ChapterPDF Available

The Affordances of Using Visibly Random Groups in a Mathematics Classroom


Abstract and Figures

Group work has become a staple in many progressive mathematics classrooms. These groups are often set objectives by the teacher in order to meet specific pedagogical or social goals. These goals, however, are rarely the same as the goals of the students visa -vis group work. As such, the strategic setting of groups, either by teachers or by students, is almost guaranteed to create a mismatch of goals. But, what if the setting of groups was left to chance? What if, instead of strategic grouping schemes, the assignment of groups was done randomly? In this chapter, I explore the implementation of just such a strategy and the downstream effects that its implementation had on students, the teacher, and the way in which tasks are used in the classroom. 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 inter-group answers, and (6) an increase in both enthusiasm for mathematics class and engagement in mathematics tasks.
Content may be subject to copyright.
Chapter 7
The Affordances of Using Visibly Random Groups
in a Mathematics Classroom
Peter Liljedahl
Abstract Group work has become a staple in many progressive
mathematics classrooms. These groups are often set objectives by the
teacher in order to meet specific pedagogical or social goals. These goals,
however, are rarely the same as the goals of the students vis-a-vis group
work. As such, the strategic setting of groups, either by teachers or by
students, is almost guaranteed to create a mismatch of goals. But, what if
the setting of groups was left to chance? What if, instead of strategic
grouping schemes, the assignment of groups was done randomly? In this
chapter, I explore the implementation of just such a strategy and the
downstream effects that its implementation had on students, the teacher,
and the way in which tasks are used in the classroom. 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 inter-group answers, and (6) an
increase in both enthusiasm for mathematics class and engagement in
mathematics tasks.
Keywords Collaboration Group work Social barriers Integration
Mobilization of knowledge Randomization
P. Liljedahl
Faculty of Education, Simon Fraser University,
8888 University Drive, Burnaby, BC, Canada V5A 1S6
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.
P. Liljedahl
Group work has become a staple in the progressive mathematics classroom
(Davidson & Lambdin Kroll, 1991; Lubienski, 2001). So much so, in fact,
that it is rare to not see students sitting together for at least part of a
mathematics lesson. In most cases, the formation of groups is either a
strategically planned arrangement decided by the teacher, or self-selected
groups decided by the studentseach of which offers different
affordances. The strategically arranged classroom allows the teacher to
maintain control over who works together and, often more importantly,
who doesn’t work together. In so doing she constructs, in her mind, an
optimal environment for achieving her goals for the lesson. Likewise, if the
students are allowed to decide who they will work with, they will
invariably make such decisions strategically in the pursuit of achieving
their goals for the lesson. In either case, the specific grouping of the
students offers different affordances in the attainment of these, often
disparate, goals.
But, what if the selection of groups was not made strategicallyby either
party? What if it was left up to chancedone randomlywith no attention
paid to the potential affordances that specific groupings could offer either a
teacher or a learner? In this chapter, I explore a different set of affordances
that result from the use of randomly assigned collaborative groupings in a
high school mathematics classroom.
Group Work
The goals for strategically assigning groups can be broken into two main
categories: educational and social (Dweck & Leggett, 1988; Hatano, 1988;
Jansen, 2006). Each of these categories can themselves be broken into sub-
categories as displayed in figure 7.1. 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. This
may necessitate, in her mind, the need to use homogenous groupings or
heterogeneous groupings where the factor that determines homo- or
heterogeneous groupings can range from ability to thinking speed to
curiosity. When she groups students in order to be productive, she is
looking for groupings that lead to the completion of more work. This may,
for example, require there to be a strong leader in a group for project work.
It may also mean that friends or weak students do not sit together, as such
pairings may lead to less productivity. Groupings designed to maintain
3 The Affordances of Using Visibly Random Groups in a Mathematics Classroom
peace and order in the classroom would prompt the teacher to not put
‘trouble-makers’ together, as their antics may be disruptive to the other
learners in the class1. Interestingly, students may self-select themselves
into groupings for the same aforementioned reasons (Cobb, Wood, Yackel,
& McNeal, 1992; Webb, Nemer, & Ing, 2006; Yackel & Cobb; 1996).
More commonly, however, students group themselves for social reasons
(Urdan & Maehr, 1995) specifically to socialize with their friends.
Teachers too, sometimes form their groups to satisfy social goals. They
may feel that a particular group of students should work together
specifically because of the diversity that they bring to a setting.
Sometimes, this is simply to force a gender mix onto the collaborative
setting. Other times, it is more complex and involves trying to get students
out of their comfort zone; to collaborate with, and get to know, students
they don’t normally associate with. A teacher may choose to create a
specific grouping to force the integration of an individual student into a
group that they are not yet a part offor example, the integration of an
international student into a group of domestic students. Finally, and less
likely, a teacher may specifically wish for their students to work with their
friendsoften as a reward for positive performance or behaviour in the
1 From a researcher's perspective each of these goals, and the accompanying use of group
work, may be predicated on an underlying theory of learning and the role that peer
interaction plays in said theory. From the teacher's perspective, however, these decisions are
less likely to be made based on theory, and more likely to be made according to what they
believe about the teaching and learning of mathematics in coordination with their beliefs
about the utility of group work (Liljedahl, 2008).
P. Liljedahl
Fig. 7.1 Goals for strategic groupings
Regardless of the goals chosen, however, there is often a mismatch
between the goals of the students and the goals of the teacher
(Kotsopoulos, 2007; Slavin, 1996). For example, whereas a teacher may
wish for the students to work together for pedagogical reasons, the
students, wishing instead to work with their friends, may begrudgingly
work in their assigned groups in ways that cannot be considered
collaborative (Clarke & Xu, 2008; Esmonde, 2009). These sorts of
mismatches arise from the tension between the individual goals of students
concerned with themselves, or their cadre of friends, and the classroom
goals set by the teacher for everyone in the room. Couple this with the
social barriers present in classrooms and a teacher may be faced with a
situation where students not only wish to be with certain classmates, but
also disdain to be with others. 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 fix this? One way would be to remove
ANY and ALL efforts to be strategic in how groups are set.
Random Groupings
Over the last six years I have done research in a number of classrooms
where I have encouraged the teachers to make group work ubiquitous,
where new groups are assigned every class, and where the assignment of
these groups is done randomly. In every one of these classrooms the lesson
begins with the teacher generating random groups for the day. The specific
method for doing this varies from teacher to teacher. Some give out
playing cards and have students group themselves according to the rank of
the card they have drawn. Others have students assigned a permanent
number and then draw groups of 3 or 4 numbered popsicle sticks or
numbered disks randomly from a jar. In other classes, the students watch
the teacher randomly populate a grid with numbers wherein each row of
the grid then forms a group. One teacher I worked with had this grid
placement done automatically by a program displayed on an interactive
whiteboard. Another teacher I worked with had laminated photographs of
all of the students and distributed these into groups by shuffling and then
randomly drawing 3 or 4 photos at a time. 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
5 The Affordances of Using Visibly Random Groups in a Mathematics Classroom
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 three to four
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 inter-group answers increases.
Engagement in classroom tasks increase.
Students become more enthusiastic about mathematics class.
Ironically, these are often the exact affordances that teachers’ strategic
groupings of students is meant, but often fails, to achieve. How is this
possible? What is it about the use of visibly random groups that allows this
to happen? Drawing on data from one classroom this chapter looks more
closely at these aforementioned observed changes as well as what it is
about visibly random groupings that occasions these changes.
The data for this study was collected in a grade 10 (ages 15-16)
mathematics classroom in an upper-middle class neighbourhood in western
Canada. The students in the class were reflective of the ethnic diversity
that exists within the school at large. Although there are students from
many different cultures and backgrounds in the school, and the class, the
majority of students (> 90%) are either first or second generation
immigrants from China or Caucasian Canadians whose families have been
in Canada for many generations. These two dominant subgroups are almost
equal in representation. This, almost bimodal, diversity is relevant to the
discussion that will be presented later.
The classroom teacher, Ms. Carley (a pseudonym), has eight years of
teaching experience, the last six of which have been at this school. In the
school year that this study took place, Ms. Carley decided to join a district
run learning team facilitated by me. This particular learning team was
organized around the topic of group work in the classroom. As the
facilitator, I encouraged each of the 13 members of the learning team to
start using visibly random groups on a daily basis with their classes. Ms.
Carley had joined the team because she was dissatisfied with the results of
group work in her teaching. She knew that group work was important to
P. Liljedahl
learning, but, until now, had felt that her efforts in this regard had been
unsuccessful. She was looking for a better way, so when I suggested to the
group that they try using visibly random groups she made an immediate
commitment to start using this method in one of her classrooms. This, in
turn, prompted me to conduct my research in her class.
The data was collected over the course of a three month period of time
from the beginning of February to the end of April. The time frame is
significant because it highlights that this was not something that was
implemented at the beginning of a school year when classroom norms
(Yackel and Cobb, 1996) are yet to be established and students are more
malleable. The fact that the change occurred mid-year allowed me the
unique opportunity to compare classroom discourse, norms, and patterns of
participation before and after implementation. Initially, I was present for
every class. This included three classes prior to implementation as well as
the first three weeks (8 classes) after initial implementation. After this, I
attended the classes every two or three weeks until the end of the project.
I became a regular fixture in the classroom and acted, not only as an
observer, but also as a participant (Eisenhart, 1988), interacting with the
students in their groups and on the tasks set by the teacher. The data
consists of: field notes from these observations, interactions, and
conversations with students during class time: interviews with Ms. Carley:
and interviews with select students. Interviews were conducted outside of
class time and audio recorded. Over the course of the study, Ms. Carley
was interviewed, if only briefly, after every observed lesson. During this
time frame 12 students were also interviewed, with two of them being
interviewed twice. These data were coded and analysed using the
principles of analytic induction (Patton, 2002). "[A]nalytic induction, in
contrast to grounded theory, begins with an analyst's deduced propositions
or theory-derived hypotheses and is a procedure for verifying theories and
propositions based on qualitative data" (Taylor and Bogdan, 1984, p. 127
cited in Patton, 2002, p. 454). In this case, the a priori proposition was that
the changes that I had observed in other classrooms were linked to the use
of a visibly random grouping scheme. This proposition became the impetus
for the collection of data in that it drove what I was looking for and how I
was looking. It became the lens for my observations and it motivated my
interview questions. It also pre-seeded the themes that I was looking for in
the coding of the data.
This is not to say that my data collection and analysis was blind to the
emergence of new themes. As a participant/observer in the classroom I was
aware of, and deliberately looking at, a great many things going on around
me. During the coding and analysis of the data I was looking for nuances
in the relationship between visibly random grouping schemes and the
changes I had observed. So, despite the fact that I had a priori themes in
7 The Affordances of Using Visibly Random Groups in a Mathematics Classroom
mind, I still coded the data using a constant comparative method (Creswell,
2008). This recursive coding allowed for the emergence of not only
nuanced themes, but also new themes.
Results and Discussion
Similar to the other classes wherein I have observed the implementation of
random grouping schemes Ms. Carley’s class exhibited the same
observable changes. In what follows I explore each of these changes more
thoroughly, illuminating the nuances of each with results from the data.
Students become agreeable to work in any group they are
placed in
Group work is not something that is foreign to the students in Ms. Carley’s
class. From time to time she allows the students to sit in pairs or threes to
work on their homework and the class had already done one group project
on graphing where the students were allowed to self-select who they
worked with. When Ms. Carley decided to implement a more ubiquitous
approach to group work in general and the use of random groups in
particular she chose to use a standard deck of playing cards to generate the
groups. She had 30 students in the class and she had decided to have the
students work in groups of three. So, she selected from the deck 3 cards of
each rank (ace ten). These were shuffled and then the students were
allowed to each select one card. Although she experimented with the
number of students per group, and had to make adjustments based on
absences, this is a grouping scheme that she stayed with for the duration of
the study.
On the first day the students were not told what was going on but just
presented with the cards as described. Later, I learned that many of the
students had thought that "it was a magic trick". When every student had a
card Ms. Carley announced that these would be the groups that they would
be working in and assigned a "station" for each group depending on their
card. This was an interesting time. Many of the students went dutifully to
their stations. However, there were a few students who I observed were
trying to fix it so that they were with their friends. I will elaborate on two
of these cases in particular.
Hunter, despite his card, went directly to the station where his friend
Jackson was sitting. This did not go unnoticed as Ms. Carley immediately
noticed that this group now had four members instead of three. When she
dealt with this she immediately challenged Hunter to see his card. When I
asked her about this later she said that "it had to be Hunter. It is always
P. Liljedahl
Hunter. He is a bit of a scammer and he likes to be with Jackson". In the
flurry of the first few minutes of class Ms. Carley had to perform a similar
check on one other group of four.
Unnoticed by Ms. Carley, however, was the situation that unfolded
immediately in front of me. Jasmine approached a group of three and took
the card out of one of the group members’ hand replacing it with her own
card and said, "you’re over there", gesturing towards one of the corners of
the room. From my initial observations of the class and my conversations
with Ms. Carley I knew that Kim, Samantha, and Jasmine, are very close
friends, are part of the "in" crowd within their grade, and tend to stick very
close together during free time and when allowed in their other classes.
The group that Jasmine approached had Samantha in it.
In general, this sort of jockeying behaviour was observed for the first three
classes after implementation. Hunter did try it again but Ms. Carley
intervened even before he got to Jackson and on the third day Hunter and
Jackson legitimately ended up togethermuch to the chagrin of Ms.
Carley. Jasmine, however, was successful each time she tried to switch
groups using the same strategy. After the first week, however, the
behaviour stopped for both Hunter and Jasmine. At this point I interviewed
both Hunter and Jasmine about their antics.
Researcher So, I noticed that last week you tried a few times to sit with
Jackson. Are you still trying to do so?
Hunter No.
Researcher Why not?
Hunter At first I thought that the teacher was trying to keep us
apart. Then, on Friday, we got to work together.
Researcher So, do you still think the teacher is trying to keep you
Hunter No. I don’t think she likes us working together, but when
the cards came up the way they did she didn’t change it. I
guess it’s up to the cards now.
Researcher I saw what you did last week.
Jasmine What do you mean?
Researcher I saw how you switched groups.
Jasmine Oh that. That’s nothing.
Researcher But you didn’t do it this week. Why not?
Jasmine I guess it doesn’t matter so much. I mean, it is just for one
class and then the groups change again.
Researcher What does that have to do with it?
Jasmine At first I was worried that I was going to be stuck with that
group for a long time, like when we worked on the project
or in my other classes.
9 The Affordances of Using Visibly Random Groups in a Mathematics Classroom
Researcher What happens in the other classes?
Jasmine My English teacher changes the seating plan every month
and then you’re stuck there forever.
For Hunter, 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 confident that the groups were temporary she
stopped trying to manipulate the groups.
I also interviewed Jennifer in the third week after implementation. I
selected Jennifer because she had shown no overt objections to the
grouping schemes used in the class.
Researcher I'm wondering what you think about all this grouping stuff
that is going on.
Jennifer Its ok I guess. It doesn't matter what I think though, it looks
like it's here to stay.
Researcher What do you mean by "it's here to stay"?
Jennifer Well, when the teacher started class on last Monday the
same way I knew that this is the way it was going to be.
When she started class today [Monday of the third week] I
was sure of it.
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. Sometimes they
occurred in isolation as in the excerpts presented above. Other times they
were present in combination with each other.
More interestingly, resignation to a new norm became the only thing
commented on by the third week. That is, regardless of what the students
thought about the introduction of visibly random groups the residual effect
was that "it is how we do things in this class". Even in the last week of the
study when I asked specific students to recall the early days of the use of
P. Liljedahl
the playing cards, their recollections of it was that it was just the
introduction of a new way to do things. That is, although the
randomization being visible, the cards being respected, and the groups
being only for one class were of great importance in the first weeks, what
endured to the end was just the norm. This is in alignment with Yackel and
Cobb's (1996) observation that norms are not something that are imposed
on a class, but are negotiated between the teacher and the students. The
fact that the grouping scheme was visibly random and that the groups were
only for one period were important elements in these negotiations.
There is an elimination of social barriers within the
As mentioned earlier, there is an almost bimodal diversity in both the class
and the school where the study took place. My observations of this "split"
are exemplified in the conversations that I had with Ms. Carley prior to her
implementation of random groups.
Researcher Can you think of any problematic situations that you think
will prevent this [random groupings] from being
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.
Researcher I have noticed that. Is that normal you think?
Ms. Carley I don’t know about normal but it is certainly not unique to
this school. I have a good friend who teaches in Surrey and
she has seen the same thing but with different groups of
students. We talk about it often and what we can do about
What Ms. Carley describes is a situation that is easily observable in both
the hallways and in the classroom. 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.
11 The Affordances of Using Visibly Random Groups in a Mathematics Classroom
We both saw this as a formidable challenge and were simultaneously
anxious and hopeful about how the random groupings would play out.
It is quite possible that some of Jasmine's antics (described in the previous
section) were motivated by this social dichotomy. On both the first and
second day of implementation she was randomly assigned to a group that
had two Asian students in them. The second time that she "stole" someone
else's card she took it from the sole Asian girl in the group where she
wanted to be in. But, as stated in the previous section, these sorts of
behaviours by Jasmine and others in the class ceased after the first two
weeks of implementation as the students settled into the new norm. This is
not to say that the social divide had disappeared yet.
After three 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 easeness" 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. This can be
seen in the interview with Melanie.
Researcher Tell me about how your group work went today?
Melanie Fine.
Researcher Who were you with?
Melanie I was with Sam and … um … the guy … I don't know his
Researcher Frank?
Melanie That's it. Frank!
Researcher Can you tell me a little bit about Sam and Frank?
Melanie Ok. Sam is smart. I worked with her one time before. She
really knows what is going on so I try to listen carefully to
her when she has something to say. She's in my Science
class as well and her sister is in my English class.
Researcher How do you know that Sam's sister is in your English
Melanie Sam told me today.
Researcher What about Frank?
Melanie I don't know Frank that well, but my friend worked with
him last week and he said that Frank is a really nice guy.
To help orient this conversation it is useful to know that Melanie is
Caucasian and that both Sam and Frank are Asian. What is remarkable
about this is that there is an awareness about each other that is forming.
Sam is aware that Melanie is in her sister's English class and Melanie is
aware that her friend worked with Frank last week. These are both strong
indicators that the two groups are now seeing each otheraware of each
P. Liljedahl
otherin a way that Ms. Carley (and I) had observed was not happening
prior to implementation. Further, Melanie's interview reveals that the two
groups are not only talking to each other, they are talking about each other.
This is not to say that race was the only social barrier at play within this
classroom prior to implementation. As in any school, there was also a more
subtle, but very real, social hierarchy at play. There were students who
were "in" and students who were "out". As already mentioned, Jasmine,
Kim, and Samantha were part of the "in" crowd. Prior to implementation
they always sat together, and as seen, Jasmine worked hard to maintain this
togetherness at initial implementation. For Jasmine, this was eased by the
realization that the groups were short lived. For Samantha, it was eased by
the fact that the nature of the group work had changed.
Researcher It's been six weeks now since Ms. Carley started moving
you around. What do you think about it?
Samantha It's ok.
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.
Researcher How do you think Jasmine and Kim feel?
Samantha Jasmine is ok with it now. She wasn't at first. And Kim
never cared. She is really easy going.
It is obvious from this transcript that Kim is also at ease, and always was,
with the random grouping scheme. More subtle, however, is the mention of
Charles. Charles is an Asian boy definitely not in the "in" crowd. I'm pretty
sure that prior to implementation Samantha did not know his name. Now
she mentions him in passing. This points to what I was observing at this
point in the study Ms. Carley's class had jelled into a cohesive whole,
absent of any social divides.
There is a lot to be seen and to be discussed in regards to the breaking
down of social barriers, both racial and non-racial, and my naïve treatment
of it is not meant to diminish the rich traditions of such research (c.f.
DeVries, Edwards, & Slavin, 1978). I merely wanted to highlight the role
that the visibly random grouping scheme played in the breaking down of
some of these barriers.
13 The Affordances of Using Visibly Random Groups in a Mathematics Classroom
Mobility of knowledge between students increases
As mentioned, prior to implementation, group work in Ms. Carley's class
was something students did as they worked on their homework or on a
project. 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.
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
four 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.
Researcher So, the students seem to be working well together.
Ms. Carley Yes … I'm still amazed at exactly how well.
Researcher We've talked a lot about the tasks you are using and how
you are using them. Do you think the tasks are responsible
for the group work we are seeing now?
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?
P. Liljedahl
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.
Ms. Carley's synopsis aligns well with my observations. Prior to
implementation, group work had a well-defined 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
The collaboration now visible in the room went beyond the intra-group
activity, however. Inter-group 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 solutions … or a combination of these as exemplified in
my observations of Kevin's group in week four of the study.
Researcher Good problem today, huh? I didn't get a chance to sit with
you today. Can you tell me how you guys solved it?
Kevin Yeah, that was a tough one. We were stuck for a long time.
Researcher We were too [referring to the group I was working with].
What did you eventually figure out?
Kevin Well, we saw that the group next to us was using a table to
check out some possibilities and we could see that there
was a pattern in the numbers they were using so we tried
that. That sort of got us going and we got an answer pretty
quickly after that.
Researcher Was it the right answer?
Kevin It was, but we weren't so sure. The group next to us had a
different answer and it took a long time working with them
before we figured out which one was correct.
Kevin's recollection of the day's activities is reflective of what I observed
between these two groups and, in fact, many groups on a daily basis. When
I asked Sam (who was in the other group) about this, she had some
interesting insights about why this coming together of the two groups
worked so seamlessly.
Researcher Your group worked pretty closely with another group
today. How did you feel about the fact that they copied
from you?
15 The Affordances of Using Visibly Random Groups in a Mathematics Classroom
Sam Did they? I didn't notice. But it isn't really copying. We are
all just working together.
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". The free and
easy mobility of knowledge results in a marked decrease in the students'
reliance on the teacher as the knower.
Researcher Have you noticed anything else that has changed over the
last five weeks?
Ms. Carley I've noticed that I'm not answering as many questions
Researcher Are you not answering them or are you not being asked
Ms. Carley Both, I think. I know there was a point where I was
deliberately trying to not answer questions, trying to push
the students back into the groups to figure it out. But now
that is not a problem. They just don't ask me questions as
much anymore. It's like that chicken and egg thing again.
Similar to the relationship between the use of random groups and the use
of more challenging tasks, the relationship between the teacher not
answering questions and the students not asking questions seems to be in
some sort of symbiosis. That is, in order for the group work to become
effective and meaningful the teacher needs to stop answering questions
P. Liljedahl
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 inter group 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 move in certain directions and moving it for
a variety of different reasons. In so doing, her role in the classroom has
Researcher So, how are you liking your classroom these days?
Ms. Carley I'm loving it. I feel like the students are completely
different. I'm completely different. It's like I have a new
job and its WAY better than my old one.
Students become more enthusiastic about mathematics class
Ms. Carley is not the only one who is enjoying her new role, however.
Many of the students I either talked to as part of my classroom
participation or in interviews alluded to the fact that Ms. Carley's
mathematics class is now an enjoyable place to be.
Frank I like this class now.
James Math is now my favourite subject.
In the fifth week of the study I spoke with Jasmine about how she was
enjoying this class.
17 The Affordances of Using Visibly Random Groups in a Mathematics Classroom
Researcher So, it's been a while since that day where you were trying
to switch groups. How are you enjoying things now?
Jasmine I love this class. I mean, math isn't my favourite subject.
But I love coming here.
Researcher Why is that? What is it about this class that you love?
Jasmine I'm never bored. There is always something going on and
time passes so quickly.
Researcher I looked at Ms. Carley's attendance book. For the last four
weeks you have never missed a class or even been late. I
only looked at four weeks, what would I have seen if I
looked further back?
Jasmine You would have seen some absences and lots of lates. I
mean, it's not like I skipped class. I don't skip. It's just that
there were reasons to be away. I guess I now try not to let
there be reasons.
Researcher What about lates?
Jasmine I'm often late for my classes. Not just math.
Researcher But you haven't been late at all lately.
Jasmine Hmm … I guess I don't want to be.
Jasmine didn't like mathematics the subject, but she loved mathematics the
class. The changes that had occurred, which began with the random
groupings, had transformed the Ms. Carley's class into something that she
didn't want to miss out on.
This was a trend that I observed in many students. In terms of attendance,
absences and lates were down across the board. Prior to implementation,
Ms. Carley had an average of 3.2 absences per class and an average of 6.7
lates per class. Between week four and week seven after implementation,
the averages were 1.6 and 2.2 respectively. Ms. Carley's class became a
place where students wanted to be. Conversations with other students
echoed Jasmine’s sentiments. In my conversations with Chad, Stacey, and
Kendra I decided to push a little further by asking them to draw
Researcher So, how is this class different from other classes?
Stacey I like this one.
Researcher Ha … do you not like other classes?
Stacey I do. But not like this one. This one is way more dynamic.
We are always doing something new and …
Kendra And the beginning of every class is a bit of an adventure
when we get to find out who we work with.
Researcher It's been six weeks. Hasn't it gotten old yet … the thing
with the random groups.
Chad No. It's still fun.
P. Liljedahl
Researcher I want to continue with Stacey's comments. In what ways
is this class different form other classes?
Chad Hmm … we need to think in this class. There really is no
other way around it. In other classes you can sort of just
tune out, but not in here.
Kendra And you have to collaborate. There is no way I could get
by just doing it on my own, even if Ms. Carley would let
Researcher It sounds like a lot of hard work.
Stacey It is, but in a good way. I mean, like I'm never bored.
These comments speak not just to enjoyment, but also engagement. The
students need to be engaged in Ms. Carley's class and they seem to enjoy
this engagement. The comments of these students confirm what both Ms.
Carley and I had observed in the class as a whole.
Researcher So, what do you think? How is it going?
Ms. Carley My sense is that it is going really well. This week all of the
students really seem to be into it. Everyone shows up ready
to go, and then we go. There are no complaints, everyone
is smiling, and we get a lot done.
I stated at the outset that the changes that I observed in Ms. Carley's class
are reflective of the changes I had seen in many of the other classes in
which I had been privileged to participate as teachers made the decision to
start using visibly random grouping schemes. But, in the past, these had
just been observations. My more focused approach to studying Ms.
Carley's class confirmed my prior (and subsequent) observations, and also
informed and enlightened them. As in the other classroom, I had observed
that the introduction of random groupings were 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 confined 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
19 The Affordances of Using Visibly Random Groups in a Mathematics Classroom
of mathematics (the class, if not necessarily the subject) increased as well
as their engagement.
Figure 7.1 (above) showed how neatly the strategic educational and social
goals could be partitioned. When non-strategic grouping methods were
used, the resulting behaviours cannot be so easily partitioned into
educational and social affordances. For example, the increased mobility of
knowledge is a direct result from the students' increased reliance on intra-
and inter-group generated results. However, this cannot be separated out
from the fact that social barriers in the room have come down. Taken
together, the data showed that the use of visibly random groupings
produces student behaviour that can be seen as being both educational and
social in nature (see figure 2). As such, the non-strategic use of visibly
random groupings turned out to be a better strategy than the
aforementioned strategic grouping schemes.
P. Liljedahl
Figure 2: Results of non-strategic groupings
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. Tasks, too,
took on a new life in the class. 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
The introduction of visibly random groupings was the impetus that both
allowed for and necessitated the many other changes that I observed.
Through the renegotiation of classroom norms (Yackel & Cobb, 1996) the
students could not continue to behave as they had earlier, Ms. Carley could
not continue being the same teacher she had been prior to implementation,
and tasks could not have avoided evolving. Change begot change.
Clarke, D., & Xu, L. (2008). Distinguishing between mathematics classrooms in Australia,
China, Japan, Korea and the USA through the lens of the distribution of responsibility
for knowledge generation: public oral interactivity and mathematical orality. ZDM,
40(6), 963-972.
Cobb, P., Wood, T., Yackel, E., & McNeal, B. (1992). Characteristics of classroom
mathematics traditions: An interactional analysis. American Educational Research
Journal, 29(3), 573-604.
Creswell, J. W. (2008). Educational research: Planning, conducting, and evaluating
quantitative and qualitative research (3rd ed.). New Jersey: Pearson Education, Inc.
Davidson, N. & Lambdin Kroll, D. (1991). An overview of research on cooperative
learning realted to mathematics. Journal of Research in Mathematics Education, 22(5),
DeVries, D., Edwards, K., & Slavin, R. (1978). Biracial learning teams and race relations in
the classroom: Four field experiments using Teams-Games-Tournament. Journal of
Educational Psychology, 70(3), 356-362.
Dweck, C. S., & Leggett, E. L. (1988). A social-cognitive approach to motivation and
personality. Psychological Review, 95, 256-273.
Esmonde, I. (2009). Mathematics learning in groups: Analyzing equity in two cooperative
activity structures. Journal of the Learning Sciences, 18(2), 247-284.
Eisenhart, M.A.: 1988, ‘The ethnographic research tradition and mathematics research’,
Journal for Research in Mathematics Education, 19(2), 99114.
Hatano, G. (1988). Social and motivational bases for mathematical understanding. New
Directions for Child Development, 41, 55-70.
Jansen, A. (2006). Seventh graders' motivations for participating in two discussion-oriented
mathematics classrooms. Elementary School Journal, 106(5), 409-428.
21 The Affordances of Using Visibly Random Groups in a Mathematics Classroom
Kotsopoulos, D. (2007). Investigating peer as "expert other" during small group
collaborations in mathematics. In Proceedings of the 29th annual meeting of the North
American Chapter of the International Group for the Psychology of Mathematics
Education. Lake Tahoe, NV: University of Nevada, Reno.
Liljedahl, P. (2008). Teachers' insights into the relationship between beliefs and practice. In
J. Maaß & W. Schlöglmann (eds.) Beliefs and Attitudes in Mathematics Education:
New Research Results. (pp. 33-44). Rotterdam, NL: Sense Publishers.
Lubienski, S. T. (2000). Problem solving as a means towards mathematics for all: An
exploratory look through a class lens. Journal for Research in Mathematics Education,
31(4), 454-482.
Slavin, R. E. (1996). Research on cooperative learning and achievement: What we know,
what we need to know. Contemporary Educational Psychology, 21, 43-69.
Urdan, T. & Maehr, M. (1995). Beyond a two-goal theory of motivation and achievement:
A case for social goals. Review of Educational Research, 65(3), 213-243.
Webb, N. M., Nemer, K. M., & Ing, M. (2006). Small-group reflections: Parallels between
Carley discourse and student behavior in peer-directed groups. Journal of the Learning
Sciences, 15(1), 63-119.
Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in
mathematics. Journal for Research in Mathematics Education, 27(4), 458-477.
... At this point, their framing of the problem of supporting student groupwork shifted from activity design to connecting the problem to tracking structures in their school and describing how tracking make students feel "dumb." Ezio and Veronica experimented with the two approaches and noticed that purposeful grouping amplified the consequences of tracking, in the shape of labeling kids as "dumb" or "awesome," while random grouping disrupts them (see Liljedahl, 2014). Their sensemaking about the problem of supporting student collaboration spanned two practices they had learned in two PD workshops (exosystem), classroom experiments and institutional practices (mesosystem), all of which they negotiated through video representations of the classroom and dynamic framing of the problem of practice (the CSPD microsystem). ...
Full-text available
Educational researchers widely acknowledge the promises and impediments of teachers’ collaborative sensemaking, illuminating the need to recognize additional resources that are salient for teachers within professional interactions. In line with this overarching goal, this study explores the role of mathematics teachers' previous professional experiences in their collaborative sensemaking. Theoretically, it is rooted in ecological theories of learning, highlighting that learning is always shaped by an interconnected set of environments. Empirically, it builds on data from a research–practice partnership with a professional development (PD) organization, where we used classroom video to support secondary mathematics teachers' teams in improving their practice. The analysis first portrays two video-based conversations to illustrate the potential of inviting and building on teachers’ previous experiences and resources to support their collaborative reasoning. Then, it looks across nine video-based conversations of the same two teams and systematically describes experiences and resources that teachers spontaneously reference through six categories: PD workshops, conferences, PD organizations, online resources, research and policy, and curricula. These categories provide a framework for designers and facilitators who want to take seriously the practice of acknowledging that teacher learning happens through a complex web of learning experiences. The study brings forth the affordances of taking a learning ecology perspective on teacher collaborative sensemaking as professional development (CSPD) and provides guidance for designing, facilitating, and analyzing CSPD conversations in ways that center on teachers’ prior knowledge and experiences.
... The study involves an initial diagnosis of attitudes towards mathematics of the group through a first questionnaire (questionnaire 1 to be described later) by individual response; the micro-experiment consists of the development of IBME practices through a Kahoot questionnaire, working in visibly randomised class sessions and groups (Liljedahl, 2014), and finally a diagnosis of the new situation through a second questionnaire (questionnaire 2 to be described later) by individual response. ...
Full-text available
Research on the relationships between the main constructs underlying inquiry-based learning is rarely reported in mathematics education research. Considering this as a complex problem which is worth to be investigated, the present study aims to provide some empirical evidences that might serve as an insight to support further investigations on the relationships between attitudes towards mathematics and inquiry-based learning approaches. Thus, this study adopts a descriptive research design where no variables are manipulated but observed and measured in order to identify changes depicted in data collection. An instructional design focusing on the nature of mathematical inquiry is carried out with the participation of 304 secondary and high school students, and a clustering approach is used to look at how participants are grouped around certain attitudinal profiles before and after such mathematical practice. The results show how the heterogeneity of attitudinal profiles present in the classroom evolves positively in terms of perceived usefulness of mathematics and mathematical self-concept as perception of competence in mathematics. This fact provides some basis that might be used for further research on the idea that certain forms of development in inquiry-based mathematics education (IBME) based on greater immersion in the nature and culture of mathematics can help students to improve their attitudes towards mathematics.
... Then, we divided the students into eight pairs, also randomly. Random groupings increase mobility of knowledge and engagement in tasks (Liljedahl, 2014), which would encourage future teachers' engagement with assessment, improving the analysis. ...
Full-text available
Assessing problem-solving remains a challenge for both teachers and researchers. With the aim of contributing to the understanding of this complex process, this paper presents an exploratory study of peer assessment in mathematical problem-solving activities. The research was conducted with a group of future Secondary mathematics teachers who first were asked to individually solve an open-ended problem and then, to assess a classmate's answer in pairs. We present a study of two cases involving two pairs of students, each of whom assessed the solution of a third classmate. The analysis was carried out in two interrelated phases: (a) individual solutions to the mathematical problem and (b) the peer assessment process. The results show that, in both cases, the assessors were strongly attached to their own solutions, which directly influenced the assessment process, focused on aspects that involve the general problem-solving process and the results. The main difference between the evaluation processes followed by the two pairs lies in the concept of assessment. While the first pair focuses on assessing the resolution process and errors, the other focuses its discussion on giving a numerical grade.
... The practice field model described above is admittedly an ideal type, and there are many variations in schools and nuances raised in the literature (e.g. techniques for forming groups, Cohen & Lotan, 2014;Liljedahl, 2014; strategies for productive scaffolding, Gonz alez & DeJarnette, 2015; van de Pol, Mercer, & Volman, 2019). However, given its prominence, it is reasonable to suggest that the model reflects the general form of the field's desired mathematics classroom, and a shared interest among scholars and educators on incorporating or at least adapting practices from professional communities to develop young people's conceptual knowledge and procedural skills for eventual entry into a "real world" beyond school. ...
Full-text available
In this paper, we revisit a long-running conversation about situated learning and the design of environments for disciplinary engagement. Throughout the 1970s and 1980s, scholars advanced an anthropological critique of the then-dominant acquisitionist paradigm of formal schooling with a situated view focused on membership in communities and participation in practices. The critique led to a practice turn in education and a consensus model for reform-oriented school classrooms as orchestrated practice fields where students engage in disciplinary practices within a structured environment. Questions remain, however, about the nature of the practices and communities that this model engenders. We join this conversation through an anthropological investigation of a self organized group of teachers who gather outside of school hours to engage in collaborative mathematical activity. Participants have the flexibility to conduct their mathematical activity however they want; yet as we show, they tend to reproduce a practice field resembling a reform-oriented school mathematics classroom. This may seem unremarkable, even desirable for many reformers. However, assuming that teachers can or should only replicate practice fields when doing mathematics may be selling them short. Our findings suggest a durability and invisibility to practice fields that may be limiting the possibilities for the production of novel learning communities within schools.
... the mobility of knowledge between students increases, -students rely less on the teacher for answers, -students rather rely on answers constructed in collaboration with peers in their group or other groups, and students' enthusiasm and engagement in mathematical tasks increases (Liljedahl, 2014). ...
Full-text available
This thesis introduces results from a design-based, task design research study in mathematics education, within which silent video tasks were defined, developed, and implemented in upper secondary school mathematics classrooms. It discusses a research problem concerning the identification of opportunities and challenges that arise from the use of silent video tasks. To tackle that problem, the researcher worked with seven teachers in six Icelandic upper secondary schools who implemented silent video tasks in their classrooms. In short, silent video tasks involve the presentation of a short silent mathematics video clip that students are asked to discuss in pairs as they prepare and record their voice-over to the video. On the basis of students’ recorded responses to the task, that are listened to by the whole group, the teacher leads a discussion with the aim to deepen and widen students’ understanding of the mathematical topic presented in the video. The idea of silent video tasks is grounded in social constructivist theories. It is considered important that interaction happens between teacher and learners and among learners themselves, who work together (support each other) toward richer understanding of mathematical content. The learner is seen as an active participant in the teaching and learning process and in the case of silent video tasks, learners get an opportunity to become aware of their own and their peers’ current ways of describing or explaining mathematical phenomena. Two implementation phases were conducted in 2017 and 2019, during which interview data on teachers’ expectations and experiences of using silent video tasks was collected and analysed. In the first phase, four mathematics teachers in randomly selected upper secondary schools in Iceland assigned a silent video task to their 17-year-old students. Results from the first phase indicated that silent video tasks might be a helpful tool for formative assessment. Thus, teachers in the second phase were purposefully selected to work at schools that aim for active use of formative assessment. One teacher assigned three silent video tasks to his 16-year-old students and two teachers assigned one silent video task to their 16-year-old students. Besides interview data, classroom observation protocols were collected during the second phase. Influenced by theory and empirical results, the process of assigning a silent video task developed. To conclude the project, some characteristics that make a video suitable for use in silent video tasks were defined and the instructional sequence of silent video tasks was described. Together with the underlying theoretical and empirical arguments, they form design principles for silent video tasks.
... A final, highly important factor in group cohesion is that students are able to feel that they have been able to interact effectively in smaller groups with the majority of the rest of the group. This feeling can be instilled by repeated, regular randomization of the members of small groups during classroom tasks (Liljedahl, 2015;Patton, 2021). ...
Group cohesion, defined simply as the feeling of camaraderie and shared goals among the members of a class, is a fundamental goal of language teachers in Japan and around the world. The body of work on group cohesion in face-to-face classes is fairly extensive, but as a result of the coronavirus pandemic, the majority of university educators in Japan were forced to shift to some form of distance learning. This study considers the issue of how instructors were able to create and maintain group cohesion in online courses, and to what extent face-to-face classes could be replicated online. Twenty-seven university instructors responded to the survey, which consisted of several open-ended questions. The results of that survey are reported here, along with recommendations the researchers were able to assemble based on survey feedback. 集団の結束力とは、端的に言って、生徒が友情を構築し目標を共有することと定義されるが、それは日本および世界中の語学教員の目標である。対面授業における集団の結束力に関する研究はかなり広範囲に及ぶが、コロナウイルスの大流行の結果、日本の大学教育者の大多数は何らかの遠隔教育を余儀なくされた。本論は、大学教員がオンライン授業で集団の結束力をどのように固め、維持することができたのか、また、対面式の授業をオンラインでどの程度再現できるのかという問題について検討した。27名の大学教員が、いくつかの自由記述の質問からなる調査に回答した。本論はその調査結果と、それに基づく提言を提示する
Getting students to think about highly engaging non-curricular tasks turns out to not be that challenging. The challenge is getting students to bring the same level of engagement and thinking to curricular tasks. In this chapter, I look at three examples of what can happen when students who have been immersed in classrooms designed specifically to not only encourage thinking but also enable thinking, are given carefully constructed sequences of curricular tasks. Results indicate that under such circumstances not only are students able to sustain high levels of thinking for long periods of time but also move through huge amounts of new-to-them curricular content in very short amounts of time with little to no direct instruction.KeywordsFlowVariation theoryEngagementThinkingProblem-solving
Many of the discourses on creativity, although not explicitly so, assumes that creativity is a solitary activity—a phenomenon that happens within an individual working in isolation away from other people and other resources. But this is not how most people work. In this chapter, I look at creativity as something that can, and does, occur within groups, working collaboratively to solve problems in and among other groups also working to solve the same problem. Using burstiness as a theoretical construct to notice and name group creativity, I look specifically at how the environment can play a role in fostering and sustaining group creativity. Results indicate that one specific environment—the thinking classroom—is particularly well suited for occasioning group creativity.KeywordsCreativityGroup creativityBurstinessThe thinking classroom
If we want to innovate mathematics education — if we want to achieve something beyond conformity and compliance in mathematics education — these institutional norms need to be challenged. In this paper, I look at the results-first research methodology in which the institutional norms are challenged by the simple goal of increasing student thinking in the classroom. I share the specific results that emerged out of this research and use it as a specific case to argue that real innovation in mathematics education can only occur if we are willing to challenge the institutional norms that have been with us for 150 years.
Full-text available
This paper sets forth a way of interpreting mathematics classrooms that aims to account for how students develop mathematical beliefs and values and, consequently, how they become intellectually autonomous in mathematics. To do so, we advance the notion of sociomathematical norms, that is, normative aspects of mathematical discussions that are specific to students' mathematical activity. The explication of sociomathematical norms extends our previous work on general classroom social norms that sustain inquiry-based discussion and argumentation. Episodes from a second-grade classroom where mathematics instruction generally followed an inquiry tradition are used to clarify the processes by which sociomathematical norms are interactively constituted and to illustrate how these norms regulate mathematical argumentation and influence learning opportunities for both the students and the teacher. In doing so, we both clarify how students develop a mathematical disposition and account for students' development of increasing intellectual autonomy in mathematics. In the process, the teacher's role as a representative of the mathematical community is elaborated.
Full-text available
A set of complex relationships exist between teachers' espoused mathematical beliefs, their plans for teaching mathematics, and their actual teaching of mathematics. Many of these relationships have been explored in previous research. In this study I look more closely at one of the relationships that has not received much attention. In particular, I examine the possible disjunctive relationship between teachers' espoused beliefs and their intentions of practice. In so doing, I challenge the research that assumes a correlative relationship between these two aspects. The results indicate that this disjunctive relationship exists and that there are very rational reasons for it to exist. As such, I add to the growing body of research that links teachers' beliefs to teachers' practice. Finally, I introduce a research methodology that proves to be a very effective method for gaining insight into the complex domain of teachers' beliefs and practice.
Full-text available
In this paper, we attempt to clarify what it means to teach mathematics for understanding and to learn mathematics with understanding. To this end, we present an interactional analysis of transcribed video recordings of two lessons that occurred in different elementary school classrooms. The lessons, which are representative of a much larger data corpus, were selected because both focus on place value numeration and involve the use of similar manipulative materials. The analysis draws on Much and Shewder's (1978) identification of five qualitatively distinct types of classroom norms and pays particular attention to the mathematical explanations and justifications that occurred during the lessons. In one classroom, the teacher and students appeared consistently to constitute mathematics as the activity of following procedural instructions in the course of their moment by moment interactions. The analysis of the other classroom indicated that the teacher and students constituted mathematical truths as they coconstructed a mathematical reality populated by experientially real, manipulable yet abstract mathematical objects. These and other differences between mathematical activity in the two classrooms characterize two distinct classroom mathematics traditions, one in which mathematics was learned with what is typically called understanding and the other in which it was not.
Although in theory ethnography has been put forward as a powerful naturalistic methodology, in practice it has rarely been used by educational researchers because of differences in assumptions, goals, and primary research questions. From my perspective as an educational anthropologist, I describe the research tradition of ethnography--its underlying assumptions, its heritage in holistic cultural anthropology, its goals and research questions, and the organization of its research methods. Throughout, I compare elements of this ethnographic tradition with more common educational research practices. In the final section, I discuss the advantages of improved communication for future research in both mathematics education and educational anthropology.
The clear and practical writing of Educational Research: Planning, Conducting, and Evaluating Quantitative and Qualitative Researchhas made this book a favorite. In precise step-by-step language the book helps you learn how to conduct, read, and evaluate research studies. Key changes include: expanded coverage of ethics and new research articles.
In this study I examined the self-reported motivational beliefs and goals supporting the participation of 15 seventh graders in whole-class discussions in 2 discussion-oriented Connected Mathematics Project classrooms. Through this qualitative investigation using semistructured interviews, I inductively identified and described the students' motivational beliefs and goals and relations among them. Results demonstrated beliefs that constrained students' participation and ones that supported their participation. Students with constraining beliefs were more likely to participate to meet goals of helping their classmates or behaving appropriately, whereas students with beliefs supporting participation were more likely to participate to demonstrate their competence and complete their work. Results illustrated how the experiences of middle school students in discussion-oriented mathematics classrooms involve navigating social relationships as much as participating in opportunities to learn mathematics.