A Commentary: Accounting-
of and Accounting-for the Engagement
of Teachers and Teaching
In this commentary, I reﬂect on the chapters by De Simone; Khalil, Lake, and John-
son; and Montoro and Gil (this volume) and how they contribute to our understanding
of engagement within mathematics education. However, to build a foundation for
this reﬂection, I will ﬁrst elaborate on the concept of engagement.
Within the context of mathematics education, the terms engaged and engage-
ment are used both colloquially and theoretically. Colloquially, engagement is used
either to refer to a person’s participation in something (verb)—she was engaged in
a problem solving exercise—or to refer to how a person participates in something
(adjective)—they were so engaged that I could not get their attention. Although these
two meanings stem from the same understanding of engagement as involvement the
ﬁrst use refers to the act of being occupied with something, while the second often
refers to how that occupation is experienced—with affect and motivation. This is not
to say that the ﬁrst is devoid of experience, but rather that the focus is on the doing
rather than the experience of the doing.
This distinction between the verb and adjective form of engagement also exists
within mathematics education research wherein the empirical and theoretical pursuit
is further bifurcated into a distinction between, what Mason (2002) refers to as, an
accounting-of and accounting-for. That is, research on engagement has been split
into a cataloguing of ways in which participants engage—an accounting-of —and
explanations for why they engage in this way—an accounting-for. In what follows
I ﬁrst explore this bifurcation through the four theoretical lenses of engagement
structures (Goldin, Epstein, & Schorr, 2007; Goldin, Epstein, Schorr, & Warner,
2011; Schorr & Goldin, 2008), studenting (Allan, 2017; Fenstermacher, 1986,1994;
Liljedahl & Allan, 2013a,b), ﬂow (Csíkszentmihályi, 1990,1996,1998), and forms
P. Liljedahl (B)
Simon Fraser University, Vancouver, Canada
© The Author(s) 2019
M. S. Hannula et al. (eds.), Affect and Mathematics Education,
ICME-13 Monographs, https://doi.org/10.1007/978-3- 030-13761- 8_14
310 P. Liljedahl
of engagement (Alvarez, 2016; Remillard, 2012). Following this I introduce the
three chapters that comprise this section of the book, again applying Mason’s (2002)
duality of accounting-of and accounting-for.
14.1.1 Engagement Structures
Engagement structures (Goldin et al., 2007,2011; Schorr & Goldin, 2008) began as
an accounting-of the differing ways in which students in a classroom setting engage
(verb) with problem solving. Through close observation and careful coding nine
behaviours emerged: Get the Job Done, Look How Smart I Am, Check This Out, I’m
Really into This, Don’t Disrespect Me, Stay Out Of Trouble, It’s Not Fair, and Let
Me Teach You.
These behaviours were accounted-for a collection of ten “simultaneously present
and dynamically interacting” components (Goldin et al., 2007) that come together
to form an engagement structure (Goldin et al., 2011). This structure sits at the
intersection of the behavioural, affective, and social. The ten individual components
that make up this “behavioral/affective/social constellation” (Goldin et al., 2011,
p. 549). These ten components are:
1. Goal or motivating desire
2. Patterns of behaviour
3. Affective pathway
4. External expressions of affect
5. Meanings encoded by emotional feelings
6. Meta-affect pertaining to emotional states
8. Interactions with systems of beliefs or values
9. Interactions with longer-term traits, characteristics, and orientations
10. Interactions with strategies and heuristics (Goldin et al., 2011, p. 549).
The ﬁrst seven of these components are based on the in-the-moment state of the
student whereas the last three components are based on the more enduring traits of
the students. Using these ten components to account-for the nine behaviours resulted
in an in-depth understanding of what motivates each of the accounted-of behaviours.
•Get the Job Done—In this behaviour, the student’s goal is to complete a mathe-
matical task or activity assigned by the teacher and to fulﬁl the explicit or implicit
obligation. The behaviour is a clear focus on carrying out the work.
•Look How Smart I Am—Here the student’s goal is to impress their peers, teacher,
or themselves with their mathematical ability, knowledge, or genius. This key
marker of this behaviour is “showing off” by trying to produce a solution faster,
or to produce a solution that is more correct, more efﬁcient, or more elegant than
14 A Commentary: Accounting-of and Accounting-for … 311
•Check This Out—The underlying desire behind this behaviour is a potential pay-
off—intrinsic or extrinsic. The potentiality of this payoff is heavily predicated on
the perceived value of what it is that the student is working on. The behaviour is
punctuated by an increased attention to the task in pursuit of the payoff.
•I’m Really into This—This behaviour is expressed in situations where the student
has entered into a state of deep engagement (Csíkszentmihályi, 1990). The goal that
drives this behaviour is the desire to remain in this state, tuning out distractions, and
seeking evolving complexities in the task as their ability increases. The physicality
of this behaviour is one of focus, rapt interest, and enjoyment.
•Don’t Disrespect Me—The motivating desire behind this behaviour is the preser-
vation of the student’s dignity, status, or sense of self-respect and well-being. The
behaviour is observable as resistance to the challenge, defensiveness, face saving,
and excessively charged discussions or arguments.
•Stay Out Of Trouble—In this case the student desires to avoid conﬂict or distress
with peers or the teacher. They achieve this through avoidance and striving to not
be noticed by others often prioritizing these goals addressing the mathematical
•It’s Not Fair—The motivating desire behind this behaviour is to redress a per-
ceived inequity and often occurs when the student perceives inequitable workload
recognition from the teacher. The behaviour is focused on correcting this inequity
rather than addressing the mathematical work itself.
•Let Me Teach You—In this behavior the student’s behaviour is guided by the
altruistic desire to help another student understand or solve the problem. The
behaviour of helping other is punctuated by the satisfaction derived by being able
to successfully help another student.
Like with engagement structures, the work on studenting began with an accounting-of
student behaviours in various learning situations. Fenstermacher (1986) ﬁrst intro-
duced the term studenting as the set of behaviours that help students learn.
The concept of studenting or pupiling is far and away the more parallel concept to that of
teaching. Without students, we would not have the concept of teacher; without teachers,
we would not have the concept of student. Here is a balanced ontologically dependent
pair, coherently parallel to looking and ﬁnding, racing and winning. There are a range of
activities connected with studenting that complement the activities of teaching. For example,
teachers explain, describe, deﬁne, refer, correct, and encourage. Students recite, practice,
seek assistance, review, check, locate sources, and access material. The teacher’s task is to
support R’s desire to student and improve his capacity to do so. Whether and how much R
learns from being a student is largely a function of how he students. (p. 39)
He later expanded this deﬁnition to also include the student behaviours that do
not help them to learn.
312 P. Liljedahl
[T]hings that students do such as ‘psyching out’ teachers, ﬁguring out how to get certain
grades, ‘beating the system’, dealing with boredom so that it is not obvious to teachers, nego-
tiating the best deals on reading and writing assignments, threading the right line between
curricular and extra-curricular activities, and determining what is likely to be on the test and
what is not. (Fenstermacher, 1994,p.1)
This notion of studenting was used by Liljedahl and Allan (2013a,b) to look
closely at student engagement (verb) across a variety of more traditional mathemat-
ics classrooms and classroom activity settings—doing tasks in class, taking notes,
homework, group work, review, and lecture. Results from this accounting-of student
behaviours showed a disturbing trend towards the later of Fenstermacher’s deﬁnition.
For example, while doing tasks in class Liljedahl and Allan (2013a) found that in
some classes upwards of 80% of the students’ behaviours did not contribute to learn-
ing. These non-learning behaviours included slacking (not doing anything), stalling
(avoiding doing anything), faking (pretending to work), and mimicking (mindlessly
following routines). In the context of homework (2013b) they found that 70–80%
of the studenting behaviour did not contribute to learning with the dominant non-
learning studenting behaviours being cheating, not doing it, or getting help. This last
behaviour was interesting because in each case it was shown that, for the most part,
the help aided in getting the homework done, but not in furthering the learning. If
the homework was marked there were more of these types of behaviours with all
three behaviours being almost equally distributed. If the homework was not marked
then cheating sharply decreased, getting help stayed about the same, and not doing
the homework increased drastically with an overall slight decline in the non-learning
studenting behaviour. In the context of note taking, Liljedahl and Allan found that
over half of the students were not attending to what they were writing and, instead,
just mindlessly copied what was on the boards. More troubling was the fact that
over 90% of the students never looked back at their notes. These passive student-
ing behaviours around notes were, like the other examples, not at all conducive to
Allan (2017) did a detailed accounting-for these studenting behaviours using the
theoretical framework of Leont’ev’s (1978) activity theory. Her results were varied
and detailed but can be summarized nicely in her analysis of two students who, from
the outside, seemed to exhibited the same behaviours, but for very different reasons.
While the ﬁrst student had a primary motive of learning and a secondary motive of
getting a good grade, the second student had the reverse—a primary motive of getting
a good grade and a secondary motive of learning. Although subtle, this difference
in motives made a world of difference in how they engaged in various classroom
activities. Although invisible to an outside observer the main difference was that the
motive of learning manifest itself as continuous engagement, while a primary motive
of getting a good grade resulted in discrete engagement. Only a close accounting-for
was able to reveal these nuanced differences in the accounted-of behaviours.
14 A Commentary: Accounting-of and Accounting-for … 313
In the early 1970s Mihály Csíkszentmihályi (1990,1996,1998) became interested
in studying the optimal experience—that moment where we are so focused and so
absorbed in an activity that we lose all track of time, we are un-distractible, and we
are consumed by the enjoyment of the activity.
… a state in which people are so involved in an activity that nothing else seems to matter;
the experience is so enjoyable that people will continue to do it even at great cost, for the
sheer sake of doing it. (Csíkszentmihályi, 1990,p.4)
In his pursuit to understand the optimal experience, Csíkszentmihályi (1990,1996,
1998) studied a population of people he thought most likely to experience this phe-
nomenon—musicians, artists, mathematicians, scientists, and athletes. Out of this
research emerged an accounting-of the elements common to every such optimal
experience (Csíkszentmihályi, 1990):
1. There are clear goals every step of the way.
2. There is immediate feedback to one’s actions.
3. There is a balance between challenges and skills.
4. Action and awareness are merged.
5. Distractions are excluded from consciousness.
6. There is no worry of failure.
7. Self-consciousness disappears.
8. The sense of time becomes distorted.
9. The activity becomes an end in itself.
While the last six elements on this list are characteristics of the internal experience
of the doer, the ﬁrst three elements can be seen as characteristics existing in the
environment of the activity and crucial to occasioning of the optimal experience—the
third of which became the central focus of Csíkszentmihályi’s (1990,1996,1998)
Csíkszentmihályi’s (1990,1996,1998) analysis of the optimal experience comes
into sharp focus when accounting-for the consequences of having an imbalance in
this system. For example, if the challenge of the activity far exceeds a person’s ability
they are likely to experience a feeling of anxiety or frustration. Conversely, if their
ability far exceeds the challenge offered by the activity they are apt to become bored.
When there is a balance in this system a state of, what Csíkszentmihályi refers to as,
ﬂow is created (see Fig. 14.1). Flow is, in brief, the term Csíkszentmihályi (1990,
1996,1998) used to encapsulate the essence of optimal experience and the nine
aforementioned elements into a single emotional-cognitive construct.
Thinking about ﬂow as existing in that balance between skill and challenge (see
Fig. 14.1), however, obfuscates the fact that this is not a static relationship. Flow is
not a collection of ﬁxed ability-challenge pairings wherein the difference between
skill and challenge are within some acceptable range. Flow is, in fact, a dynamic
process (see Fig. 14.2). As students engage in an activity their skills improve. In
314 P. Liljedahl
Fig. 14.1 Graphical representation of the balance between challenge and skill (Csíkszentmihályi,
Fig. 14.2 Graphical representation of the balance between challenge and skill as a dynamic process
order for these students to stay in ﬂow the challenge of the task must similarly
increase (Liljedahl, 2016,2018).
However, this theory of ﬂow did not always match up with the data Csíkszentmi-
hályi (1998) was analyzing. In particular, he found that a balance of challenge and
skill did not always produce ﬂow and became “a frustrating puzzle in an otherwise
fruitful research program” (Csíkszentmihályi & Csíkszentmihályi, 1988, p. 260).
Massimini and Carli (1988) eventually found that ﬂow only occurred if there was a
balance between challenge and skill and when both of these were in an elevated state.
Otherwise it would produce apathy (see Fig. 14.3). This realization eventually led to
an even more nuanced model (see Fig. 14.4) involving eight distinct states resulting
from different challenge-skill ratios (Massimini, Csikszentmihalyi, & Carli, 1987).
Csíkszentmihályi’s (1990,1996,1998) notion of the optimal experience and the
resulting framework of ﬂow is one of the only ways in which we can account-for the
adjective form of engagement.
14 A Commentary: Accounting-of and Accounting-for … 315
Fig. 14.3 The four-channel model of ﬂow (Massimini & Carli, 1988)
Fig. 14.4 Visual representation of the eight states of the challenge-skills ratio (Massimini et al.,
14.1.4 Modes of Engagement
Drawing on Ellsworth’s (1997) work on ﬁlm study and Rosenblatt’s (1982,1980)
theory of transactions, Remillard (2012) developed a model of how textual resource
developers position their audience and their resource, and how teachers, in turn,
interact with that resource. This theory begins with the mode of address, which is
about positioning the reader in a particular way so as to know how to initiate the
interaction between the reader and the resource. This is followed by the forms of
address, which are the particular looks and formats embedded in the resource that
manifest the mode of address. A reader will interact with this resource through a
particular stance called a mode of engagement. This stance, in turn, guides the forms
of engagement, which are the various ways in which the reader engages with the
resource and what they look for.
Looking at Remillard’s (2012) theory through the lens of Mason’s (2002)
accounting-of and accounting-for, the idea that the authors of textual resources posi-
tion the reader and that the reader brings a stance to their interaction with the resource
is an accounting-of the fact that both the production of a resource and how that
resource is utilized is guided by goals and volitions and that in order for the inter-
316 P. Liljedahl
action between author and reader to begin some assumptions about these goals and
volitions needs to be made. The modes and forms of the address of the resource and
the modes and forms of the engagement of the reader are an accounting-for the ways
in which these assumptions manifest themselves.
Alvarez (2016) extended and re-purposed Remillard’s (2012) theory to both
account-of and account-for the interaction between a professional developer and
a group of teachers. In this regard, the modes of address referred to the ways in
which the professional developer positioned the teachers and the forms of address
referred to the different pedagogical approaches the professional developer used with
the teachers. From the other side, the modes of engagement were the stances that the
teachers brought to the experience and the forms of engagement were the different
ways in which the teachers engaged in the professional development sessions. In
this sense, the notion of engagement (verb) in both Remillard’s (2012) theory and
Alvarez’s (2016) re-purposing is different in nature than in the aforementioned three
theories in that engagement is not the primary goal of this framework, but rather a
by-product of the efforts to account-for the interaction between a textual resource
and its reader.
14.2 Engagement of Teachers and Teaching
Although situated within a broad set of contexts, the aforementioned theories of
engagement, can be appropriated for use within the speciﬁc context of the teaching
and learning of mathematics. In what follows, I summarize the ways in which the
authors of the three chapters that comprise this section of the book have done so
and draw attention to how their work is both an accounting-of and -for engagement.
Although each of these chapters are, ostensibly, about understanding engagement,
each of the results also makes a contribution to our understanding of how to create
engagement. Further, although each chapter is explicitly about teachers, two of the
chapters also offer us subtly insights into the engagement of the learner. In my
summary I try to pull out these more elusive aspects of the contributions.
14.2.1 Teachers’ Classroom Engagement Structures:
A Comparative Study of a Novice US
and an Experienced UK Mathematics Teacher
To begin with Khalil, Lake, and Johnson co-opt the framework of engagement struc-
tures (Goldin et al., 2007,2011; Schorr & Goldin, 2008), developed and used to
account-of and -for student engagement (verb), to do both an accounting-of and an
accounting-for teachers’ engagement (verb). This novel application of the frame-
work, coupled with the notion of learning trajectories (Desimone, 2009), enables
14 A Commentary: Accounting-of and Accounting-for … 317
the authors to account-for the in-the-moment affect of two teachers and eventually
reveals that the teacher’ in-the-moment behaviours may mirror their students’ in-
the-moment behaviours and underpins the teachers’ emotional needs.
There are three implications emerging from this research. First, and foremost, the
framework of engagement structures (Goldin et al., 2007,2011; Schorr & Goldin,
2008) can be used to “connect teachers’ prior ‘in-the-moment’ behaviour as math-
ematics learners with their ‘in-the-moment’ behaviour as mathematics teachers”
(Khalil, Lake, & Johnson, this volume). Further, this framework can be used as a lens
to connect their affective and motivational domains to these same behaviours—both
past and present. Finally, and moving beyond the framework as a tool for the
accounting-for teachers behaviour, the framework of engagement structures can be
used to actively help in teachers’ affective development and to connect their emo-
tional development to their behaviours in-the-moment and across their overarching
learning trajectory as teachers.
The novel appropriation of the engagement structures framework (Goldin et al.,
2007,2011; Schorr & Goldin, 2008) coupled with an explicit desire to not only
understand teacher behaviour, but also to change their behaviour brings to the fore
the importance of explicitly linking teachers’ practice to their past experiences as
a learner and teacher with the in-the-moment emotions. Khalil, Lake, and Johnson
have done an exceptional job of theorizing this complex conﬂuence and actualizing
it as both research and intervention.
14.2.2 Exploring Flow in Pre-service Primary Teachers
Doing Measurement Tasks
Montoro and Gil chapter sits at this same intersection between research and interven-
tion with their dual intention of trying to provide pre-service primary teachers ﬂow
experiences (Csíkszentmihályi, 1990,1996,1998) while at the same time looking
closely at the elements present when these experiences occur. Mobilizing the theory
of ﬂow (Csíkszentmihályi, 1990,1996,1998) within a mixed-method approach, the
authors were able to account-for the difference in the ﬂow experiences of 230 partici-
pants and to hypothesize a way towards creating engaging (adjective) experiences for
this population of learners. The methodological approach used in this paper is both
rigorous and illuminating, giving us a much needed glimpse into the occasioning of
The authors begin their work by looking closely at the participants’ enjoyment
and concentration on each of ﬁve measurement tasks. Exploratory and conﬁrmatory
factor analysis reveals that 70% of the participants experienced elements of ﬂow in
tasks 1, 2, 3, and 5, with only 50% having the same experience with task 4. A cursory
analysis of the ﬁve tasks reveals that, while all the tasks are situated in the context
of measurement, only task 4 has a problem solving focus. The authors conjecture
that this results in an imbalance between the challenge of the task and the ability of
318 P. Liljedahl
the doer (see Fig. 14.1) and results in a decrease in the occasioning of ﬂow. This is
conﬁrmed through video analysis of one group of pre-service teachers working on
This video analysis further illuminates, not only why task 4 does not promote
ﬂow to the same extent as the other four tasks, but also why the other four tasks do.
Drawing heavily on ﬂow’s dimension of balancing challenges and skills (Csíkszent-
mihályi, 1990,1996,1998), Medina and Caudra’s analysis reveals the importance
of tasks being easy to begin and that an early entry proves to be vital to ensuring
that ﬂow is initiated. Even if the initial entry proves to be incorrect it is more fruitful
for occasioning ﬂow than to begin correctly with something difﬁcult. This result
contradicts Massimini et al. (1987) who predicted that low challenge would produce
apathy (see Fig. 14.4). However, the result is in line with Csíkszentmihályi’s earlier
work and offers us a more nuance understanding of how to engage (adjective) not
only pre-service elementary teachers in particular, but also learners in general.
14.2.3 The Intertwinement of Rationality and Emotions
in the Mathematics Teaching: A Case Study
Finally, De Simone looks at one teacher, Carla, and her engagement (verb) while
teaching. Although not drawing on the aforementioned theories of engagement, she
does account-for Carla’s decision-making process through the joint and intertwined
theories of rationality (Habermas, 1982,1998) and affect (Brown & Reid, 2006). De
Simone argues that with only “the pure rationality à la Habermas, [she] could only
have described the decisions of the teacher, without accounting-for [my emphasis]
their underlying reasons”. To get at the reasons she needed emotions—more specif-
ically, emotional orientation (Brown & Reid, 2006) and its resultant and visible
emotionality—to see the expectations in Carla’s decision-making. In so doing, De
Simone makes a powerful contribution to our understanding of the interplay between
teacher knowledge and teacher practice—something that is often overlooked in our
rush to improve teaching.
Although her work is descriptive in nature, De Simone offers us a means to think
about how her results can be used to improve teacher decision-making. Whereas emo-
tions and emotionality are situated and ﬂeeting, the rationality is grounded and can
be “prepared apriori”. Although not the topic of this chapter, De Simone’s method-
ological use of emotionality gives us insight into the reciprocal relationship between
teacher and student in the decision-making process. Whereas Carla’s emotionality
signaled her intentions to the researcher, it also signaled these same intentions to
the students—helping to inform their own decision-making process within the class-
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