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? 

The instructor's role in an IBL class, Part 2

In my last post, I described three of five aspects of an IBL instructor’s role: managing expectations, managing emotions, and keeping the students engaged. In this post, I take up the remaining aspects.

Finding out what students know is an ongoing task. For those who follow such things, this is also called formative assessment, and it is a critical part of a successful teaching-and-learning experience. There are a couple of purposes for formative assessment. One is to formulate responses as the instructor that will help students move forward in understanding the mathematics. Another is to identify opportunities where specific students may benefit from working on particular problems, or to find opportunities for students to share what they know at a time when it will benefit the class. One of the great benefits of teaching via IBL is that there are so many opportunities to hear from students and to develop a picture of where they are in their mathematical development. By listening to discussions between and among students in pairs or groups, and during presentations and the ensuing discussions, the instructor should have a good idea of when students might have something especially productive to contribute, or when a discussion from one group should be shared with the whole class, for instance. Notice that while formal quizzes or exams remain a source of information, as an IBL instructor the opportunities to find out what students are thinking go far beyond this, and are embedded in the everyday tasks of the class. Also notice that grades are not really a purpose of formative assessment. The focus is on student learning, and how to enhance it.

Fitting the problems to the students is a task that begins before the semester, but continues to occur through the semester. Before the semester, the major task of an IBL instructor is to determine the main course content goals, which could be particular theorems, skill with specific kinds of problems, or facility with certain techniques. Sources for beginning this work on your first attempt with a class might be the department course syllabus, and/or standard textbooks. From these, the instructor’s job is to put a priority on the central ideas. Then, the instructor works on developing a sequence of problems, lemmas, etc., that will carry the students from their anticipated starting point through to the goal results. As the semester gets underway, the IBL instructor works (1) to find problems to engage particular students (often the highest students or the ones struggling the most), or (2) to use to the students' advantage what they know and are thinking about, and to respond with a set of problems that provide an alternate path to the results, and (3) to modify the difficulty of the problems as the students may be more or less advanced than anticipated and more or fewer lemmas are needed between the main results to keep the majority of the class moving in a positive direction. 

I hope this captures at least some of the key ingredients in the recipe for a successful IBL course. Let me know your thoughts.

The instructor's role in an IBL class, Part 1

In discussing inquiry-based learning (IBL) with college faculty and K-12 teachers, I find that one of the difficult things to do is to communicate what the instructor’s role is, as opposed to what it is not. Many people are familiar with such mantras as, Teaching is not telling, or, Don’t lecture. These are helpful, but then instructors are left wondering what to do. In this post, I want to briefly describe a few important duties of an instructor in an IBL classroom. The roles I am going to describe are not mutually exclusive categories, but interwoven threads. Nonetheless, I call these out because I think they capture some critical aspects of the flavor of teaching an IBL course. These duties are: managing expectations, managing emotions, keeping the students engaged, finding out what students know, and fitting the problems to the students. In this post, I will deal with the first three aspects, and deal with the last two in my next post.

Managing expectations is a primary duty in an IBL classroom. Students come to class, and especially, come to math class, with expectations, including unconscious ones, about what is going to happen. These expectations are often something like, "The teacher will show me a formula and examples, and I just have to memorize and repeat what the teacher does on similar examples." In contrast, in an IBL classroom, students are expected to bring their ideas to problems for which the path to solution may not be clear. Students are not used to being asked to think things through for themselves in math class, and this leads to frustration. The teacher’s first duty is to make it clear to students that they will need to bring their own ideas, and that they will often not know what to do, or they will do things that turn out not to work, but as a class, they will make progress in understanding the mathematics. In class, the IBL instructor can say things like, “This is going to be different, but you will learn a lot,” or, “You’re going to experience mathematics the way that mathematicians do,” or, “You will get stuck a lot in this class. That’s ok. You can even write ‘STUCK!’ on your work when that happens. The important thing is to learn from what you try, both what works and what doesn’t."

In tandem with managing expectations is managing emotions. As mathematicians, we experience frustration as we search for a solution, and we take wrong turns, or the path to the solution is longer than we hoped. Students feel this frustration. If you are managing expectations properly, then students should know that frustration is normal and expected. However, there is more to the instructor’s role than that. If the entire class is boiling over with frustration, the instructor has a duty to respond. If the students are left to flounder, a mutiny can begin to brew. The instructor may say things like, “It seems like this problem/theorem is really stumping us. Let’s brainstorm how we can find new ways to attack it,” or, “I am glad to see everyone is showing persistence on this problem. Sometimes the best way to get past a roadblock is to go around it. So why don’t we look at this {example, related theorem, special case} for now and then come back to the main problem,” or, “This problem is really giving us a rough ride. Let me tell you a quick story about this time when I was frustrated and how I got through it…”

Keeping the students engaged is a multifaceted task. I have written on this blog before about a specific engagement strategy, and about how to reach out to students who are reluctant to participate. I will add to what I’ve written before by mentioning the importance of managing classroom participation. When there is a student presenting work to the class, the instructor’s role is to ensure that the rest of the class is engaging with the presenter and/or the presenter’s work. This can be done in several ways. One of them is to give the students question/comment stems and to call on students to use the stem to offer a question or comment. Some of my stems include, “I like how you…” “What led you to think of…” “I’m not sure I follow how you got from … to …” Another way is to have the presenter pause and to do a check for understanding or a think-pair-share. When the presentation is done, the instructor then pushes the students to ensure that they agree with all aspects of the mathematics, and also see to it that they understand insofar as possible how the student came up with that solution. When students are working in pairs or small groups, the instructor should make sure that all group members are contributing to the conversation. When necessary, this can be actively managed when the instructor inserts him or herself into a group conversation to make statements such as, “I see that Ann and Bob are sharing their ideas. Carol, is what they’re saying making sense to you?” or, “I appreciate that you’re all giving Darryl your attention. Could someone else try to restate what Darryl has shared so far?” or, “It seems like this group has spent some time working independently, each person with his or her own ideas. I think now would be a good time to share some of your progress and see what you can learn from each other’s approaches to this problem."

In the next post, I will take up the other two aspects of the instructor's role.

Let me know your thoughts or other aspects of the instructor's role I have left out.

Formative Assessment

An idea that has been coming up a lot in several different contexts for me is formative assessment. Let me start by stating what formative assessment is, and what it is not. Formative assessment is information gathered from students to assess their current understanding, with the purpose of using that information to make instructional decisions. Formative assessment need not be a formal exam, and cannot be an exam if it will not impact instruction after the exam. Thus, when I am speaking with folks in school districts, sometimes I say “formative assessment” and this is interpreted to mean something like quarterly (or whatever frequency) benchmark tests. In fact, my experience is that teachers rarely are given the time or resources to use benchmark tests as formative assessment. Instead of informing further instruction, they disrupt instruction, as teachers interrupt their regular lessons to review for the benchmark exam, and after the exam, hurry to move on to whatever is next on the overstuffed curriculum pacing guide. As described here, benchmark tests are NOT formative assessment. Instead, when I think of formative assessment, I think of day-to-day tasks that allow the teacher to gather information about what students know, and give the teacher the chance to address gaps and other issues students are having in understanding the ideas of the course.

This semester, one of the best things I learned was how to build questions that would serve as good formative assessment. The questions I have been using were described in my earlier post discussing how I deal with misconceptions. Since this post is about formative assessment, I want to describe how I react to students’ responses to the questions. The questions I have been using are frequently true-false questions, or sometimes multiple choice. Students are first given time to respond to the questions alone (most commonly I have been using Google Forms to collect their initial response), and then to discuss their responses with their peers. Since the discussion may alter their opinions, I then ask for a show of hands for each answer choice. I have seen a few things happen. 

1. Sometimes there seems to be broad consensus on the correct answer. In that case, I will ask one or two students to summarize the reasons for the correct choice, record the answer for the class, and move on.
2. Sometimes, the hand votes are close to equally split between two choices. In this case, I try to get at least one person on each side to articulate the reasons for their answer choice. Then I either ask follow-up questions or I ask other students to add to the arguments for each side. Sometimes this is enough for students to see which is the correct choice. I hear students saying things like, “Oh, I didn’t think of that example,” or, “I changed my mind.” If I feel that there is consensus, then I will record the correct answer at the board and summarize the discussion. If the discussion is not progressing, then I usually prompt students to come up with one or more examples or to draw a graph or diagram related to the statement. Since the topics in these questions are not new, students generally have enough knowledge to resolve the questions. Lastly, I may refer them back to previous work that we did, or pull up the work of a student from the previous class meeting. One of these moves is generally enough to push the discussion toward the correct answer.
3. The third thing that sometimes happens is that I see very few hand votes. I generally take this as a sign of confusion. In that case, I will do one of two things. Sometimes, I tell the students that I see very few votes, and that they need to go back to the discussion with their partner for a couple more minutes to settle things before we can have a class discussion. Then we vote again. More typically, I call on students who raised their hand to explain their thinking, and then call on those who did not vote to see if they are following the argument. After enough students have participated, and I am satisfied that the main ideas have been discussed and reiterated sufficiently, I will ask students for any final questions, and then summarize the discussion and record our answer.  

Formative assessment can be a powerful tool. How do you assess students and use that to inform your instructional moves?