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Abstract and Figures

In this chapter I first introduce the notion of a thinking classroom and then present the results of over ten years of research done on the development and maintenance of thinking classrooms. Using a narrative style I tell the story of how a series of failed experiences in promoting problem solving in the classroom led first to the notion of a thinking classroom and then to a research project designed to find ways to help teacher build such a classroom. Results indicate that there are a number of relatively easy to implement teaching practices that can bypass the normative behaviours of almost any classroom and begin the process of developing a thinking classroom.
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© Springer International Publishing Switzerland 2016
P. Felmer et al. (eds.), Posing and Solving Mathematical Problems,
Research in Mathematics Education, DOI 10.1007/978-3-319-28023-3_21
Building Thinking Classrooms: Conditions
for Problem-Solving
Peter Liljedahl
In this chapter, I rst introduce the notion of a thinking classroom and then present
the results of over 10 years of research done on the development and maintenance
of thinking classrooms. Using a narrative style, I tell the story of how a series of
failed experiences in promoting problem-solving in the classroom led fi rst to the
notion of a thinking classroom and then to a research project designed to fi nd ways
to help teachers build such a classroom. Results indicate that there are a number of
relatively easy-to-implement teaching practices that can bypass the normative
behaviours of almost any classroom and begin the process of developing a thinking
My work on this paper began over 10 years ago with my research on the AHA!
experience and the profound effects that these experiences have on students’ beliefs
and self-effi cacy about mathematics (Liljedahl, 2005 ). That research showed that
even one AHA! experience, on the heels of extended efforts at solving a problem or
trying to learn some mathematics, was able to transform the way a student felt about
mathematics as well as his or her ability to do mathematics. These were descriptive
results. My inclination, however, was to try to fi nd a way to make them prescriptive.
The most obvious way to do this was to fi nd a collection of problems that provided
enough of a challenge that students would get stuck, and then have a solution, or
solution path, appear in a fl ash of illumination. In hindsight, this approach was
overly simplistic. Nonetheless, I implemented a number of these problems in a
grade 7 (12–13 year olds) class.
P. Liljedahl (*)
Simon Fraser University , Burnaby , BC , Canada
The teacher I was working with, Ms. Ahn, did the teaching and delivery of prob-
lems and I observed. Despite her best intentions the results were abysmal. The stu-
dents did get stuck, but not, as I had hoped, after a prolonged effort. Instead, they
gave up almost as soon as the problem was presented to them and they resisted any
effort and encouragement to persist. After three days of constant struggle, Ms. Ahn
and I both agreed that it was time to abandon these efforts. Wanting to better under-
stand why our well-intentioned efforts had failed, I decided to observe Ms. Ahn
teach her class using her regular style of instruction.
That the students were lacking in effort was immediately obvious, but what took
time to manifest was the realization that what was missing in this classroom was
that the students were not thinking. More alarming was that Ms. Ahn’s teaching was
predicated on an assumption that the students either could not or would not think.
The classroom norms (Yackel & Rasmussen, 2002 ) that had been established had
resulted in, what I now refer to as, a non-thinking classroom. Once I realized this, I
proceeded to visit other mathematics classes—fi rst in the same school and then in
other schools. In each class, I saw the same basic behaviour—an assumption,
implicit in the teaching, that the students either could not or would not think. Under
such conditions, it was unreasonable to expect that students were going to spontane-
ously engage in problem-solving enough to get stuck and then persist through being
stuck enough to have an AHA! experience.
What was missing for these students, and their teachers, was a central focus in
mathematics on thinking. The realization that this was absent in so many class-
rooms that I visited motivated me to fi nd a way to build, within these same class-
rooms, a culture of thinking, both for the student and the teachers. I wanted to build,
what I now call, a thinking classroom —a classroom that is not only conducive to
thinking but also occasions thinking, a space that is inhabited by thinking individu-
als as well as individuals thinking collectively, learning together and constructing
knowledge and understanding through activity and discussion.
Early Efforts
A thinking classroom must have something to think about. In mathematics, the
obvious choice for this is a problem-solving task. Thus, my early efforts to build
thinking classrooms were oriented around problem-solving. This is a subtle depar-
ture from my earlier efforts in Ms. Ahn’s classroom. Illumination-inducing tasks
were, as I had learned, too ambitious a step. I needed to begin with students simply
engaging in problem-solving. So, I designed and delivered a three session workshop
for middle school teachers (ages 10–14) interested in bringing problem-solving into
their classrooms. This was not a diffi cult thing to attract teachers to. At that time,
there was increasing focus on problem-solving in both the curriculum and the text-
books. The research on the role of problem-solving as both an end unto itself and as
a tool for learning was beginning to creep into the professional discourse of teachers
in the region.
P. Liljed a hl
The three workshops, each 2 h long, walked teachers through three different
aspects of problem-solving. The fi rst session was focused around initiating problem-
solving work in the classroom. In this session, teachers experienced a number of
easy-to-start problem-solving activities that they could implement in their class-
rooms—problems that I knew from my own experiences were engaging to students.
There were a number of mathematical card tricks to explain, some problems with
dice, and a few engaging word problems. This session was called Just do It , and the
expectation was that teachers did just that—that they brought these tasks into their
classrooms and had students just do them. There was to be no assessment and no
submission of student work.
The second session was called Teaching Problem-Solving and was designed to
help teachers emerge from their students’ experience a set of heuristics for problem-
solving. This was a signifi cant departure from the way teachers were used to teach-
ing heuristics at this grade level. The district had purchased a set of resources built
on the principles of Pólya’s How to Solve It ( 1957 ). These resources were pedantic
in nature, relying on the direct instruction of these heuristics, one each day, fol-
lowed by some exercises for students to go through practicing the heuristic of the
day. This second workshop was designed to do the opposite. The goal was to help
teachers pull from the students the problem-solving strategies that they had used
quite naturally in solving the set of problems they had been given since the fi rst
workshop, to give names to these strategies and to build a poster of these named
strategies as a tool for future problem-solving work. This poster also formed an
effective vocabulary for students to use in their group or whole class discussions as
well as any mathematical writing assignments.
The third workshop was focused on leveraging the recently acquired skills
towards the learning of mathematics and to begin to use problem-solving as a tool
for the daily engagement in, and learning of, mathematics. This workshop involved
the demonstration of how these new skills could intersect with the curriculum in
general and the textbook in particular.
The series of three workshops was offered multiple times and was always well
attended. Teachers who came to the fi rst tended, for the most part, to follow through
with all three sessions. From all accounts, the teachers followed through with their
‘homework’ and engaged their students in the activities they had experienced within
the workshops. However, initial data collected from interviews and fi eld notes were
mixed. Teachers reported things like:
“Some were able to do it.”
“They needed a lot of help.”
“They loved it.
“They don’t know how to work together.
“They got it quickly and didn’t want to do anymore.
“They gave up early.”
Further probing revealed that teachers who reported that their students loved
what I was offering tended to have practices that already involved some level of
problem-solving. If there was already a culture of thinking and problem-solving in
the classroom, then this was aided by the vocabulary of the problem-solving posters,
Building Thinking Classrooms: Conditions for Problem-Solving
and the teachers got ideas about how to teach with problem-solving. It also revealed
that those teachers who reported that their student gave up or didn’t know how to
work together mostly had practices devoid of problem-solving and group work. In
these classrooms, although some students were able to rise to the task, the majority
of the class was unable to do much with the problems—recreating, in essence, what
I had seen in Ms. Ahn’s class. In short, the experiences that the teachers were having
implementing problem-solving in the classroom were being fi ltered through their
already existing classroom norms (Yackel & Rasmussen, 2002 ).
Classroom norms are a diffi cult thing to bypass (Yackel & Rasmussen, 2002 ),
even when a teacher is motivated to do so. The teachers that attended these work-
shops wanted to change their practice, but their initial efforts to do so were not
rewarded by comparable changes in their students’ problem-solving behaviour.
Quite the opposite, many of the teachers I was working with were met with resis-
tance and complaints when they tried to make changes to their practice.
From these experiences, I realized that if I wanted to build thinking classrooms—
to help teachers to change their classrooms into thinking classrooms—I needed a set
of tools that would allow me, and participating teachers, to bypass any existing
classroom norms. These tools needed to be easy to adopt and have the ability to
provide the space for students to engage in problem-solving unencumbered by their
rehearsed tendencies and approaches when in their mathematics classroom.
This realization moved me to begin a program of research that would explore
both the elements of thinking classrooms and the traditional elements of classroom
practice that block the development and sustainability of thinking classrooms. I
wanted to fi nd a collection of teacher practices that had the ability to break students
out of their classroom normative behaviour—practices that could be used not only
by myself as a visiting teacher but also by the classroom teacher that had previously
entrenched the classroom norms that now needed to be broken.
Thinking Classroom
As mentioned, a thinking classroom is a classroom that is not only conducive to
thinking but also occasions thinking, a space that is inhabited by thinking individu-
als as well as individuals thinking collectively, learning together and constructing
knowledge and understanding through activity and discussion. It is a space wherein
the teacher not only fosters thinking but also expects it, both implicitly and explic-
itly. As such, a thinking classroom, as I conceive it, will intersect with research on
mathematical thinking (Mason, Burton, & Stacey, 1982 ) and classroom norms
(Yackel & Rasmussen, 2002 ). It will also intersect with notions of a didactic con-
tract (Brousseau, 1984 ), the emerging understandings of studenting (Fenstermacher,
1986 , 1994 ; Liljedahl & Allan, 2013a , 2013b ), knowledge for teaching (Hill, Ball,
& Schilling, 2008 ; Shulman, 1986 ) and activity theory (Engeström, Miettinen, &
1999 ).
P. Liljed a hl
In fact, the notion of a thinking classroom intersects with all aspects of research
on teaching and learning, both within mathematics education and in general. All of
these theories can be used to explain aspects of an already thinking classroom, and
some of them can even be used to inform us how to begin the process of build a
thinking classroom. Many of these theories have been around a long time, and yet
non-thinking classrooms abound. As such, I made the decision early on to approach
my work not from the perspective of a priori theory but from existing teaching
General Methodology
The research to fi nd the elements and teaching practices that foster, sustain and
impede thinking classrooms has been going on for over 10 years. Using a frame-
work of noticing (Mason, 2002 ), 1 I initially explored my own teaching, as well as
the practices of more than 40 classroom mathematics teachers. From this emerged
a set of nine elements that permeate mathematics classroom practice—elements that
account for most of whether or not a classroom is a thinking or a non-thinking class-
room. These nine elements of mathematics teaching became the focus of my
research. They are:
1. the type of tasks used and when and how they are used
2. the way in which tasks are given to students
3. how groups are formed, both in general and when students work on tasks
4. student workspace while they work on tasks
5. room organization, both in general and when students work on tasks
6. how questions are answered when students are working on tasks
7. the ways in which hints and extensions are used, while students work on tasks
8. w hen and how a teacher levels 2 their classroom during or after tasks
9. assessm ent, both in general and when students work on tasks
Ms. Ahn’s class, for example, was one in which:
1. practice tasks were given after she had done a number of worked examples
2. students either copied these from the textbook or from a question written on the
3. students had the option to self-group to work on the homework assignment when
the lesson portion of the class was done
1 At the time, I was only informed by Mason ( 2002 ). Since then, I have been informed by an
increasing body of literature on noticing (Fernandez, Llinares, & Valls,
2012 ; Jacobs, Lamb, &
2010 ; Mason, 2011 ; Sherin, Jacobs, & Philipp, 2011 ; van Es, 2011 ).
2 Levelling (Schoenfeld, 1985 ) is a term given to the act of closing of, or interrupting, students’
work on tasks for the purposes of bringing the whole of the class (usually) up to certain level of
understanding. It is most commonly seen when a teacher ends students work on a task by showing
how to solve the task.
Building Thinking Classrooms: Conditions for Problem-Solving
4. students worked at their desks, writing in their notebooks
5. students sat in rows with the students’ desk facing the board at the front of the
6. students who struggled were helped individually through the solution process,
either part way or all the way
7. there were no hints, only answers, and an extension was merely the next practice
question on the list
8. when ‘enough time’ time had passed, Ms. Ahn would demonstrate the solution
on the board, sometimes calling on ‘the class’ to tell her how to proceed
9. assessment was always through individual quizzes and tests
This was not, as determined earlier, a thinking classroom. Each of these elements
was something that needed exploring and experimenting with. Many were steeped
in tradition and classroom norms (Yackel & Rasmussen, 2002 ).
Research into each of these was done using design-based methods (Cobb,
Confrey, diSessa, Lehrer, & Schauble, 2003 ; Design-Based Research Collective,
2003 ) 3 within both my own teaching practice as well as the practices of a number of
teachers participating in a variety of professional development opportunities. This
approach allowed me to vary the teaching around each of the elements, either inde-
pendently or jointly, and to measure the effectiveness of that method for building
and/or maintaining a thinking classroom. Results fed recursively back into teaching
practice , each time leading either to refi ning or abandoning what was done in the
previous iteration.
This method, although fruitful in the end, presented two challenges. The fi rst had
to do with the measurement of effectiveness . To do this, I used what I came to call
proxies for engagement —observable and measurable (either qualitatively or quan-
titatively) student behaviours. At fi rst, this included only behaviours that fi t the a
priori defi nition of a thinking classroom. As the research progressed, however, the
list of these proxies grew and changed depending on the element being studied and
teaching method being used.
The second challenge had to do with the shift in practice need ed when it was
determined that a particular teaching method needed to be abandoned. Early results
indicated that small shifts in practice did little to shift the behaviours of the class as
a whole. Larger, more substantial shifts were needed. These were sometimes diffi -
cult to conceptualize. In the end, a contrarian approach was adopted. That is, when
a teaching method around a specifi c element needed to be abandoned, the new
approach to be adopted was, as much as possible, the exact opposite to the practice
that had shown to be ineffective for building or maintaining a thinking classroom.
When sitting showed to be ineffective, we tried making the students stand. When
levelling to the top failed, we tried levelling to the bottom. When answering ques-
tions proved to be ineffective, we stopped answering questions. Each of these
3 This research is now informed also by Norton and McCloskey ( 2008 ) and Anderson and Shattuck
2012 ).
P. Liljed a hl
approaches needed further refi nement through the iterative design-based research
approach, but it gave good starting points for this process .
In what follows, I will fi rst present the results of the research done on two of
these elements—student workspace and how groups are formed—both indepen-
dently and jointly. I then present, in brief, the results of the research done on the
remaining seven elements and discuss how all nine elements hold together as a
framework to build and maintain thinking classrooms. All of this research is
informed dually by data and analysis that looks both on the effect on students and
the uptake by teachers.
Student Workspace
The research on student workspace began by looking at the default—students sit-
ting in their desks. It became obvious early in this work that this was not conducive
to the building of a thinking classroom. As such, almost immediately, a new space
was explored. Following the contrarian approach established early on, the next
space to test was to have students standing and working somewhere other than at
their desks. The shift to having students work on whiteboards and blackboards was
then an obvious extension.
In many classrooms where the research was being done, however, there were not
enough whiteboards and blackboards available for all groups to work at. Some stu-
dents would have to still be seated in their desks. This led to a phase of experimenta-
tion with alternative work surfaces, including poster board or fl ipchart paper
attached to the walls and smaller whiteboards laying on desks—with some class-
rooms using all three at the same time. Whenever this occurred, there was a general
sense shared between whatever teachers were in the room, as well as myself, that
the vertical whiteboards were superior to any of the other options available to stu-
dents. These observations led to the following pseudo-quantitative study focusing
on this phenomenon.
The participants for this study were the students in fi ve high school classrooms; two
grade 12 ( n = 31, 30), two grade 11 ( n = 32, 31) and one grade 10 ( n = 31). 4 In each
of these classes, students were put into groups of two to four and assigned to one of
ve work surfaces to work on while solving a given problem-solving task.
4 In Canada, grade 12 students are typically 16–18 years of age, grade 11 students 15–18 and grade
10 students 14–17. The age variance is due to a combination of some students fast-tracking to be a
year ahead of their peers and some students repeating or delaying their grade 11 mathematics
Building Thinking Classrooms: Conditions for Problem-Solving
Participating in this phase of the research were also the fi ve teachers whose classes
the research took place in. Most high school mathematics teachers teach anywhere
from three to seven different classes. As such, it would have been possible to have
gathered all of the data from the classes of a single teacher. In order to diversify the
data, however, it was decided that data would be gathered from classes belonging to
ve different teachers.
These teachers were all participating in one of several learning teams which ran
in the fall of 2006 and the spring of 2007. Teachers participated in these teams vol-
untarily with the hope of improving their practice and their students’ level of
engagement. Each of these learning teams consisted of between 4 and 6, a 2-h meet-
ing spread over half a school year. Sessions took teachers through a series of activi-
ties modelled on my most current knowledge about building and maintaining
thinking classrooms. Teachers were asked to implement the activities and teaching
methods in their own classrooms between meetings and report back to the team how
it went.
The teachers, whose classrooms this data was collected in, were all new to the
ideas being presented and, other than having individual students occasionally dem-
onstrate work on the whiteboard at the front of the room, had never used them for
whole class activity.
As mentioned, the students, in groups of 2–4, worked on one of fi ve assigned work
surfaces: a wall-mounted whiteboard, a whiteboard laying on top of their desks or
table, a sheet of fl ipchart paper taped to the wall, a sheet of fl ipchart paper laying on
top of their desk or table, and their own notebooks at their desks or table. To increase
the likelihood that they would work as a group, each group was provided with only
one felt or, in the case of working in a notebook, one pen. To measure the effective-
ness of each of these surfaces, a series of proxies for engagement were established.
It is not possible to measure how much a student is thinking during any activity,
or how that thinking is individual or predicated on and with the other members of
his or her group. However, there are a variety of proxies for this level of engage-
ment that can be established— proxies for engagement . For the research presented
here, a variety of objective and subjective proxies were established.
1 . Time to task
This was an objective measure of how much time passed between the task being
given and the fi rst discernable discussion as a group about the task.
2 . Time to fi rst mathematical notation
This was an objective measure of how much time passed between the task being
given and the fi rst mathematical notation was made on the work surface.
3 . Eagerness to start
This is a subjective measure of how eager a group was to start working on a
task. A score of 0, 1, 2 or 3 was assigned with 0 being assigned for no enthusiasm
P. Liljed a hl
to begin and a 3 being assigned if every member of the group were wanting to
4 . Discussion
This is a subjective measure of how much group discussion there was while
working on a task. A score of 0, 1, 2 or 3 was assigned with 0 being assigned for
no discussion and a 3 being assigned for lots of discussion involving all mem-
bers of the group.
5 . Participation
This is a subjective measure of how much participation there was from the group
members while working on a task. A score of 0, 1, 2 or 3 was assigned with 0
being assigned if no members of the group were active in working on the task
and a 3 being assigned if all members of the group were participating in the
6 . Persistence
This is a subjective measure of how persistent a group was while working on a
task. A score of 0, 1, 2 or 3 was assigned with 0 being assigned if the group gave
up immediately when a challenge was encountered and a 3 being assigned if the
group persisted through multiple challenges.
7 . Non-linearity of work
This is a subjective measure of how non-linear groups work was. A score of 0, 1,
2 or 3 was assigned with 0 being assigned if the work was orderly and linear and
a 3 being assigned if the work was scattered.
8 . Knowledge mobility
This is a subjective measure of how much interaction there was between groups.
A score of 0, 1, 2 or 3 was assigned with 0 being assigned if there was no interac-
tion with another group and a 3 being assigned if there were lots of interaction
with another group or with many other groups.
These measures, like all measures, are value laden. Some proxies (1, 2, 3, 6)
were selected partially from what was observed informally when being in a setting
where multiple work surfaces were being utilized. Others proxies (4, 5, 7, 8) were
selected specifi cally because they embody some of what defi nes a thinking class-
room—discussion, participation, non-linear work, and knowledge mobility.
As mentioned, these data were collected in the fi ve aforementioned classes dur-
ing a group problem-solving activity . Each class was working on a different task.
Across the fi ve classes, there were ten groups that worked on a wall-mounted white-
board, ten that worked on a whiteboard laying on top of their desks or table, nine
that worked on fl ipchart paper taped to the wall, nine that worked on fl ipchart paper
laying on top of their desk or table, and eight that worked in their own notebooks at
their desks or table. For each group, the aforementioned measures were collected by
a team of 3–5 people: the teacher whose class it was, the researcher (me), as well a
number of observing teachers. The data were recorded on a visual representation of
the classroom and where the groups were located with no group being measured by
more than one person.
Building Thinking Classrooms: Conditions for Problem-Solving
Results and Discussion
For the purposes of this chapter, it is suffi cient to show only the average scores of
this analysis (see Table 1 ).
Th e data confi rmed the informal observations. Groups are more eager to start and
there is more discussion, participation, persistence and non-linearity when they
work on the whiteboards. However, there are nuances that deserve further attention.
First, although there is no signifi cant difference in the time it takes for the groups to
start discussing the problem, there is a big difference between whiteboards and
ipchart paper in the time it takes before groups make their fi rst mathematical nota-
tion. This is equally true whether groups are standin g or sitting. This can be attrib-
uted to the non-permanent nature of the whiteboards. With the ease of erasing
available to them, students risk more and risk sooner. The contrast to this is the very
permanent nature of a felt pen on fl ipchart paper. For students working on these
surfaces, it took a very long time and much discussion before they were willing to
risk writing anything down. The notebooks are a familiar surface to students, so this
can be discounted with respect to willingness to risk starting.
Although the measures for the whiteboards are far superior to that of the fl ipchart
paper and notebook for the measures of eagerness to start, discussion, and participa-
tion, it is worth noting that in each of these cases, the vertical surface scores higher
than the horizontal one. Given that the maximum score for any of these measures is
3, it is also worth noting that eagerness scored a perfect 3 for those that were stand-
ing. That is, for all ten cases of groups working at a vertical whiteboard, ten inde-
pendent evaluators gave each of these groups the maximum score. For discussion
and participation, eight out of the ten groups received the maximum score. On the
same measures, the horizontal whiteboard groups received 3, 3, and 2 maximum
scores, respectively. This can be attributed to the fact that sitting, even while work-
ing at a whiteboard, still gives students the opportunity to become anonymous, to
hide and to not participate. Standing doesn’t afford this.
Table 1 Average times and scores on the eight measures
paper Notebook
N (groups) 10 10 9 9 8
1. Time to task 12.8 s 13.2 s 12.1 s 14.1 s 13.0 s
2. Time to fi rst
20.3 s 23.5 s 2.4 min 2.1 min 18.2 s
3. Eagerness 3.0 2.3 1.2 1.0 0.9
4. Discussion 2.8 2.2 1.5 1.1 0.6
5. Participation 2.8 2.1 1.8 1.6 0.9
6. Persistence 2. 6 2.6 1.8 1.9 1.9
7. Non-linearity 2.7 2.9 1.0 1.1 0.8
8. Mobility 2. 5 1.2 2.0 1.3 1.2
P. Liljed a hl
With respect to non-linearity, it is clear that the whiteboards, either vertical or
horizontal, allow a greater freedom to explore the problem across the entirety of the
surface. Although the whiteboards provide an ease of erasing that is not afforded on
the fl ipchart, work is rarely erased by the students working on whiteboard surfaces.
It seems that rather than erasing to make room for more work, the workspace
migrates around the whiteboard surface, representing the chronological nature of
problem-solving. In contrast, the groups working on fl ipchart paper tended to not
write any work down until they were clear it would contribute to the logical devel-
opment of a solution.
Finally, it is worth noting that groups that were standing also were more likely to
engage with other groups that were standing close by. Although not measured, it
was clear that this was more true for the vertical whiteboard groups. There are a
number of reasons for this. Most obvious, vertical surfaces are more visible.
However, there were very few observed instances of groups that were sitting down
looking up to see what the groups that were standing were doing. Likewise, there
were no instances of the students standing, looking at the work of the groups that
were sitting. Amongst those that were standing, there was a lot of interaction
between those working on whiteboards, and almost none between those working on
ipchart paper. Finally, there was very little interaction between those working on
ipchart paper and those working on whiteboards. Part of this can be explained by
proximity—the whiteboard groups were clustered on one or two whiteboards, while
the fl ipchart people were clustered elsewhere. But it also is the case that the white-
board groups had little reason to look to the fl ipchart groups. They worked slower
and had little written on their work surfaces. This was also true between the fl ipchart
groups—there was little to look at.
In short, groups that worked on vertical whiteboards demonstrated more think-
ing classroom behaviour—persistence, discussion, participation and knowledge
mobility—than any of the other types of work surfaces. The next most conducive
was a horizontal whiteboard. The remaining three were not only not conducive to
promoting thinking classroom behaviour but they may actually have inhibited it.
From this it is clear that the non-permanence of surfaces is critical for decreasing
time to task, as well as improving enthusiasm, discussion, participation, and persis-
tence. It also increases the non-linearity of work which mirrors the actual work of
thinking groups. Making these non-permanent surfaces vertical further enhances all
of these qualities, as well as fostering inter-group collaboration, something that is
needed to move the class from a collection of thinking groups to being a thinking
Vertical Non-permanent Surfaces: Teacher Uptake
Having this evidence that vertical non-permanent surfaces (VNPS) are so instru-
mental in the fostering of thinking classroom behaviour, a follow-up study was
done with teachers vis-à-vis the use of this work surface. The goal of this follow-up
Building Thinking Classrooms: Conditions for Problem-Solving
study was to see the degree to which teachers, when presented with the idea of non-
permanent vertical surfaces, were keen to implement it within their teaching, actu-
ally tried it, and continued to use it in their teaching.
Participants for this portion of the study were 300 in-service teachers of mathemat-
ics—elementary, middle and secondary school. They were drawn from three
sources over a four-year period (2007–2011): participants in variety of single work-
shops, participants in a number of multi-session workshops, and participants in
learning teams. The breakdown of participants, according to grade levels, and form
of contact is represented in Table 2 .
There were a number of teachers who attended a combination of learning teams,
multi-session workshops and single workshops. In these cases, their data was regis-
tered as belonging to the group with the most contact. That is, if they attended a
single workshop, as well as being a member of a learning team, their participation
was registered as being a member of a learning team.
These participants are only a subset of all the teachers that participated in these
learning teams, multi-session workshops, and single workshops. They were selected
at random from each group I worked with by approaching them at the end of the fi rst
(and sometimes only) session and asking them if they would be willing to have me
contact them and, potentially, visit their classrooms.
Data consists primarily of interview data. Each participant was interviewed imme-
diately after a session where they were fi rst introduced to the idea of vertical non-
permanent surfaces, 1 week later, and 6 weeks later. These interviews were brief
and, depending on when the interview was conducted, was originally designed to
gauge the degree to which they were committed to trying, or continuing to use,
vertical non-permanent surfaces in their teaching and how they were using them.
However, participants wanted to talk about much more than just this. They wanted
to discuss innovations they had made, the ways in which this was changing their
Table 2 Distribution of participants in VNPS study
Elementary Middle Secondary Total
Learning team 21 43 41 105
Multi-session workshops 12 28 42 82
Single workshops 35 24 54 113
Total 68 95 137 300
P. Liljed a hl
teaching practice as a whole, the reactions of the students and their colleagues, as
well as a variety of other details pertaining to vertical non-permanent surfaces. With
time, these impromptu conversations changed the initial interview questions to
begin to also probe for these more nuanced details. For the purposes of this chapter,
however, only the data pertaining to the original intent will be presented.
In addition to the interview data, there were also fi eld notes from 20 classroom
visits. These visits were implemented for the purposes of checking the fi delity of the
interview data—to see if what teachers were saying is actually what they were
doing. In each case, this proved to be the case. It was clear from these data that
teachers were true to their words with respect to their use of vertical non-permanent
surfaces. However, these visits, like the interviews, offered much more than what
was expected. I saw innovations in implementation, observed the enthusiasm of the
students, and witnessed the transformational effect that this was having on the
teaching practices of the participants .
Results and Discussion
In general, almost all of the teachers who were introduced to the notion of vertical
non-permanent surfaces were determined to try it within their teaching and were
committed to keep doing it, even after 6 weeks (see Fig. 1 ). This is a signifi cant
uptake rarely seen in the literature. This is likely due, in part, to the ease with which
it is modelled in the various professional development settings. During these ses-
sions, not only are the methods involved easily demonstrated but the teachers
immediately feel the impact on themselves as learners when they are put into a
group to work on a vertical non-permanent surface.
An interesting result from this aggregated view is that there were more teachers
using non-permanent vertical surfaces after 6 weeks than there was after 1 week.
This has to do with access to these vertical non-permanent surfaces. Many teachers
struggled to fi nd such surfaces. There were some amazing improvisations in this
regard, from using windows to bringing in a number of novel surfaces, from shower
curtains to glossy wall boards. One teacher even stood her classroom tables on end
to achieve the effect. As time went on, teachers were able to convince their admin-
istrators to provide them with enough whiteboards that these improvisations no lon-
ger became necessary. For some teachers, this took more time than others and
speaks to the delayed uptake seen in Fig. 1 . However, it also speaks to the persis-
tence with which many teachers pursued this idea with.
A disaggregated look at the data shows that neither the grade levels being taught
(see Fig. 2 ) or the type of professional development setting in which the idea was
presented (see Fig. 3 ) had any signifi cant impact on the uptake.
Literature on teacher change typically implies that sustained change can only be
achieved through professional development opportunities with multiple sessions
and extended contact. That is, single workshops are not effective mediums for pro-
moting change (Jasper & Taube,
2004 ; Little & Horn, 2007 ; Lord, 1994 ; McClain
Building Thinking Classrooms: Conditions for Problem-Solving
& Cobb, 2004 ; Middleton, Sawada, Judson, Bloom, & Turley, 2002 ; Stigler &
Hiebert, 1999 ; Wenger, 1998 ). The introduction of vertical non-permanent surfaces
as a workspace doesn’t adhere to these claims. There are many possible reasons for
this. The fi rst is that the introduction of non-permanent vertical surfaces was
achieved in a single workshop could be, as mentioned, due to the simple fact that it
is a relatively easy idea for a workshop leader to model and for workshop partici-
pants to experience. Forty fi ve minutes of solving problems in groups standing at a
whiteboard coupled with a whole group discussion on the affordances of recreating
this within their own classrooms is enough to convince teachers to try it. And trying
it leads to a successful implementation. Unlike many other changes that can be
made in a teacher’s practice, vertical non-permanent surfaces (as demonstrated in
immediate intension tried it in the first week still doing it after six
intends to keep doing it
after six weeks
elementary (n=68) middle (n=95) secondary (n=137)
Fig. 2 Uptake of VNPS by grade levels ( n = 300)
immediate intension tried it in the first week still doing it after six weeks
intends to keep doing it
after six weeks
Fig. 1 Uptake of VNPS ( n = 300)
P. Liljed a hl
the fi rst study) was well received by students, was easy to manage at a whole class
level, and had an immediate positive effects on classroom thinking behaviour.
Together, the ease of modelling coupled with a successful implementation meant
that vertical non-permanent surfaces did not need more than a single workshop to
change teaching practice.
These possible reasons are supported by the comments of teachers from the
interviews after week 1 and week 6. The following comments were chosen from the
many collected for their conciseness.
I will never go back to just having students work in their desks .”
How do I get more whiteboards ?”
The principal came into my class now I’m doing a session for the whole staff on Monday .”
My grade - partner is even starting to do it .”
The kids love it. Especially the windows .”
I had one girl come up and ask when it will be her turn on the windows .”
Not only is the implementation of vertical non-permanent surfaces immediately
effective for these teachers, it is also infectious with other teachers quickly latching
on to it and administrators quickly seeing the affordances it offers.
But if vertical non-permanent surfaces are the solution, what was the problem?
When I began the research on students’ workspace, the default was students sitting
in desks—sometimes individually in rows, other times clustered in groups. The
move from the desks to the vertical workspaces was made, not because I saw some-
thing specifi cally wrong with students being in desks, but rather through adherence
to the contrarian approach that was adopted early on in the more general research
project. Looking back now at students working in desks, from the perspective of the
affordances that having them stand at a non-permanent vertical surface offers, I see
immediate intension tried it in the first week still doing it after six
intends to keep doing it
after six weeks
learning team (n=105) multi-session workshops (n=82) single workshop (n=113)
Fig. 3 Uptake of VNPS by professional development setting ( n = 300)
Building Thinking Classrooms: Conditions for Problem-Solving
more clearly the problems that desks introduced into my efforts to build and main-
tain thinking classrooms. Primarily, this has to do with anonymity and how desks
allow for and even promote this. When students stand at a whiteboard or a window,
they are all visible. There is nowhere to hide. When students are in their desks, it is
easy for them to become anonymous, hidden and safe—from participating and
from contributing. It is not that all students want to be hidden, to not participate, but
when the problems gets diffi cult, when the discussions require more thinking, it is
easy for a student to pull back in their participation when they are sitting. Standing
in a group makes this more diffi cult. Not only is it immediately visible to the teacher
but it is also clear to the student who is pulling back. To pull back means to step
towards the centre of the room, towards the teacher, towards nothing. There is no
anonymity in this.
Forming Groups
The research into how best to form groups began, like it did with student work sur-
faces, by looking at how groups are typically formed in a classroom. In most cases,
this is either a strategically planned arrangement decided by the teacher or self-
selected groupings of friends as decided by the students. Teachers tend to make
groupings in order to meet their educational goals. These may include goals around
pedagogy, student productivity, or simply the construction of a peaceful work envi-
ronment. Meanwhile, students, when given the opportunity, tend to group them-
selves according to their social goals. This mismatch between educational and
social goals in classrooms creates conditions where, no matter how strategic a
teacher is in her groupings, some students are unhappy in the failure of that group-
ing to meet their social goals (Kotsopoulos, 2007 ; Slavin, 1996 ).
This disparity results in a decrease in the effectiveness of group work. This led to
the exploration of alternative grouping methods. The fact that strategic grouping
strategies were often not working, coupled with the contrarian approach of action in
such instances, meant that random grouping methods needed to be explored.
Working with the same type of population of teachers described above, a variety of
random grouping methods were implemented and studied. This preliminary research
showed, very quickly, that there was little difference in the effectiveness of strategic
groupings and randomized groupings when the randomization was done out of sight
of the students. The students assumed that all groupings had a hidden agenda, and
merely saying that they were randomly generated was not enough to change class-
room behaviour.
However, when the randomization was done in full view of the students, changes
were immediately noticed. When randomization was done frequently—twice a day
in elementary classrooms and every class in middle and secondary classrooms—the
changes in classroom behaviour was profound. Within 2–3 weeks:
Students became agreeable to work in any group they were placed in.
There was an elimination of social barriers within the classroom.
P. Liljed a hl
Mobility of knowledge between students increased.
Reliance on the teacher for answers decreased.
Reliance on co-constructed intra- and inter-group answers increased.
Engagement in classroom tasks increased.
Students became more enthusiastic about mathematics class.
To confi rm these observations, one grade 10 (age 15–16) was studied. The details
and results of this research have already been published in a chapter entitled The
Affordances of Using Visibly Random Groups in a Mathematics Classroom
(Liljedahl, 2014 ). What follows is a summary of this research.
The class in which the study was done belonged to Ms. Carley, a teacher with
eight years experience who was a participant in one of the learning teams I was
leading. Ms. Carley had joined the team because she was dissatisfi ed with the results
of group work in her teaching. She knew that group work was important to 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 vis-
ibly random groups she made an immediate commitment to start using this method
in one of her classrooms.
Data consisted of interview transcripts and fi eld notes collected over a 3-month
period immediately prior to and during an implementation of visibly random groups
in Ms. Carley’s class. These data were analysed using analytic induction (Patton,
2002 ) anchored in the a priori and grounded observations from my initial experi-
mentation with random groupings.
These results both confi rmed and nuanced the initial observations. Students very
quickly shed their anxieties about what groups they were in. They began to collabo-
rate in earnest. After three weeks, a porosity developed between group boundaries
as both intra- and inter-group collaboration fl ourished. With this heightened mobi-
lization of knowledge came a decrease in the reliance on the teacher as the knower
in the room. In the end, there was a marked heightening of enthusiasm and engage-
ment for problem-solving in particular, and in mathematics class in general. In
short, Ms. Carley’s class became a thinking classroom .
Visibly Random Groupings: Teacher Uptake
Similar to the research on the vertical non-permanent surfaces a pseudo-quantitative
study was done on the uptake by teachers on the idea of visibly random groupings
(VRG). Tapping into the similar populations of teachers engaged in learning teams,
multi-session workshops, and single workshops between 2009 and 2011, a popula-
tion of 200 teachers were selected to participate (see Table 3 ).
These teachers were introduced to the idea of visibly random groupings in a
similar fashion as above—through modelling and immersion. They were likewise
interviewed immediately after their professional development experience, 1 week
after their experience, and 6 weeks after. The results of this analysis can be seen in
4 .
Building Thinking Classrooms: Conditions for Problem-Solving
The dip in the uptake between week 1 and week 6 was minor. What was interest-
ing was the uptick in intension after week 6. From the interviews, it became clear
that the teachers who had come away from using visibly random groups did so
because, after 3–4 weeks, things were working so well that they thought they could
now allow the students to work with who they wanted. Once they saw that this was
not as effective, they recommitted to going back to random groupings.
Like with vertical non-permanent surfaces, there was no discernible difference in
uptake between elementary, middle or secondary teachers. However, unlike the pre-
vious study, there was a slight difference depending on the nature of the profes-
sional development environment they were participating in (see Fig. 5 ).
From the interviews, it seemed that although the immediate delivery of the idea
was accomplished within a single session, the support of the learning team helped
teachers to get on board late if they hesitated in implementing in the 1st week. This
explains the uptick in the number of learning team members who started using ran-
domized groups in between the fi rst and the sixth week. This also explains why
there was no such uptick amongst the single workshop participants who had no
follow-up session, or amongst the multi-session participants who did not have a
second session until 8 weeks after the initial idea was presented.
Regardless, there was still a signifi cant uptake by those teachers who only expe-
rienced one 90 min session on the use of visibly random groupings. This can be
explained in the same way as it was for the vertical non-permanent surfaces—it was
easily modelled and the affordances became immediately apparent. As well, the
students took to it quickly with little resistance once the participants implemented it
within their own classrooms.
Table 3 Distribution of participants in VRG study
Elementary Middle Secondary Total
Learning team 15 22 31 68
Multi-session workshops 25 19 14 58
Single workshops 10 25 39 74
Total 50 66 84 200
immediate intension tried it in the first week still doing it after six weeks
intends to keep doing it
after six weeks
Fig. 4 Uptake of VRG ( n = 200)
P. Liljed a hl
As with the research on the vertical non-permanent surfaces, the research on vis-
ibly random groupings included 14 classroom visits. Unlike the research on VNPS,
however, the purpose of these visits was not to check the fi delity of the interview
data. Rather, it was to see if teachers were continuing to use VRG’s even 6–9
months after their last work with me. In each of the 14 visits, I saw a continued use
of VRG strategies. And like with my visits in the VNPS research, these visits
offered much more than what was expected. I saw innovations in implementation,
observed the enthusiasm of the students, and witnessed the transformational effect
that this was having on teaching practices.
VNPS and VRG Taken Together: Teacher Uptake
Once it was established that both vertical non-permanent surfaces and visibly ran-
dom groupings were effective practices for building aspects of a thinking classroom
and that these methods had good uptake by teachers, it was easy to bring them
together. From a professional development perspective, this is no more diffi cult
than presenting each one separately. VNPS and VRG are easily modelled together,
with the participants being put into visibly random groupings to work on vertical
non-permanent surfaces. So, this is what was done with a population of teachers
similar to the ones described above. From this, 124 participants were followed to
gauge the uptake of being exposed to both of these methods simultaneously. The
results can be seen in Fig.
6 .
immediate intension tried it in the first week still doing it after six
intends to keep doing it
after six weeks
learning team (n=105) multi-session workshops (n=82) single workshop (n=113)
Fig. 5 Uptake of VRG by professional development setting ( n = 200)
Building Thinking Classrooms: Conditions for Problem-Solving
Like with visibly random groupings, there was no signifi cant difference in uptake
by grade level and a slight difference in uptake as disaggregated by the professional
development setting in which the combined methods were presented. Like with vis-
ibly random groupings, the teachers in the learning team setting were more consis-
tently implementing the methods presented, whereas those teachers in the single
workshop sessions were less likely to get on board late and more likely to drop off
early (see Fig. 7 ). Despite these differences, however, the uptake across for each
group was impressive with much enthusiasm for it.
With respect to the effect on students, my observations during ten classroom
visits showed the combined benefi ts of the two interventions. The fact that the stu-
dents were so comfortable working with each other, coupled with the high visibility
immediate intension tried it in the first week still doing it after six weeks
intends to keep doing it
after six weeks
Fig. 6 Uptake of both VNPS and VRG ( n = 124)
immediate intension tried it in the first week still doing it after six
intends to keep doing it
after six weeks
learning team (n=53) multi-session workshops (n=26) single workshop (n=45)
Fig. 7 Uptake of VNPS and VRG by professional development setting ( n = 124)
P. Liljed a hl
of the work afforded by the vertical surfaces, allowed for enhanced intra-group
knowledge mobilization. The teachers often commented that they saw huge
improvements in the classroom community.
“I used to think I had a community in my classroom. Now I see what a community can look
like .
My observation of the student actions during these ten classroom visits confi rmed this .
General Findings: All Nine Elements
The results from research on students’ workspace and grouping methods are indicative
of the fi ndings of research into each of the nine aforementioned elements. From the
design-based research on each of these—independently or in conjunction with
others—emerged a set of teaching practices that are conducive to either the building,
or maintenance, of a thinking classroom. In what follows briefl y, these are:
1 . The type of tasks used and when and how they are used
Lessons need to begin with good problem-solving tasks. At the early stages of
building a thinking classroom, these tasks need to be highly engaging,
collaborative tasks that drive students to want to talk with each other as they try
to solve them (Liljedahl, 2008 ). Once a thinking classroom is established, the
problems need to permeate the entirety of the lesson and emerge rich mathe-
matics (Schoenfeld, 1985 ) that can be linked to the curriculum content to be
‘taught’ that day.
2 . The way in which tasks are given to students
Tasks need to be given orally. If there are data or diagrams needed, these can be
provided on paper, but the instructions pertaining to the activity of the task need
to be given orally. This very quickly drives the groups to discuss what is being
asked rather than trying to decode instructions on a page.
3 . H ow groups are formed, both in general and when students work on tasks
As presented above, groupings need to be frequent and visibly random. Ideally,
at the beginning of every class, a visibly random method is used to assign stu-
dents to a group of 2–4 for the duration of that class. These groups will work
together on any assigned problem-solving tasks, sit together or stand together
during any group or whole class discussions.
4 . Student workspace while they work on tasks
As discussed, groups of students need to work on vertical non-permanent sur-
faces such as whiteboards, blackboards, or windows. This will make visible all
work being done, not just to the teacher but to the groups doing the work. To
facilitate discussion, there should be only one felt pen or piece of chalk per
5 . Room organization, both in general and when students work on tasks
The classroom needs to be de-fronted. The teacher must let go of one wall of the
classroom as being the designated teaching space that all desks are oriented
Building Thinking Classrooms: Conditions for Problem-Solving
towards. The teacher needs to address the class from a variety of locations within
the room and, as much as possible, use all four walls of the classroom. It is best
if desks are placed in a random confi guration around the room.
6 . How questions are answered when students are working on tasks
Students only ask three types of qu estions: (1) proximity questions—asked when
the teacher is close; (2) stop-thinking questions—most often of the form ‘is this
right’; and (3) keep-thinking questions—questions that students ask so they can
get back to work. Only the third of these types should be answered. The fi rst two
types need to be acknowledged but not answered.
7 . The ways in which hints and extensions are used while students work on tasks
Once a thinking classroom is established, it needs to be nurtured. This is done
primarily through how hints and extensions are given to groups as they work on
tasks. Flow (Csíkszentmihályi 1990 , 1996 ) is a good framework for thinking
about this. Hints and extensions need to be given so as to keep students in a per-
fect balance between the challenge of the current task and their abilities in work-
ing on it. If their ability is too high, the risk is they get bored. If the challenge is
too great, the risk is they become frustrated.
8 . When and how a t eacher levels their classroom during or after tasks
Levelling needs be done at the bottom. When every group has passed a minimum
threshold, the teacher needs to engage in discussion about the experience and
understanding the whole class now shares. This should involve a reifi cation and
formalization of the work done by the groups and often constitutes the ‘lesson’
for that particular class.
9 . Assessment, both in general and when students work on tasks
Assessment in a thinking classroom needs to be mostly about the involvement of
students in the learning process through efforts to communicate with them where
they are and where they are going in their learning. It needs to honour the
activities of a thinking classroom through a focus on the processes of learning
more so than the products and it needs to include both group wo rk and individual
However, this research also showed that these are not all equally impactful or pur-
poseful in the building and maintenance of a thinking classroom. Some of these are
blunt instruments capable of leveraging signifi cant changes while others are more
refi ned, used for the fi ne-tuning and maintenance of a thinking classroom. Some
are necessary precursors to others. Some are easier to implement by teachers than
others, while others are more nuanced, requiring great attention and more practice
as a teacher. And some are better received by students than others. From the whole
of these results emerged a three-tier hierarchy that represent not only the bluntness
and ease of implementation but also an ideal chronology of implementation (see
Table 4 ).
P. Liljed a hl
In the aforementioned research, I presented the results of research into teachers
implementing teaching practices from stage one, either separately or together.
However, the effect on these teachers is more profound than the numbers and graphs
indicated above. This experience with elements in stage one propels them to thirst
for more, both in particular and in general. They want more tasks, more examples
of how to make random groupings, how to fi nd vertical surfaces. But they also want
to know more about assessment, how to ask and answer questions, how to organize
their rooms, how to give instructions and how to sustain the engagement they have
experienced while at the same time feeling like they are getting through the curricu-
lum. In short, their experience with the teaching methods associated with stage one
elements is quite naturally propelling them into wanting to engage in the elements
in stages two and three.
These results are not defi nitive, exhaustive or unique. The teaching methods that
emerged as effective for each of these elements emerged as a result of an a priori
commitment to make change in a contrarian fashion. This continued until positive
effects began to emerge, at which point refi nements were recursively explored. It is
possible that a different approach to the research would have yielded different
methods. Different methods could, likewise, emerge a different set of stages opti-
mal for the development of thinking classrooms.
The main goal of this research is about fi nding ways to build thinking classrooms.
One of the sub-goals of this work on building thinking classrooms was to develop
methods that not only fostered thinking and collaboration but also bypassed any
classroom norms that would potentially inhibit this from happening. Using the
methods in stage one while solving problems, either together or separately, was
almost universally successful. They worked for any grade, in any class and for any
teacher. As such, it can be said that these methods succeeded in bypassing whatever
norms existed in the over 600 classrooms in which these methods were tried.
Further, they not only bypassed the norms for the students but also the norms of the
Table 4 Nine elements as chronologically implemented
Stage one Stage two Stage three
• Begin lessons with problem-solving
• Oral instructions • Levelling
• Vertical non-permanent surfaces • De-fronting the room • Assessment
• Visibly random groups Ans wering questions • Managing fl ow
Building Thinking Classrooms: Conditions for Problem-Solving
teachers implementing them. So different were these methods from the existing
practices of the teachers participating in the research that they were left with what I
have come to call rst-person vicarious experiences . They are fi rst person because
they are living the lesson and observing the results created by their own hands. But
the methods are not their own. There has been no time to assimilate them into their
own repertoire of practice or into the schema of how they construct meaningful
practice. They simply experienced the methods as learners and then were asked to
immediately implement them as teachers. As such, they experienced a different way
in which their classroom could look and how their students could behave. They
experienced, through these other ly methods, an other ly classroom—a thinking
The results of this research sound extraordinary. In many ways, they are. It
would be tempting to try to attribute these to some special quality of the profes-
sional development setting or skill of the facilitator. But these are not the source of
these remarkable results. The results, I believe, lie not in what is new but what is not
old. The classroom norms that permeate classrooms in North America, and around
the world, are so robust, so entrenched, that they transcend the particular classrooms
and have become institutional norms (Liu & Liljedahl, 2012 ). What the methods
presented here offer is a violent break from these institutional norms, and in so
doing, offer students a chance to be learners much more so than students (Liljedahl
& Allan, 2013a , 2013b ).
By constructing a thinking classroom, problem-solving becomes not only a
means but also an end. A thinking classroom is shot through with rich problems.
Implementation of each of the aforementioned methods associated with the nine
elements and three stages relies on the ubiquitous use of problem-solving. But at the
same time, it also creates a classroom conducive to the collaborative solving of
Since this research was completed, I have gone back to visit several of the class-
rooms of teachers who fi rst took part in the research. These teachers are still using
VNPS and VRG as well as having refi ned their practice around many of the other
nine aforementioned elements. Unlike many other professional development initia-
tives and interventions I have seen implemented over the years, these really seemed
to have had a lasting impact on teacher practice. The reason for this seems to come
from two sources. First, teachers talk about how much their students like the ‘new’
way of doing mathematics. So much so, in fact, that when they go back to using
direct instruction, even for brief periods of time, the students object. The second and
more intrinsic reason is that they feel more effective as teachers. Their students are
exhibiting the traits that they had been striving for but were unable to achieve
through nuanced changes to their initial teaching practice.
P. Liljed a hl
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P. Liljed a hl
... Furthermore, by engaging students in discussion based on their responses it becomes visible that it is expected of learners to think for themselves and in collaboration with peers. Working in such a way in the mathematics classroom requires a certain classroom culture that Peter Liljedahl calls a thinking classroom: a classroom where students are expected to think and given opportunities to think, think individually or collectively, constructing knowledge and learning to understand together via activities and discussion (Liljedahl, 2016). ...
... Although silent video tasks are different from problem solving tasks, I could relate to the teachers' struggles and intentions as I read a chapter (see Liljedahl, 2016) on Building Thinking Classrooms. I had an aha! moment realizing that what I experienced when my students worked on the silent video task was a thinking classroom. ...
... • Give short oral instructions. This is a preferred way in which tasks are given to students due to results indicating that rather than decoding instructions from paper, -students will discuss their interpretation of the task among themselves, within their groups (Liljedahl, 2016). ...
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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.
... Students can see solutions or parts of solutions written on the different vertical writing surfaces. Students can also hear mathematical communication from other students and new student-content interactions evolve (Johnston, 2012;Liljedahl, 2016). The geometrical aspects of the classroom, school building, and the surroundings can serve as mathematical content in the classroom and increase the students' awareness of the built physical environment and surroundings (Soygenis & Erktin, 2010). ...
... Recognition of the significance of PE during the educational activity can also be found in mathematics education research where PE is not primarily targeted. One such example is where student workspace and room organization are two out of nine focus elements of mathematics teaching practices (Liljedahl, 2016). Another such example is where the teacher's and the students' spatial positions and the classroom seating arrangement are included in a proposed model for social classroom climate (Kuzle & Glasnović Gracin, 2021). ...
... In essence, this can be formulated as when looking to change teaching practices, always consider PE but avoid relying solely on PE for changes to take place. One example for such undertakings are the nine focus elements of mathematics teaching practices in Liljedahl's (2016) framework. ...
Conference Paper
The physical environment (PE) affects the teaching and learning in school. Research is conclusive that different characteristics of PE can be enabling or hindering learning activities. Still, we need to know more about the role of PE in mathematics education to utilize what a good PE can offer and to avoid the hindering situations. The aim of this study is to characterize the different roles PE play in relation to the teacher, the student, the learning content, and their interactions. For this, mathematics teachers' stories about their experiences of PE in teaching are analysed. The results show that teachers often try to prevent disturbances or distractions from insufficiencies in PE. The results also suggest that aspects of classroom PE, such as classroom layout, sustain classroom norms whereas other elements in PE can be an aid in breaking norms.
... The term 'flow' has been used in teaching mathematics to study improvement in mathematics skills among gifted students (e.g. Heine, 1997), students' motivation and feelings regarding mathematics (Schweinle et al., 2006), teachers' instruction in mathematics (Liljedahl, 2016), students' engagement in problem-solving (Williams, 2001) and the use of software in connection with 2D geometric figures, rotations and symmetries (Sedig, 2007). Flow has been studied at both the individual and group level (e.g. ...
... A student experiences flow if a balance arises between challenges and skills (i.e. the level of the challenges corresponds to the level of the skills), and the student is then in the flow zone (Liljedahl, 2016). An optimal challenge (Engeser & Rheinberg, 2008) is present when both challenges and skills are high (i.e. ...
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This article is intended to contribute to greater knowledge regarding the importance of flow and the time used to perform an activity, with a focus on students’ mathematical experiences of 3D bodies. Thirty-one 9th-grade students took part in the study. Flow and variation theory was used in the analysis of lesson observations, submission tasks, audio recordings, logbooks, tests and nationwide tests. The results indicate that the selected mathematics problem is characterized by seven components, which serve as the basis for identifying intended critical aspects; a variation is evident in the balance between skills and challenges that is characterized by the critical aspects that the students discern; a variation is evident in the experience of flow that is dependent upon the students’ approach to their work on various activities; the students’ mathematical experiences are based, both short- and long-term, on discerned critical aspects and on the time spent on the activity that generates flow. Theoretical contributions as well as implications for teaching are presented at the end of the article.
... The use of high cognitive tasks is advocated, involving complex problems or procedures, in particular multiple-solutions tasks, as they allow more students to solve the proposed tasks and can contribute to the development of flexibility, one of the characteristics of creativity, which requires a change in the way we "see" the situations, opposing fixation. The intuitions that lead to short and elegant problem solutions are called Aha! experiences (Liljedahl, 2016;NCTM;2014, Vale, Pimentel & Barbosa, 2018. We can use tasks with multiple solutions in classroom through two approaches: inviting students to solve each task in more than one way; or wait the occurrence of different approaches naturally within the classroom. ...
... We defend the first approach, because it is richer for the discussions, otherwise we risk that only the most common solutions connected to the concepts involved in the task appear and normally students became comfortable when they find a way to solve a task, not looking for another way that could be more intuitive and easier (Barbosa, Vale & Palhares, 2012;Vale & Barbosa, 2018a;Vale & Barbosa, 2018b;Vale et al., 2018). Liljedahl (2016) has defended the idea that the classroom must provide students Aha! Experiences. That is, to establish opportunities where students think individually, but also think collectively, learning together and building knowledge and understanding through action and discussion (Lildejahl, 2016). ...
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The aim of this paper is to share part of an ongoing study in which we are interested in introducing a Gallery Walk (GW) as an instructional strategy to contemplate in the classroom, in the context of preservice teacher training for elementary education (6-12 years old), to promote students' mathematical knowledge and skills, through problem solving abilities. In this study we intend, in particular, to identify the strategies used by students when solving challenging tasks with multiple approaches, using a GW, as well as characterize their reaction during their engagement in the GW as a teaching and learning strategy. A qualitative and interpretive study, with an exploratory approach, was adopted and the collected data included classroom observations and written productions. The results allowed to identify the strategies used by the participants and to verify the potential of the GW in the quality of the written productions and discussions, which proved to be more effective than in more traditional discussions, allowing to increase the repertoire of solving strategies of each student and communication and collaborative skills; it had a positive effect on the participants‟ achievements and it was an enjoyable and rewarding experience for all of them.
... To promote students' thinking, teachers should provide students with ample opportunities to engage actively in thinking activities in different contexts (Liljedahl, 2016;Zohar & Schwartzer, 2005). In the context of 3DP, the learner's attention focuses on the created artifact as an "objectto-think-with" (Papert, 1980). ...
Conference Paper
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Educators concur that schools should prepare students for the digital transformation era. 3D printing is a disruptive digital technology that can be integrated into education as a learning subject as well as educational technology. The potential of 3D design and printing (3DP) activities for learning different subjects and developing higher-order thinking skills is recognized. However, in practice, these activities fall short of fulfilling the potential. Our study was conducted in conjunction with a professional development program in which 38 teachers learned to integrate 3DP in education. Here we present a workshop focused on developing analytical thinking (AT) and applied mathematical skills (AMS). In the workshop, the teachers were assigned to design a spinning-top that could fit into a Kinder egg and analyze its geometry. We inquired whether and how the teachers employed AT and AMS during the activity by a post-workshop questionnaire. Nearly all teachers reported that they used analytical skills and specifically noted identifying the problem, breaking it into parts, and finding functional relationships. They also applied the mathematical skills of understanding and mathematizing real-world problems. The program helped teachers uncover the potential of 3DP technology to advance teaching mathematics and technology and foster higher-order thinking skills.
... One voice that stood in opposition to this idea was Guskey (1986), who showed that teacher beliefs could change as a result of changes to their practice, and the mechanism of that change was evidence of students' improved learning. Liljedahl (2016) extended this idea through his notion of a first-person vicarious experience by showing that changes in teachers' practice can lead to changes in beliefs, not only through evidence of student learning, but also through evidence of student enjoyment and behavior in the learning setting. In this work, we push these ideas further and document the changes in teacher's beliefs and knowledge within the recent COVID-19 upheaval, where circumstances necessitated changes in practice. ...
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In the spring of 2020, schools and universities around the world were closed because of the COVID-19 pandemic. The relative lockdown affected more than 1.5 billion learners as teachers and students sheltered at home for several weeks. As schooling moved online, teachers were forced to change how they taught. In the research presented here, we focus on university mathematics professors, and we analyze how their practice, knowledge, and beliefs intertwine and change under these circumstances. More specifically, the context of the pandemic and the relative lockdown provides us with the experimental basis to argue that the new practice affected both knowledge and beliefs of mathematics teachers and that practice, knowledge, and beliefs form a system. Being part of a system, the reactions to change in practice can be of two types, namely, the system as a whole tries to resist change, or the system as a whole changes — and it changes significantly. The research presented here proposes a model for describing and analyzing what we called a teaching system and examines three cases that help to better depict the systemic nature of teaching.
... Thus, the activities can take place in authentic primary mathematics classrooms. This study can, therefore, offer practical advice on DM implementation in mathematics classrooms, especially considering the growing concern over student disengagement from learning mathematics (Liljedahl, 2016) and calls for a technological approach to problem-solving to meet the need for STEM expertise (Cui & Ng, 2021). Future research directions include exploring other creative uses of DM as complementary to K-12 curricular learning and how best to orient students' learning trajectories to improve mathematics education with computation. ...
We report on a case study of eight grade 5 to 6 students (ages 12–14) involved in a three-day “digital-making summer camp” focused on mathematical problem-solving through block-based programming combined with programmable electronics. Data analysis focused on the computational thinking (CT), mathematics, and problem-solving challenges that surfaced during the activities and on the students’ developing perspectives about themselves as computational thinkers and problem solvers. Our results suggest the students developed various CT and problem-solving competences and dispositions during the digital-making activities. However, they experienced difficulties with ill-structured problems. This study supports the creative uses of digital making as complementary to K-12 mathematics learning.
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Algebraic thinking is an important part of mathematical thinking, and researchers agree that it is beneficial to develop algebraic thinking from an early age. However, there are few examples of what can be taken as indicators of young students' algebraic thinking. The results contribute to filling that gap by analyzing and exemplifying young students' early algebraic thinking when reasoning about structural aspects of algebraic expressions during a collective and tool-mediated teaching situation. The article is based on data from a research project exploring how teaching aiming to promote young students' algebraic thinking can be designed. Along with teachers in grades 2, 3, and 4, the researchers planned and conducted research lessons in mathematics with a focus on argumentation and reasoning about algebraic expressions. The design of teaching situations and problems was inspired by Davydov's learning activity, and Toulmin's argumentation model was used when analyzing the students' algebraic thinking. Three indicators of early algebraic thinking were identified, all non-numerical. What can be taken as indicators of early algebraic thinking appear in very short, communicative micro-moments during the lessons. The results further show that the use of learning models as mediating tools and collective reflections on a collective workspace support young students' early algebraic thinking when reasoning about algebraic expressions.
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Delivering an effective online mathematics course at the university level is more challenging than the offline mode because of the method course content's nature. One of the strategies to answer the challenges faced in an online learning environment is the utilization of Open Educational Resources (OER). This descriptive case study research investigates students' satisfaction and performance in the Open Educational Resources (OER)- integrated online calculus course. This study involved eight students who took a calculus course entirely online. The lecturer used various OER, both integrated into Canvas and during the online meetings. Students were satisfied with the online calculus course since it has met their expectations. The online course content and use of OER, classroom environment, and lecturer facilitation were three components that students find valuable. Likewise, students' performance in the online calculus course is excellent, where all students got grades A. Although students' performances in Unit Test 1 and Unit Test 2 are not as excellent as other assessments, they demonstrate active participation during the online meetings, frequently use OER, and consistently fulfill all assignments. They succeed in developing digital learning activities as their pair works and learning videos about the application of integral as their group projects. Keywords: Students’ satisfaction, Performance, OER, Online, Calculus
Teaching is a complex endeavor that necessarily requires teachers to attend to some activities and ignore others. This case study focuses on prospective teachers’ learning to notice student mathematical thinking. We frame our view of noticing with the professional noticing framework (Jacobs, Lamb, & Philipp, in Journal for Research in Mathematics Education 41:169–202, 2010), and our view of student mathematical thinking with the MOST analytical framework (Leatham, Peterson, Stockero, & Van Zoest, in Journal for Research in Mathematics Education 46:88–124, 2015). We share evidence that a research experience that focused prospective teachers in a sustained, intense experience focused on articulating student mathematical thinking through focused video analysis influenced their ability to notice in-the-moment student mathematical thinking during their student teaching experience.