ways of handling student misconceptions in the moment. In this post, I will discuss how I follow through to make the course better, both from class to class and from semester to semester. I will draw on my Transition to Proof course as the primary example.

I am teaching Transition to Proof for about the fifth time this semester. At this point, I have a good idea of which proofs will cause students the most trouble, and I have specific ideas of what attempts I am likely to see. I have built up this mental cache of ideas by noting what sorts of proofs I have gotten in the past, keeping track of the activities and problems I have used with students, and remembering what kind of effect those items had on the students.

I use this information from class to class to make decisions about whether a misconception from one class needs to be dealt with in the next class. For instance, students had some misconceptions and general confusion around the logical terms "contrapositive," "converse," and "negation." One way I deal with misconceptions is to have multiple problems that center on the same topic, so that we see the same idea come up repeatedly in different ways. In addition, for the first time, I am trying to make use of this information by creating concept questions and using interactive engagement alongside IBL. 

Interactive engagement (IE) has been around for some time, and is probably best established in physics as a mode of instruction that produces significant gains in students' conceptual understanding. More recently, some evidence has emerged that IE has a significant impact in students' understanding of calculus as well (Epstein).

The concept questions I have created are short, multiple-choice or true-false questions that attempt to elicit students' misconceptions, so as to create a space for dialogue that leads students to confront the error in their thinking, and therefore come away with a more robust understanding. In the case of the logical terms, I created a few items that asked questions like, "Which of the following statements is true exactly when the statement, If A, then B, is true?" Or, "Which statement has the opposite truth value to, If A, then B?"

I have been using the questions by projecting them at the front of the class (and making them available for students to view as a Blackboard quiz or Google Form), asking students to answer each question, and then discuss their answers with a partner. Then we go through solutions as a class, usually by having a student explain their answer verbally, but sometimes we may draw a diagram to assist in the explanation. This has led to some good discussions, and has allowed us to zero in on specific issues students are having, without having to prove another theorem or proposition. It's relatively quick and focused on the issues students are having. 

In a recent class, there was disagreement about which choice was an equivalent expression of the definition of one-to-one. Because there were a number of students holding each opinion, there was a lively discussion among students, in pairs or small groups, attempting to decide which option was correct. The room was abuzz with mathematical discussion. As a whole class, I called on students and found three different answer choices that students thought might be correct. Eventually, a couple of students were able to remind the class of what we have learned about the contrapositive, and thereby convince the class of the correct option.

These kinds of discussions are exactly the kind of interactions that let students overcome misconceptions and solidify their understanding of key ideas from the course.

How else do you follow up to address student learning issues?