Book: Transforming Mathematics Instruction

Liljedahl, P. (2014). The affordances of using visually random groups in a mathematics classroom. In Y. Li, E. Silver, & S. Li (eds.) Transforming Mathematics Instruction: Multiple Approaches and Practices. New York, NY: Springer.  (pp. 127-144). ISBN: 978-3-319-04993-9

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Chapter 7: The Affordances of Using Visibly Random Groups in a Mathematics Classroom – Peter Liljedahl

Pg 127 What if, instead of strategic grouping schemes, the assignment of groups was done randomly?

Results indicate that the use of visibly random grouping strategies, along with ubiquitous group work, can lead to:
(1) students becoming agreeable to work in any group they are placed in,
(2) the elimination of social barriers within the classroom,
(3) an increase in the mobility of knowledge between students,
(4) a decrease in reliance on the teacher for answers,
(5) an increase in the reliance on co-constructed intra- and intergroup answers, and
(6) an increase in both enthusiasm for mathematics class and engagement in mathematics tasks.

Group Work

Pg 128. The goals for strategically assigning groups can be broken into two main categories: educational and social (Dweck and Leggett 1988 ; Hatano 1988 ; Jansen 2006 ).

When a teacher groups her students for pedagogical reasons, she is doing so because she believes that her specific arrangement will allow students to learn from each other.

In essence, the diversity of potential goals for group work and the mismatch between educational and social goals in a classroom almost ensures that, no matter how strategic a teacher is in her groupings, some students will be unhappy in the failure of that grouping to meet their individual goals. How to fi x this? One way would be to remove any and all efforts to be strategic in how groups are set.

Random Groupings

Pg 130 Regardless of the particulars of the method, however, the norm that was established in each of the classes that I worked in was that the establishment of groups at the beginning of class was not only random, but visibly random. Once in groups, students were then universally assigned tasks to work on, either at their tables or on the whiteboards around the room. The students stayed in these groups throughout the lesson, even if the teacher was leading a discussion, giving instructions, or demonstrating mathematics.

Although often met with resistance in the beginning, within 3–4 weeks of implementation, this approach has consistently led to a number of easily observable changes within the classroom:

• Students become agreeable to work in any group they are placed in.
• There is an elimination of social barriers within the classroom.
• Mobility of knowledge between students increases.
• Reliance on the teacher for answers decreases.
• Reliance on co-constructed intra- and intergroup answers increases.
• Engagement in classroom tasks increases.
• Students become more enthusiastic about mathematics class.

Results and Discussion

Students Become Agreeable to Work in Any Group They Are Placed In

pg 134 …the defining quality of Ms. Carley’s grouping scheme was the random nature of it. Once he came to see that it was both random and that the random outcomes would be respected, he became more relaxed about it. For Jasmine, however, the defining quality was the short-term commitment that the grouping strategy demanded. When I had first observed Jasmine’s antics, I had assumed that it had to do with trying to be close to her friends when, in reality, she was trying to avoid being “stuck” with a group she didn’t like. Once she became confi dent that the groups were temporary, she stopped trying to manipulate the groups.

…Jennifer’s observation coincides perfectly with the subsiding of any residual visible opposition to the random groupings. Although she was not overtly opposed to the groupings in the first week, her mention of the practice continuing in the second and third weeks and how that was a sign that “it’s here to stay” indicates a resignation to the new classroom norm (Yackel and Cobb 1996 ) that is likely shared by many of her peers. This is a different phenomenon from Hunter, who saw the randomness in the cards, or Jasmine, who focused on the temporariness in each grouping.
These three themes occurred and reoccurred in many of the conversations I overheard, conversations I was part of, and in interviews.

Pg 135

There Is an Elimination of Social Barriers Within the Classroom

Researcher: Can you think of any problematic situations that you think will prevent this [random groupings] from being successful?
Ms. Carley: The obvious one is the split between the Asian and Caucasian students.
Researcher: What do you mean split ?
Ms. Carley: It’s almost as though we have two distinct cultures in this school with almost no overlap. The Caucasian students have their own social groupings, not all together. And the Asian students have their own. And there is almost no mixing between the two. In fact, it’s almost as though they aren’t even aware of each other.

When Ms. Carley allowed the students to self-select who they wanted to work with, the selections were always guided by this “split”. This is not to say that there were any racial tensions in the group. I observed no evidence of dislike or disdain for each other. It really was just as Ms. Carley had described—two distinct social groupings.

pg 136

After 3 weeks of implementing visibly random groups, some interesting phenomena began to emerge. Whereas in the first few days after implementation there was an awkwardness present in the first few minutes of group work, now there was an “at ease ness ” about the way the students came together. This was more than comfort with a process, however. It was more akin to a familiarity between students.

…the two groups are not only talking to each other, they are talking about each other…

pg 137

…Researcher I know that you used to like to sit with Jasmine and Kim a lot. How is it being away from them?
Samantha I’m not away from them. I still see them all the time and I did sit with Kim and Charles the other day. But it’s different now. Before we would just sit and talk. Now we are working on stuff at the boards and stuff.
There isn’t a lot of time to just socialize anyway.

Mobility of Knowledge Between Students Increases

After implementation, group work became ubiquitous. The main activity in these groups was to work through a series of tasks that Ms. Carley set during her lessons. These were originally “try this one” tasks that followed direct instruction. But as the study went on, Ms. Carley began to also use tasks as a way to initiate discussions. The tasks also became more challenging, requiring the students to do more than just mimic the examples already presented on the boards. This “ramping up” of the use of tasks was accompanied by a number of easily observable changes in the way in which the groups worked, with the most obvious of which was the way in which the knowledge moved around the room.

pg 138
Immediately after implementation, group work looked very much like it did prior to implementation—the students worked largely independent of each other, interacting only to check their answers with their group members or to ask one or another to explain how to do something. After 4 weeks, however, group work looked very different. Students now spent almost no time working independently. Instead, they spent their time working collaboratively on the tasks set by Ms. Carley. This collaboration consisted of discussion, debate, and the sharing and demonstration of ideas. In part this was due, of course, to the increasing demand and frequency of the tasks set by the teacher. But it was also due to the coalescing of the groups into collaborative entities.

Ms. Carley: You know, I’ve thought a lot about that lately. At first I thought it was all due to the tasks. In fact, I was talking to a colleague who was asking about my class. She was asking for a copy of the tasks so she could start using them with her students and that’s when I realized that it’s sort of a chicken and egg thing. If we spring the tasks on the students before they know how to work in groups, then it won’t work. At the same time, if we try to teach them how to work in groups without having something to work on, then it won’t work either.

Researcher: So, how did you manage it in this class? What came first?
Ms. Carley: I think the random groups came first. That broke the mould on what group work had looked like in the past and gave me room to introduce a new way of working.

Prior to implementation, group work had a well-defi ned set of actions and behaviours associated with it.
These norms were not conducive to the collaborative skills and affordances necessary to increase the demand on students vis-à-vis the ubiquitous use of tasks. The introduction of random groups into the classroom shattered the existing norm and allowed for a new set of classroom norms to be established that were more conducive to collaboration.

The collaboration now visible in the room went beyond the intra-group activity, however. Intergroup collaboration also became a natural and anticipated part of every class. This often took one of three forms: (1) members of a group going out to other groups to “borrow an idea” to bring back to their group, (2) members of a group going out to compare their answer to other answers, (3) two (or more) groups coming together to debate different solution or a combination of these…

pg 139

Researcher In other classes I have been in, I don’t see that happening. You know, groups sharing with each other. Sam That’s probably because they don’t work together as much as we have. I mean, we are always together with different people. I think I have worked with everyone in this room now. If you asked me who I worked with yesterday I’m not sure I could tell you. And if you asked the teacher to tell you who was in which group today, I don’t think she could tell you either. When we were trying to figure out which answer was correct, we were like one big group.

What Sam is describing is what I have come to call the porosity of groups. Although group boundaries are defined for the period, these boundaries are clearly temporary and arbitrary. This allows for them to also be seen as open and allowing for the free movement of members from one group to another to extend the collaborative reach of the group. When asked about this, many students mention that they feel that they are free to move around the room as necessary to “get the job done”. Along with this mobility of groups and group members comes mobility of knowledge— the movement of ideas, solution strategies, and solutions around the room. In fact, it is the need to move knowledge that prompts the movement of individuals as they go out “to borrow an idea”.

Pg 140

…in order for the group work to become effective and meaningful, the teacher needs to stop answering questions and, as the group work becomes effective and meaningful, the students stop needing to ask questions. Ms. Carley’s class has become a collective making use of both intra- and intergroup collaborations.

This is not to say that the role of the teacher is diminished. Ms. Carley still sets the tasks, the groups, and the expectations. More importantly, however, she monitors the flow of knowledge around the room:

Researcher: I noticed that you were forcing some groups together today. What were you trying to achieve?
Ms. Carley: It depends. Sometimes I am trying to crash ideas together. Other times I am trying to help a group get unstuck. Which groups do you mean?
Researcher: I mean when you sent one whole group from over there to over here.
Ms. Carley: Ah. Well, that group over there had gotten an answer pretty quickly. As it turned out, it was the right answer, but I didn’t think they had done enough work checking their answer so I sent them over to that group to shake their confidence a little bit.
Researcher: How so?
Ms. Carley: Well, that group had a different answer and that would force the two groups to figure out what was going on.

Not only is Ms. Carley monitoring the flow of knowledge in the room, she is manipulating it—forcing it to move in certain directions and moving it for a variety of different reasons. In so doing, her role in the classroom has changed.

Students Become More Enthusiastic About Mathematics Class

Pg 142

I had observed that the introduction of random groupings was pivotal in producing broad changes in the classroom. However, these changes were more than just changes to the way the class was run. The introduction of random groupings led to, and allowed for, changes in the students, the teacher, and what was possible in this new setting.

The students became open to working with anyone. The social barriers that existed in the classroom came down and the classroom became a collaborative entity that was not defined by, or confi ned to, the boundaries set by the teacher. As these barriers came down and the class coalesced into a community, their reliance on the teacher as the knower diminished and their reliance on themselves and each
other increased. Their enjoyment of mathematics (the class, if not necessarily the subject) increased as well as their engagement.

…the nonstrategic use of visibly random groupings turned out to be a better strategy than the aforementioned strategic grouping schemes.

Student change aside, Ms. Carley altered the way she used tasks as well as the way she answered questions. She found that she no longer needed to be the knower or the teller in the room. She changed the timing and the method of her direct instruction, and she began to rely much more on her ability to manipulate groups and move ideas around the room.

Their role changed from “try this one” to objects around which group work was organized.
They increased in frequency and difficulty, and they became the objects and objectives of lessons.