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DOI 10.1007/s11858-015-0703-6
ZDM Mathematics Education (2015) 47:625–637
ORIGINAL ARTICLE
Numeracy task design: a case of changing mathematics teaching
practice
Peter Liljedahl1
Accepted: 5 June 2015 / Published online: 16 June 2015
© FIZ Karlsruhe 2015
… efforts to intensify attention to the traditional math-
ematics curriculum do not necessarily lead to increased
competency with quantitative data and numbers. While
perhaps surprising to many in the public, this conclu-
sion follows from a simple recognition—that is, unlike
mathematics, numeracy does not so much lead upwards
in an ascending pursuit of abstraction as it moves out-
ward toward an ever richer engagement with life’s
diverse contexts and situations. (Orrill 2001, p. xviii)
What is needed is not more abstraction, but more contextu-
alization—and an increased ability to deal with this contextu-
alization. As such, numeracy is generally seen as some com-
bination of mathematical knowledge, tools, and dispositions
(Goos, Geiger, & Dole, 2013), and to be numerate means to
be willing and able to use this knowledge, tools, and disposi-
tions across a wide variety of contextual (even real) situations.
Although varying across geographic jurisdiction, educa-
tional agencies, and individual researchers, all definitions
of numeracy draw in some way on these aforementioned
attributes while simultaneously aligning themselves with,
and distancing themselves from, mathematics.
… mathematical literacy1 is not only about arithmetic
and higher mathematics but also about a general skill
(or habit of mind) that is required in many subjects
across the curriculum. (Cohen, 2001, p. 28)
1 Although it can be argued that there is a distinction between the
terms numeracy and mathematical literacy (McAskill, Holmes, Fran-
cis-Pelton, & Watt, 2004), the fact is that preferential use of the terms
seems to be geographic, with some countries choosing to use the
former while other opt for the latter (Hoogland, 2003). Even within
countries there is sometimes a geographic disparity. For example,
in Canada, all of Canada uses mathematical literacy, except British
Columbia, which uses numeracy. As such, for the purposes of this
article, the terms will be used interchangeably.
Abstract Over the last 15 years, numeracy has become
more and more prominent in curriculum initiatives around the
world. Yet, the notion of numeracy is still not well defined,
and as such, often not well understood by the teachers who
are charged with the responsibility of helping our students
to develop their numeracy skills. In this article I explore the
work of a team of mathematics teachers brought together for
the purpose of developing a set of numeracy tasks for use
within district wide numeracy assessments. Results indicate
that these teachers’ experience designing these tasks, and pilot
testing them in their own classrooms, propelled them to make
massive changes in their own mathematics teaching practice.
Through a lens of Rapid and Profound Change (Journal of
Mathematics Teacher Education 13:411–423, 2010) the mech-
anism and catalyst behind these changes are revealed.
Keywords Numeracy · Task design · Teacher change
1 The numeracy movement
Around the world it has long been recognized that students
are completing their compulsory education without the
mathematical skills to cope with the demands that life and
work require of them. This recognition has launched, simul-
taneously around the world, what is commonly called the
Numeracy Movement (Hillyard, 2012; Wiest, Higgins, &
Frost, 2007). This movement is driven by a recognition that
mathematics alone is not helping students to achieve their,
and society’s, goals. What is needed is not more mathemat-
ics, deeper mathematics, or higher standards in mathematics.
* Peter Liljedahl
liljedahl@sfu.ca
1 Simon Fraser University, Burnaby, Canada
626 P. Liljedahl
1 3
That is, implicit in all the varying understandings of
numeracy, numeracy is not mathematics. While rooted in
mathematics, numeracy aspires to mobilize the skills and
knowledge of mathematics in such a ways that it transcends
the goals of mathematics.
… that will enable a typical member of the culture or
subculture to participate effectively in activities that
they value. (Evans, 2000, p. 236)
… gives students the tools to think for themselves, to
ask intelligent questions of experts, and to confront
authority confidently. (Cuban, 2001, p. 87)
… enable the learner to become a self-managing per-
son, a contributing worker and a participating citizen
in a developing democracy. (South Africa Department
of Education, 2003, p. 10)
Taken together, numeracy can be, and has been, defined
as:
Mathematical literacy is an individual’s capacity to
identify and understand the role that mathematics
plays in the world, to make well-founded mathemati-
cal judgments and to engage in mathematics, in ways
that meet the needs of that individual’s current and
future life as a constructive, concerned and reflective
citizen. (OECD, 2003, p. 14)
But this definition, like the hundreds of definition
existing in curriculum documents, white papers, and
research reports are not actionable. Instead, they are
idealizations—idealizations of what numeracy can
be, and what numeracy can achieve. They are more a
call to action than a blueprint for action. Yet action is
expected.
1.1 Numeracy and curriculum
This call for action, driven largely by the OECD’s focus on math-
ematical literacy (as manifest in their PISA and PIAAC assess-
ments) has resulted in a massive restructuring of curriculum all
over the world (Biedermann, 2014). For example, in Australia,
numeracy has been declared a cross-curriculum commitment,
naming numeracy as one of seven competencies applicable to all
school subjects (Council of Australian Governments, 2008).
That all systems and schools recognise that, while
mathematics can be taught in the context of mathe-
matics lessons, the development of numeracy requires
experience in the use of mathematics beyond the
mathematics classroom, and hence requires an across
the curriculum commitment. (p. 7)
However, the new Australian national curriculum (Aus-
tralian Curriculum, Assessment and Reporting Authority,
2012), like so many curricula around the world, does “not
provide teachers with detailed guidance in recognising the
numeracy demands of the subjects they teach, in designing
tasks and learning sequences that embed numeracy across
the curriculum, or in making decisions about pedagogies
that support numeracy learning” (Goos, Geiger, & Dole,
2013).
Although lacking a national curriculum, Western Canada
(like Australia) has declared numeracy as one of the cross-
curricular key areas of learning from Kindergarten to grade
9, and as one of the aims of mathematics education within
the 10–12 curriculum. And like Australia, the only real
support that teachers receive in realizing these curriculum
goals is a definition of numeracy.
Numeracy can be defined as the combination of
mathematical knowledge, problem solving, and com-
munication skills required by all persons to function
successfully within our technological world. Numer-
acy is more than knowing about numbers and number
operations. (Mathematics K to 12: Mathematics cur-
riculum documents, 2008, p. 11)
Regardless, since 2002, the British Columbia Minis-
try of Education has required each school district in Brit-
ish Columbia to provide an annual report on the numeracy
performances of students within their district. These reports
must include an explanation of how numeracy is being
measured within the district, what populations are being
measured, what improvements have been seen over time,
and what the district plans are to improve numeracy perfor-
mance in the coming year.
In short, much is required of teachers in these changing
environments. They are being expected to implement the
ideals of numeracy, not as a distinct topic in mathematics,
but in more subtle and dispositional ways, both within their
mathematics curriculum and across all curricula. And often
with little more support than a definition. In short, they are
being asked to change their practices in expansive and fun-
damental ways.
2 Teacher change
Current research on mathematics teacher change is situ-
ated largely within professional development literature and
can be sorted into three main categories: content, method,
and effectiveness. The first of these categories, content, is
meant to capture all research pertaining to teachers’ knowl-
edge and beliefs including teachers’ mathematical con-
tent knowledge, both as a discipline (Ball, 2002; Davis
& Simmt, 2006) and as a practice (Hill, Ball, & Schil-
ling, 2008). Recently, this research has been dominated
by a focus on the mathematical knowledge teachers need
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Numeracy task design: a case of changing mathematics teaching practice
1 3
for teaching (Ball & Bass, 2000; Ball, Hill, & Bass, 2005;
Davis & Simmt, 2006) and how this knowledge can be
developed within preservice and inservice teachers. Also
included in this category is research on teachers’ beliefs
about mathematics and the teaching and learning of math-
ematics and how such beliefs can be changed within the
preservice and inservice setting (Liljedahl, 2009, 2007;
Liljedahl, Rolka, Rösken, 2007). Some of the conclusions
from this research speaks to the observed discontinuities
between teachers’ knowledge/beliefs and their practice
(Cooney, 1985; Karaagac & Threlfall, 2004; Skott, 2001;
Wilson & Cooney, 2002) and, as a result, calls into ques-
tion the robustness and authenticity of these knowledge/
beliefs (Lerman & Zehetmeir, 2008).
The second category, method, is meant to capture the
research that focuses on a specific professional develop-
ment model such as action research (Jasper & Taube,
2004), lesson study (Stigler & Hiebert, 1999), communities
of practice (Little & Horm, 2007; McClain & Cobb, 2004;
Wenger, 1998), or more generally, collegial discourse about
teaching (Lord, 1994). This research is “replete with the
use of the term inquiry” (Kazemi, 2008, p. 213) and speaks
very strongly of inquiry as one of the central contributors
to teachers’ professional growth. Also prominent in this
research is the centrality of collaboration and collegiality in
the professional development of teachers and has even led
some researchers to conclude that reform is built by rela-
tionships (Middleton, Sawada, Judson, Bloom, & Turley,
2002).
More accurately, reform emerges from relationships.
No matter from which discipline your partners hail,
no matter what financial or human resources are
available, no matter what idiosyncratic barriers your
project might face, it is the establishment of a struc-
ture of distributed competence, mutual respect, com-
mon activities (including deliverables), and personal
commitment that puts the process of reform in the
hands of the reformers and allows for the identifica-
tion of transportable elements that can be brokered
across partners, sites, and conditions. (p. 429).
Finally, work classified under effectiveness is meant to
capture research that looks at changes in teachers practice
as a result of their participation in some form of a profes-
sional development program. Ever present in such research,
explicitly or implicitly, is the question of the robustness of
any such changes (Lerman & Zehetmeir, 2008).
Conclusions from all this research shows that with
time and continued intervention, support, and collabora-
tion, teachers can make significant and robust changes to
their practice. As a mathematics inservice teacher educator
working in a variety of professional development settings
I have witnessed teacher change of the form exemplified
in the aforementioned research. But I have also witnessed
change of a different kind—rapid and profound change in
practice—examples of which are not often found in the lit-
erature (Liljedahl, 2010).
2.1 Rapid and profound change
In prior research (Liljedahl, 2010) I identified and articu-
lated the phenomenon of mathematics teachers’ Rapid and
Profound Change. This research showed that this phenom-
enon can be nuanced into five distinct and non-hierarchical
mechanisms of change which I have come to call: (1) con‑
ceptual change; (2) accommodating outliers; (3) reifica‑
tion; (4) leading belief change; and (5) push–pull rhythm
of change. These five mechanisms of change stand in stark
contrast to the more pedestrian mechanisms of change
articulated above.
Conceptual change comes from the science educa-
tion literature, but has recently been applied to the learn-
ing of mathematics (Greer, 2004; Tirosh & Tsamir, 2004;
Vosniadou, 2006; Vosniadou & Verschaffel, 2004). This is
not a theory that applies to learning in general. Rather, it
is highly situated, applicable only in contexts where: (1)
misconceptions are formed through lived experiences and
in the absence of formal instruction; (2) there is a phenom-
enon of concept rejection; and (3) there is a phenomenon
of concept replacement (Vosniadou, 2006). In my previous
work (Liljedahl, 2010) this theory was able to explain the
rapid transformation of several teachers I had encountered.
Accommodating outliers is born from the work of Piaget
(1968) and explains instances in which teachers are able
to keep various aspects of their practice and experiences
disjoint from each other. Sometimes when, in professional
development settings, they are asked to consider these
experiences in unison, a process of accommodation occurs,
from which emerges a new view on mathematics and what
it means to teach and learn mathematics.
Reification is borrowed from the work of Wenger (1998)
and explains the observed phenomenon wherein a teacher
makes rapid changes to their practice after participating in
a process in which they “give form to [their] experiences by
producing objects that congeal this experience into thing‑
ness” (Wenger, 1998, p. 58).
Leading Belief Change acknowledges that teachers prac-
tice is not distinct from their beliefs about teaching and
learning mathematics (Ball, 1988; Chapman, 2002; Fosnot,
1989; Lortie, 1975; Skott, 2001) and that these beliefs do
not exist in isolation from each other. Instead, they cluster
together along lines of relevance to form robust systems
of beliefs (Chapman, 2002; Green, 1971). These belief
clusters can, however, exist distinct from each other, or in
tension with each other. Sometimes, when changes to one
of these clusters is made, it cause a significant and rapid
628 P. Liljedahl
1 3
reorganization of beliefs about teaching and learning
of mathematics, resulting in complementary significant
changes in teaching practice.
Finally, the Push–Pull Rhythm of Change articulates
the phenomenon where, in some cases, teacher change
goes through a period of tension wherein teachers first
resist changes by railing against the institutional systems
in which they are working and their inability to change it,
only to then suddenly collapse their focus down to the con-
fines of their classroom—a domain in which they do have
the ability to make change.
3 Research question
Considering the high expectation on mathematics teacher
change that is necessitated by the numeracy movement,
I have become curious about the professional growth of
teachers within this context. In particular, what are the
experiences of mathematics teachers as they struggle to
make sense of these emergent, and ill-supported, demands?
What is the nature of any of the resultant changes they may
undergo in their thinking about the teaching and learning
of mathematics, and what are the mechanisms of these
changes? Answers to these questions will go a long way
towards informing the mathematics education community
at large about teacher change in general, and the implemen-
tation of numeracy initiative in particular.
4 Methodology
The data for the research presented her comes from pro-
ject that was initiated in an effort to generate the afore-
mentioned district numeracy performance data required by
the British Columbia Ministry of Education. More specifi-
cally, the data comes a team that was assembled to craft the
numeracy tasks that would be used to gather this numeracy
performance data. This team, called the Numeracy Design
Team, consisted of four grade 5 and six grade 8 teachers.
The objective, for this team of teachers, was to develop two
numeracy tasks—one at each grade level—that would then
be administered to all of the grade 5 and 8 students in the
district.
4.1 The numeracy design team
This team met every 3 weeks from September until Febru-
ary, for a total of six meetings. Each meeting was 3 h long
and took place during school time. The terms of reference
for the team were scant with the only requirement being
that at the end of the six meeting the two numeracy tasks
were to be ready for district wide use. The only resources
afforded the team to help them in this endeavour were the
release time to meet, the aforementioned definition in use
in Western Canada, and a facilitator.
My initial role within this project was as this facilitator.
I was asked, as an academic, to take on this role in an effort
to help the teachers bridge, what the district administra-
tors referred to as, the gulf between theory and practice. I
accepted this role, in part, because I was curious how, given
the emergent importance of numeracy, the team of teach-
ers were going to negotiate this directionless space I saw as
endemic of the numeracy movement.
As the facilitator I organized these six meetings around
the general work of task design. In the first meeting I put
three questions to the team. The first was simply what they
thought numeracy was. The second question asked them to
think of a student that they had taught that they believed
to be exceptionally numerate and to then articulate for the
group what qualities that students possessed that made
them numerate. I intentionally asked for qualities rather
than knowledge as I wanted to get them away from any
assumption that numeracy was some collection of facts.
My third question was to now think about what a numeracy
task should look like. From these discussions a working
definition of numeracy emerged:
Numeracy is the willingness and ability to apply and
communicate mathematical understanding and proce-
dures in novel and meaningful problem solving situ-
ations.
Although not a perfect definition, it does capture the
qualities of our discussions on the first day.
In the second meeting the details of the previous day’s
discussions were refined as we began to design a number
of preliminary numeracy tasks for the teachers to take away
and test with their own students. In the third through fifth
meetings we debriefed the teachers’ experiences in imple-
menting the tasks and refined (or redesigned) the tasks
based on their feedback. Through this process the task grew
from being just about solving a task to also being able to
justify that solution. Eventually we produced two tasks, one
of which was eventually named Giving out Bonuses (Fig. 1).
The last meeting was used to finalize the wording, for-
matting the two tasks, and to write a script to help teach-
ers across the district implement the tasks within their own
classrooms.
4.2 Participants
As mentioned, the design team was comprised of four
grade 5 and six grade 8 teachers. These teachers joined the
design team as a result of a district request to all teachers of
these grades. Initially 13 teachers responded to this request
and from this the district administrator overseeing the
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Numeracy task design: a case of changing mathematics teaching practice
1 3
project selected 10 teachers to participate. This selection
was primarily done on the basis of teaching experience.
Of the 10 teacher who were part of the design team, six
were female and four were male, four had taught for over
15 years and 2 had taught for less than four. None of the
teachers had a university education in mathematics. In short,
the design team was a diverse collection of elementary and
middle school teachers representative of the gender, experi-
ential, and educational makeup of the school district.
4.3 Method
This research is ethnographic in nature in that, despite being
the facilitator, I was an observer within the design team com-
munity. However, having the role of facilitator made the col-
lection of data difficult as I found myself too embroiled in the
task design activities, impromptu conversations, commitments,
and actions, to adopt the removed stance of observer. Instead,
I adopted a stance of noticing (Fernandez, Llinares, and Valls
2012; Jacobs, Lamb, and Philipp 2010; Liljedahl, 2010;
Sherin, Jacobs, and Philipp 2011; van Es 2011). This stance
allowed me to work within the design team setting to achieve
our explicit goals, while at the same time staying tuned to the
experiences of the teachers involved. If something of interest
occurred, I was able to subtly shift from facilitator to observer
to researcher, and to begin to probe more deeply the comments,
conversations, and experiences of the teachers through ques-
tioning, individual interviews, and classroom visits.
As the design team was not primarily a research initiative it
would have been intrusive to introduce more formal data collec-
tion tools—such as audio and video recorders—into the design
team setting. As such, data consists primarily of the field notes
Fig. 1 The eventual grade 8 task
630 P. Liljedahl
1 3
taken during and immediately after each meeting, audio record-
ings of interviews with individual participants (and the resultant
transcripts), as well as field notes from classroom visits.
4.4 Analysis of data
These data were coded using a process of analytic induction (Pat-
ton, 2002). This process, like grounded theory (Charmaz, 2006),
relies on the use of a constant comparative method. The differ-
ence, however, is that with analytic induction the process starts
with a set of a priori codes. In the case of these data, these codes
came from two fundamentally different sources. The first was
from the framework of Rapid and Profound Change (Liljedahl,
2010) described earlier. The reason this framework was chosen
was that as a participant observer in the design team I began to
become aware that every member of the team was making sig-
nificant changes to their teaching practice, seemingly as a conse-
quence of being part of the project. This observation moved me
to shift my aforementioned stance from the more passive notic‑
ing to the more active data gathering. The framework of Rapid
and Profound Change (Liljedahl, 2010) gave access into this
phenomenon and afforded me very specific codes to analyse the
nature of the teacher changes I was observing.
The other source for a priori codes came from the larger
context in which this design team was situated—from the
local context of the school district all the way up to the
global context of the numeracy movement. This provided
me, not so much a set of codes, as a set of characteristics
for me to look for in the data. For example, characteristics
such as definitions, lack of support, and other than math-
ematics proved to be useful when coding the data.
At the same time, analytic induction, through its con-
stant comparative method, allows for the emergence of new
themes (Patton, 2002). This was the case with the research
presented here.
5 Results and discussion
The analysis of the data confirmed my initial observation that
the teachers in the design team were making significant changes
in their teaching. It further became clear that the mechanism of
this change was more or less the same for all 10 teachers. Rather
than discuss these changes across all of the members of the
design team I have chosen, instead, to represent the results in
the form of cases, written in the form of narratives (Clandinin,
1992). The three teachers—Lara, Frank, and Victoria—were
selected because the changes they underwent are reflective of
the changes the rest of the design team experienced.
In what follows, I present these three cases and then dis-
cuss them through the framework of Rapid and Profound
Change (Liljedahl, 2010). This is then followed by the pres-
entation of six further themes that emerged from the analysis.
5.1 The case of Lara
Lara is a middle school teacher (grades 6–8) with
15 years of teaching experience. She has always prided
herself on her progressive views on teaching mathemat-
ics and, as such, is often involved in the district math-
ematics initiatives. She has pilot tested new curriculum,
new textbooks, and has been part of the district learn-
ing team on assessment for the past 2 years. Lara was
recruited to the numeracy design team by one of the dis-
trict administrators.
Despite her progressive inclinations, Lara came to the
numeracy design team with very little a priori understand-
ing of what numeracy is. She was aware that there was an
increasing focus on numeracy in the curriculum and in their
district. But, for the most part, she assumed that this was
some sort of holistic description of mathematics the way
that literacy seemed to encapsulated reading and writing.
Isn’t it just that thing that people have that allows
them to work with numbers easily, like the cashier
who can give change without using the register? But
not by doing subtraction in their head, but by just
knowing all sorts of little tricks and shortcuts.2
However, when asked to think about a numerate student
she had taught and the qualities they possessed Lara very
quickly fixated on the importance of communication abili-
ties and then grounded this in her own experiences.
My best students are usually very good at explain-
ing their thinking and discussing things with their
classmates. [..] Almost every time I find myself doing
mathematics in real life it is always with another per-
son. Well, maybe that isn’t entirely true, but I can
think of a number of examples where I have been in
debate with someone about what the right answer is.
Even this morning I was arguing with my husband
about what time he has to leave the house to get the
girls to dance this afternoon.
From that point on, communication became an impor-
tant focus for Lara, both within the project and beyond.
After pilot testing the numeracy task Lara came to the
third meeting talking about the “best way for the students
to do the task”. Lara had noticed that the most effective
way for students to do this task was to have them work in
groups of two to three students for 15 min before complet-
ing the task individually.
2 This is not a direct quote of what Lara said. Rather, it is an as
accurate as possible paraphrasing of her direct comment recorded in
the field notes. This is true of many of the quotes in presented in this
article.
631
Numeracy task design: a case of changing mathematics teaching practice
1 3
This is the best way to make sure they are practicing
the communication that is so important in numeracy.
Lara had been part of many different professional devel-
opment initiatives in the district. As such, problem solving
and collaboration were already regular parts of her class-
room teaching. This is why she was so quick to try having
the students collaboratively discuss the task before solving
it individually. Because of this prior focus with collabora-
tion she was expecting the students to do well on the pilot
tests. But they did not. In particular, Lara she was unhappy
with her students’ ability to justify their solutions in writ-
ing. She realized that this was something she had never
before focused on in her teaching that she now wanted to
improve.
I realized that what was missing was their ability to
articulate their thinking. At first I just had them writ-
ing stuff at the end of every problem solving activ-
ity. This didn’t do much. Then I realized that it maybe
had more to do with literacy than numeracy so I vis-
ited one of my friends who teaches grade 2. [..] Now
I have a numeracy word wall, a problem solving strat-
egy poster, and an articulating thinking rubric that I
am using in my class and it is making a huge differ-
ence in their writing.
For Lara, this was a significant change in her teaching
and towards the end of the numeracy design team project
Lara was seeing big improvements.
I am actually thinking now that the more my students
improved their ability to articulate their thinking the
more they improved their ability to think. They are
getting better and better at being able to do numeracy
tasks. Not just the writing part, but the number parts
as well. And more and more of them are pushing at
the ceiling in the tasks.
5.1.1 Analysis of the case of Lara
The changes in Lara’s teaching began between the second
and third meeting. However, the mechanism of this change
began in the first meeting when she was asked to think
about the qualities of past students that she considered to
be numerate. This brought to mind, for Lara, their ability
to communicate, and through a process of accommodating
outliers, communication immediately became something
that she believed to be an important quality to measure
through the numeracy tasks. During the task design and
redesign process this belief was reified to be more spe-
cifically about students ability to justify their solutions.
This change in Lara’s belief structures, through a process
of leading belief change, led to a wholesale change in her
beliefs about the teaching of mathematics as something
that is now “all about articulating thinking”. These changes
in beliefs resulted in a massive restructuring of her teach-
ing practice to incorporate a word wall, a problem solv-
ing poster, and an assessment rubric—all to drive her new
focus across the whole of her mathematics curriculum.
5.2 Frank
Frank is also a middle school teacher with 12 years of
teaching experience. Unlike Lara, Frank has never been
part of a mathematics initiative at the district. He classi-
fies himself as a social studies and language arts special-
ist who has spent his career honing his teaching in these
areas. Frank came to the numeracy design team because he
has become increasingly unhappy with his students perfor-
mance in mathematics and his inability to “put his finger on
what is wrong”.
Frank had no initial ideas of what numeracy was—or,
at least, none that he would offer. From the beginning he
positioned himself within the group as the person who has
never attended a mathematics workshops or been part of a
design team and that he was “just keen to learn”. When the
discussions about the qualities of numerate persons began
Frank eventually began to talk about students who “just get
it”. He was very clear that he was not talking about gifted
students, but students who just had a very good sense of
what was going on when they were working on problems.
Towards the end of this discussion Frank became very ani-
mate, almost upset, at the realization that these skills are so
important to life and yet our K-12 curriculum does nothing
to foster these within students.
What are we doing in math if we are not working on
these things? I see students every year who have all
the facts, but if I ask them what 2 + 3 ½ is they can’t
answer without writing something down. What are
we doing to kids when they can sit and multiply out
three digit numbers but they can’t think clearly about
simple everyday concepts?
Like Lara, this became a real mantra for Frank.
During the pilot testing process, Frank became con-
cerned with two things he was observing in his students.
First, he was bothered by the poor performance of the
majority of his students. At the same time, he was worried
by the lack of challenge for some of his top students. The
design team had been working hard on ensuring that the
task allowed every student to start.
All my students were able to start. This is not the
problem. The problem is that my weak students were
too challenged and my really strong students are
not being challenged enough. Somehow, we need to
632 P. Liljedahl
1 3
make the task harder without making it harder. Does
that make sense?
The task design process reified Frank’s thinking into
the activity of problem solving, which Frank realized his
students had been missing in their experiences. He began
asking me for problems and resources of problems which
he then began using almost exclusively within his teaching.
At the same time he began problematizing everyday occur-
rences in his classroom.
I realize that I make lots of numeracy-like decisions
every day within my teaching, and that if I start get-
ting my students to make some of these then they will
start to really experience numeracy at its best. So, for
example, last week it was time to start basketball in
PE. Normally, I make teams at the beginning of a unit
and then those teams stay together for the whole unit.
Well, rather than making the teams myself I put the
students in groups and told them to come up with a
proposal for who should be on what team so that the
teams are fair. It turned into a whole week project that
included a whole bunch of data gathering about each
student in class. [..] In the end I had 8 proposals and
I chose one.
5.2.1 Analysis of the case of Frank
Like Lara, the changes in Frank’s teaching began with
the first meeting when, despite not having any particular
thoughts about numeracy, he accommodated the outliers
of some of his past students. This resulted in his focus on
students who “just get it”. The pilot testing after the second
meeting shifted this focus towards challenging his students,
which was further reified through the task design process
into a belief that problem solving was important. Like Lara,
this belief was the leading belief change catalysing Frank
to change his beliefs about the teaching and learning of
mathematics. As a result, Frank began to expansively use
problem solving in his day to day teaching of mathematics,
as well as other subjects.
Although following the same profound change mecha-
nism of accommodating outliers—reification—leading
belief change as Lara, this change followed a different tra-
jectory for Frank. Rather than focusing on communication
as Lara did, Frank focused on challenging his student, cul-
minating in a profound implementation of problem solving
into his teaching.
5.3 Victoria
Victoria is a former high school teacher with 8 years teach-
ing experience. At the time of the research, she had recently
started teaching grade 8 at a middle school where her
teaching partners had gladly given her all of the mathemat-
ics and science courses to teach. Victoria is the only mem-
ber of the design team who has a degree in mathematics
and has been trained as a mathematics teacher. Victoria has
come to the design team with very strong traditional views
about what numeracy is and what a numeracy assessment
should look like.
Isn’t numeracy just basic number facts? So, a numer-
acy assessment should just be a test of basic facts. We
really need this in this district. That is why I am here.
Interestingly, when I asked the group to think of quali-
ties of a numerate student Victoria’ answer did not mention
basic facts.
I taught a boy last year who was so good. He could
solve things in more than one way. He could explain
his thinking. And he was always trying to make con-
nections to other things we had learned.
This was a significant departure from her initial stance.
As much as Victoria valued fluency of basic facts, she also
seemed to value the more diverse skills of “making con-
nections” and “solving things in more than one way”. Like
Frank and Lara, this initial focus became a steadfast focus
for Victoria as the project evolved.
During the task design work of the design team the
diverse skills that Victoria valued were reified into a desire
to have students produce multiple solutions. This view was
refined further after pilot testing the numeracy tasks with
her own students—something she mistakenly thought
would go well.
I guess my students are used to more structured prob-
lems. My problems tend to be linked more closely to
specific things I am teaching.
In this moment, Victoria realized that the connectedness
she was looking to impart in her students was not possible
through her current teaching methods.
When I said that in our first meeting I was think-
ing about the way I teach. I really do value multi-
ple solution methods and I want my students to see
that there is often more than one way to do things.
So, I always teach them how to do things in more
than one way. And then what I want to see is that
my students can do these multiple ways that I have
shown them. This numeracy task requires some-
thing completely different form the students. This
isn’t about me having shown multiple ways. This is
about students being able to identify how to solve it
for themselves. They just don’t have any experience
doing this.
633
Numeracy task design: a case of changing mathematics teaching practice
1 3
The numeracy tasks we were developing, with their
openness and ambiguity, were asking for different skills
from the students. After this, Victoria began to notice that
one of the things that her students lacked was an ability to
deal with the freedom that the tasks offered. This realiza-
tion shifted Victoria’s thinking to a new belief of teaching
in which students needed to “identify how to solve [prob-
lems] for themselves”. This new belief about what students
need to be able to do, led in turn, to new beliefs about what
it means to teach and learn mathematics. In particular, that
she needs to stop being so directed in her teaching and
offer, instead, opportunities for students to “figure it out on
their own”. Over the course of the design team this led to a
wholesale reformulation of Victoria’s teaching style.
Instead of teaching first and then giving them ques-
tions second I give them the questions first. I just do
my lesson backwards.
5.3.1 Analysis of the case of Victoria
Like Frank and Lara, Victoria went through a process
of accommodating outliers—reification—leading belief
change. During the first meeting the accommodation of out‑
liers led to Victoria placing importance on solving “things
in more than one way”. The task design process reified this
idea into a focus on multiple solutions and the pilot testing
created a leading belief change that restructured her beliefs
around what it means to teach and learn mathematics.
5.4 Emergent themes
Emerging out of, and cutting across, these aforementioned
three cases in particular, and all 10 cases in general are a
number of themes. Whereas the aforementioned framework
Rapid and Profound Change (Liljedahl, 2010) explains the
mechanism of change that the teachers underwent as partic-
ipants of the numeracy design team, the themes articulated
below are the fuel that drove this mechanism. In what fol-
lows I briefly present each of these themes.
5.4.1 Past students
As seen in the three cases of Lara, Frank, and Victoria, the
activity of considering the qualities of a past ‘numerate’
student was a powerful trigger for each of them to accom‑
modate an outlier. Through this process they brought into
the main qualities that they may not see in their students on
a daily basis. In each case, these qualities were identified as
something worth assessing through the eventual numeracy
tasks being developed. Interestingly, for each member of
the design team the particular quality that they championed
in the first meeting shifted subtly along the way from being
something that they wanted to measure in their students to
being something that they wanted to foster within their stu-
dents—both through the numeracy task and in mathematics
more generally.
In making this shift two things are happening. First, the
teachers are disaggregating the nuanced qualities of ‘good
students’. Because of their association with successful stu-
dents these qualities are automatically seen as important.
Second, they are seeing these qualities as something within
their domain of influence—as something that they can fos-
ter within their classrooms and through their teaching.
5.4.2 Locally constructed definition
From the responses to the first question during our first
meeting it was clear that the teachers in the numeracy
design team had a very poor understanding of what Numer-
acy is. The majority of responses had something to do with
number sense, a few thought it had something to do with
real life, two thought it was related to literacy, and Victoria
thought it was basic facts.
Ironically, the extraction and fostering of the quali-
ties of a “numerate” student had no resemblance to these
aforementioned initial conceptions of numeracy (the noun).
They emerged, instead, from an intuitive understanding of
what it meant to be possess the qualities of numeracy—
to be numerate (the adjective). The aggregation of these
adjective understandings led to the taken-as-shared defini-
tion of numeracy as
… the willingness and ability to apply and commu-
nicate mathematical understanding and procedures in
novel and meaningful problem solving situations.
Although naively constructed, there is much overlap
with more thoughtfully constructed definitions from around
the world.
5.4.3 Task design
However, this definition is an idealization. It exists, like
all numeracy definitions, in the abstract plane, somewhere
between the intuitive understanding of what it means to be
numerate and a will for it to be something other than math-
ematics. It lacks concreteness. That concreteness comes
through the process of task design. That is, it is not until the
participants began to actually articulate these ideas in a rei‑
fied form of a task that the embodied qualities of the defini-
tion could be seen clearly. And like the process of thinking
of a numerate past student, the process of task design was
the impetus behind profound changes. In each case, it was
at this stage of the process that the ideas emerging form the
accommodated outliers changed, and took on a form that
was more articulate.
634 P. Liljedahl
1 3
Lara shifted from the abstract notion of communica-
tion to the concrete notion of justifying an answer. Frank
moved from the intangible articulation of students who
“just get it” to the more actionable idea of challenging stu-
dents. And Victoria transitioned from wanting students to
do “things in more than one way” to looking for more than
one solution.
Without the task design to reify the definition, and the
participants’ initial intuitive notions, the idea of numeracy
would have remained vague and abstract, much the way the
local curriculum documents represents it.
5.4.4 Poor student performance
The unexpected poor student performance on the pilot
testing also acted like a catalyst of change. On the heels
of surprising and poor student performance each of the
teachers subtly shifted their beliefs about numeracy—
and what it meant to be numerate—to mathematics more
broadly. The realization that their students were wholly
incapable of what they deemed to be an important quality
in numeracy meant, to each participant, that these quali-
ties were absent in their teaching practice in general. This
realization created the impetus to expand their new beliefs
from the context of the numeracy task in particular to
their mathematics teaching practice in general. And in the
process, these new beliefs caused a perturbation of their
existing beliefs about mathematics teaching and learning.
Lara’s focus on justifying an answer, when applied to
her practice in general took on an expansive goal of artic-
ulating thinking. Frank shifted his belief that a numeracy
task should challenge every student to the importance of
problem solving in mathematics, and beyond. And Victoria,
realizing that her students could only mimic what she gave
them, turned her lessons upside down and started each day
with a problem for them to solve.
5.4.5 For others—For themselves
The unexpected poor performance of students during the
pilot testing phase marks another significant transition
for the teachers on the design team. With the exception
of Frank, the teachers joined the numeracy design team
because they felt they had something to offer. These were
confident and capable teachers who, despite being gen-
eralist teachers, were very good mathematics teachers.
They joined this team not to help themselves, but to help
others, to use some of their expertise to develop resources
for other teachers. Even despite the initial changes in
their thinking as a result of the accommodation of outli‑
ers from considering past “numerate” students, and the
reification of their abstract notions into more concrete
ones as a result of the task design process, this view that
this was all for others was still pervasive. This can be
seen in the subtle comments made by some of the teach-
ers towards the end of our second meeting—just prior to
pilot testing the first draft of the numeracy tasks.
It will be interesting to see what teachers will think of
these tasks.
I think the district scores are going to be very interest-
ing.
Although the qualities that these draft tasks embodied
were seen as important, at some level, the teachers seemed
to assume that their students already possessed these quali-
ties. This made the poor performance of the students even
more surprising.
I couldn’t believe it. My students have no tolerance
for this kind of work.
My students can’t work in groups.
My students are useless at explaining their thinking.
The sudden and unexpected failure of their students was
a frank reminder that the qualities the tasks expected were,
in fact, absent in their own students. This realization, then,
shifted the for others nature of their work to for themselves.
Each meeting thereafter was partially occupied with talk about
their own emerging practices and only scantly on the work of
the task design. Even the last meeting, where our work shifted
to the scripting of implementation instructions for teachers,
contained significant discussion of their teaching in general.
5.4.6 The subversive goal of the task design process
With the substantive changes to each of the teachers’ prac-
tice, coupled with the time at each meeting spent discuss-
ing this, the teachers began to suspect that the explicit goal
of the numeracy design team was really just a diversion and
that the real goal for this team was to encourage substantive
changes in their teaching practice. This sentiment is nicely
summarized by Victoria.
I don’t think this team was about numeracy tasks at
all. I think it was all a trick to get us to change our
teaching. I mean, we hardly talk about the task any-
more. But we keep meeting and we now talk almost
only about our own teaching.
Even Frank, who came to the team looking for change
was suspicious that the numeracy task design goals were
the only goals for the team.
I believe that this team was about numeracy tasks.
We definitely need these in this district and we have
635
Numeracy task design: a case of changing mathematics teaching practice
1 3
designed a good one. But, I think it was also about
making us see deficits in our own teaching. By mak-
ing us try these tasks with our students you were
making sure that we would see things we didn’t like.
It was like a reality check for us … and I think it was
planned.
Lara had a slightly different take on this, extending the
groups theory of subversion to the students.
I don’t think it just about us. I think it is also about
our students. My students have changed like crazy
over the last few months just by having played with
the numeracy tasks that we experimented with.
Having them be part of the process has been really
impactful for them and I think that was also part of
the grand scheme.
5.4.7 The role of numeracy in the K‑12 mathematics
curriculum
There is a gulf between the idyllic and pragmatic aspects
of numeracy within the local curriculum. While it is highly
coveted by the ministry of education, numeracy is ill-
defined and poorly supported. As a result, it is easily dis-
missed as something not important and not relevant for
teachers and school districts alike. The district in which this
research took place felt otherwise. They felt that numeracy
was important and were not deterred by the lack of available
resources. They provided the support and trusted the numer-
acy design team to hone the parameters of what numeracy
is, what it means for a student to be numerate, and to find
ways to assess this. And we did. However, along the way the
members of the design team came to see numeracy, not as
something idyllic and unobtainable, but something that can
be enacted and achieved within their classrooms.
At the end of the project I asked the teachers to com-
ment on the position of numeracy within the curriculum.
From the perspective of looking back through the recent
transformation of their teaching practices the teachers in
the design team offered some insightful observations.
Although Frank came to the group much more con-
cerned about his mathematics teaching than anything to do
with numeracy, numeracy turned out to not only be the cat-
alyst for this change, but the context through which he now
viewed all his mathematics teaching.
Numeracy is a pretty big deal in this district and when
I came to this team I had it in my mind that numeracy
was just a subset of mathematics. Now, I’m starting
to think that it is actually the other way around—that
mathematics is a subset of numeracy. [As he spoke
Frank drew the diagram in Fig. 2 on the board] For
me, everything we have talked about here, all the
changes I have made, apply to my mathematics teach-
ing much more so than my numeracy teaching. In
fact, what we have done here is bleeding into all of
my teaching.
Frank’s diagram, although juxtaposing curricular areas,
was speaking about his own personal view of these rela-
tionships between numeracy and mathematics. Lara, on the
other hand, situated this view within the curriculum, not
from an individual perspective, but from a perspective that
views the curriculum as something for all teachers.
Right now numeracy is so easily ignored. The cur-
riculum mentions it, but I ignored it … we all ignored
it. Now when I read the curriculum documents it is
almost as if it is written in bold and that numeracy
IS the primary goal for mathematics teaching. But I
worry how others read it. It is too easy to ignore if
you don’t know what to look for.
In all, the teachers acknowledged that, despite the
prevalence of numeracy in the curriculum documents and
the ministry mandate, they had found it easy to ignore it
within their mathematics teaching. Their participation in
the numeracy design team, and the experiences this par-
ticipation entailed, made it not only impossible to ignore
Math
Numeracy
Numeracy
Math
Fig. 2 Frank’s Venn diagrams
636 P. Liljedahl
1 3
numeracy, but fundamentally changed the way it was posi-
tioned for them vis-à-vis the mathematics curriculum.
6 Conclusion
It is clear from the results presented above that the teach-
ers in this project made significant—even rapid and pro-
found—changes in their mathematics teaching practice.
The mechanism of this change was through a chained
sequence of accommodating outliers, reification, and lead‑
ing belief change. As such, this research extends my ear-
lier work on rapid and profound change (Liljedahl, 2010)
which looked at 42 cases wherein only one mechanism of
change was at play for each teacher.
These mechanisms of change were fuelled by three dis-
tinct experiences within the design team: a consideration
of a past numerate student, numeracy task design, and the
unexpected poor results of their own student during pilot
testing. Unlike the previous research (Liljedahl, 2010) on
rapid and profound teacher change, wherein the catalys-
ing experiences were treated descriptively, the potentially
prescriptive nature of the catalysing experiences in the
research presented here offers a means by which change
may be occasioned.
However, these catalysing experiences were effec-
tive only in that they occurred within a context that was
largely unfamiliar to the teachers. The lack of pragmatic
clarity as to what numeracy is, coupled with a lack of
resources around this important construct, afforded the
emergence of a more intuitive and grounded entry into
numeracy. The initial for others approach to the design
team set the stage for a heightened response to the unex-
pected and poor student performance while pilot testing
the tasks within their own classrooms. This shocking
experience instantly transformed the for others activ-
ity into the for themselves activity and helped drive the
teachers to make significant changes within their math-
ematics teaching in general.
Taken together, the idealized and ill-supported defini-
tions of numeracy present in the local context, combined
with the expectation that the teachers design a compre-
hensive tool for assessment, created within the numeracy
design team a perfect storm highly conducive to teacher
change. However, this storm would not have been possible
had there not been, at the outset of the project, a tension
between numeracy and mathematics wherein numeracy
stood, not beside (or inside) of mathematics as is often the
case in our definitions, but in opposition to it—as some-
thing new, as something different. And in so doing, numer-
acy offered these participants a different context in which
to think about, and experiment with, the teaching of numer-
acy (née mathematics).
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