Hybrids between rituals and explorative routines: opportunities to learn through guided and recreated exploration

Abstract

Within the commognitive perspective, ritual and explorative routines are used in a very particular way to distinguish students’ routines according to whether they are driven by social reward or by generating a substantiated narrative. Explorative routines in this theorisation may refer not to inquiry-based activity but to the result of a student’s routine moving from being process-oriented to becoming outcome-oriented, a deritualisation. Choice of tasks as well as a teacher’s moves offer students different opportunities to engage in rituals, explorative routines and deritualisations. Through nuancing the space spanned by opportunities to engage in rituals and explorative routines respectively, we describe and contrast classroom practices in three lessons from three contexts. The lessons share a commonality in encouraging explorative routines as a starting point, yet being adapted towards ritual activity through decreased openings for student agentivity, fewer invitations for students’ own substantiations or both. We argue that such adaptations are driven by the teachers’ commitment to reach mathematical closure in a lesson, to balance considerations of the classroom community and individual students and to meet curricular requirements. Our model helps interrogate the nature and relevance of hybrids of explorative routines and rituals.

Full text

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https://doi.org/10.1007/s10649-022-10167-z
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Hybrids betweenrituals andexplorative routines:
opportunities tolearn throughguided andrecreated
exploration
IbenMajChristiansen1 · ClaudiaCorriveau2 · KerstinPettersson1
© The Author(s) 2022
Abstract
Within the commognitive perspective, ritual and explorative routines are used in a very par-
ticular way to distinguish students’ routines according to whether they are driven by social
reward or by generating a substantiated narrative. Explorative routines in this theorisation may
refer not to inquiry-based activity but to the result of a student’s routine moving from being
process-oriented to becoming outcome-oriented, a deritualisation. Choice of tasks as well as
a teacher’s moves offer students different opportunities to engage in rituals, explorative rou-
tines and deritualisations. Through nuancing the space spanned by opportunities to engage in
rituals and explorative routines respectively, we describe and contrast classroom practices in
three lessons from three contexts. The lessons share a commonality in encouraging explorative
routines as a starting point, yet being adapted towards ritual activity through decreased open-
ings for student agentivity, fewer invitations for students’ own substantiations or both. We argue
that such adaptations are driven by the teachers’ commitment to reach mathematical closure in
a lesson, to balance considerations of the classroom community and individual students and to
meet curricular requirements. Our model helps interrogate the nature and relevance of hybrids
of explorative routines and rituals.
Keywords Teacher decision-making· Mathematics teaching· Ritual· Exploration·
Opportunities to learn
* Iben Maj Christiansen
Iben.christiansen@su.se
Claudia Corriveau
claudia.corriveau@fse.ulaval.ca
Kerstin Pettersson
kerstin.pettersson@su.se
1 Stockholm University, Stockholm, Sweden
2 Laval University, Quebec, Canada
Educational Studies in Mathematics (2023) 112:49–72
Accepted: 18 June 2022/ Published online: 8 August 2022
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1 3
1 Introduction
Mathematics is characterised not only by a particular terminology and accepted claims,
but also by accepted actions (Kitcher, 1984). Sfard (e.g., 2008) has characterised math-
ematics as a discourse with a particular terminology and visuals, accepted statements—
referred to as “endorsed narratives”—and routines. To learn mathematics means to learn
the discourse in all these respects. This has implications for how learning, and by implica-
tion teaching, is described and analysed. We engage with the latter in this study.
Researchers have drawn on commognition and its concept of routine to study students’
engagement in mathematical discourse in different contexts (e.g., Tabach & Nachlieli, 2015;
Viirman & Nardi, 2019). Within the commognitive tradition, routines are described not as
the actions themselves, but as “a set of metarules that describe a repetitive discursive action”
(Sfard, 2008, p. 208) or as “repetition-generated patterns of our actions” (Lavie etal., 2019,
p. 153). A routine is considered to have “three subsets that specify, respectively, the applica-
bility conditions, the course of action (procedure) and the closing conditions of the routine”
(Sfard, 2008, p. 221). There are three types of routines, determined by their purpose and
whether they are practical or discursive: rituals, deeds and explorative routines. Later work
within this school of thought has focused on rituals and explorative routines. Within the
commognitive perspective, what distinguishes these two types of routines is whether they
are driven by social reward or by generating a substantiated narrative (Sfard, 2008, passim).
Hence, carrying out an algorithm can be a ritual for one student, an explorative routine for
another and likewise for an investigative activity. Therefore, we have decided to use the term
explorative routine in the current paper, to distinguish it from the use of exploration to refer
to investigative activity; for the latter, we will use investigation. Choice of task and teacher
moves can make engaging in explorative routines more required of students, or alternatively
enable more ritual activity (Heyd-Metzuyanim etal., 2019; Nachlieli & Tabach, 2019).
The current study emerged from concerns shared amongst the authors. Using commognitive
concepts to analyse teaching, we experienced difficulties distinguishing how and explaining why
some lessons that started with interactions which encouraged explorative routines seemed, at
first sight, to “go awry.” To go beyond the first impression that lessons were undergoing adapta-
tions counter to their initial aim, we undertook a further analysis to enable a description of differ-
ent adaptations of lessons, in the space between ritual-enabling and exploration-requiring OTLs.
This further aided identifying learning opportunities generated by the teachers’ moves.
In the next section, we provide some background that frames our research objectives. Those
objectives prompt us to propose an analytical framework based on the concept of “opportunity
to learn” as presented by Nachlieli and Tabach (2019), combined with the concept of deritu-
alisation from Lavie etal. (2019). The latter concept enables us to identify changes in oppor-
tunities to learn between ritual and explorative routines. Thereafter, we explicate the methodo-
logical stance and the process of analysis. The subsequent section of the article presents the
analysis of the lessons, and the last an interpretation and discussion of our findings. We close
the article with our conclusions.
2 Earlier work
The focus of this article is to inform the unpacking of situations when teachers’ invita-
tions for students to engage in formulating endorsed mathematical narratives are altered
during lessons. This is not a new concern; therefore, we begin by situating the study
50 I. M. Christiansen et al.
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within the broader debate around this question. We distinguish two main trends. First,
we refer to studies on tasks which invite students to engage in constructive struggle.
Second, we direct our attention to studies on how teachers adapt lessons, by deviating
from plans or from how the lesson was introduced.
2.1 Tasks andteachers’ moves thatinvite students inconstructive struggle
Outside of the commognitive tradition, features such as problem solving, open tasks, rich
problems, discovery learning, investigations, opportunity to struggle and engaging in math-
ematical activity are often argued to constitute good teaching practices (e.g., Anthony &
Walshaw, 2009). For instance, productive failure—where students struggle to get the desired
answer in an investigation—is argued to lead to more desirable learning (Loibl & Leuders,
2018). Through such approaches, students are argued to develop more connections across
mathematical topics, develop a more relational view of mathematics, increase their motiva-
tion, learn the ways of mathematicians, learn mathematical concepts, better retain learning
or better learn algorithms (e.g., Boaler, 2022; Henningsen & Stein, 1997; Jäder etal., 2017;
Moyer etal., 2018; Smith & Stein, 2018; however, see also Munter etal., 2015).
There is a large body of work on the nature of tasks which can facilitate productive
struggle, often referred to as “high-level tasks” (e.g., Smith etal., 2008), tasks with
“high cognitive demand” (Henningsen & Stein, 1997) or “rich problem tasks” (Sch-
oen & Charles, 2003). Letting students in the experimental group practise creative tasks
even for very short periods of time has led to better test performance (Norqvist, 2018;
Wirebring etal., 2015).
Tasks but also interactions that require students to “use their head” (Corey etal.,
2010) are important in generating opportunities to learn. Hence, it is important to con-
sider teaching moves, from the perspective of which openings they generate for stu-
dents. A substantial body of work proposes guidelines for teachers’ interventions in
students’ work. These can be summarised as Hofmann and Mercer did, that “teachers’
interventions in small-group work need to be contingent on any difficulties that the stu-
dents are encountering, but without inducing dependence on the teacher” (2016, p. 402).
Warshauer (2015) suggests a continuum of teacher responses from telling students to
generating affordances, distinguished amongst others according to cognitive demand. Oth-
ers have offered more specific suggestions such as asking students to explain, listening to
students at task, revoicing, probing students’ reasoning, encouraging students to compare
ideas or directing students to resources (Amador & Carter, 2018; Hofmann & Mercer, 2016;
Olawoyin etal., 2021). The nature of the class discussions following students’ independent
work, in particular the extent to which the teacher manages to draw attention to key math-
ematical ideas, has also been seen as a main factor in facilitating learning from cognitively
demanding tasks (Asami-Johansson etal., 2020; Ceron, 2019; Kazemi & Hintz, 2014).
Tasks utilised in many classrooms may not be open-ended investigative tasks, yet
contribute to opportunities to learn. It is important to be able to describe the effort that
is involved in students’ sense making in connection to more commonly used types of
tasks. Within the commognitive perspective, Sfard has argued convincingly (2008, pas-
sim) that learning of mathematics often starts with rituals—“routines performed for the
sake of social rewards or in an attempt to avoid a punishment” (Lavie etal., 2019, p.
166) and as “imitating someone else’s former performance” (Nachlieli & Tabach, 2019,
p. 255)—that students with time and effort can change into discursive, product-oriented
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routines that lead to the production of substantiated narratives, or explorative routines
for short. Lavie etal. (2019, p. 157) argue that “helping students in transforming initial
rituals into explorations is amongst the principal challenges in teaching mathematics.”
This contrasts with the research above that mainly addresses teaching which starts from
rich situations, not from rituals which then need to undergo a deritualisation.
Through a commognitive analysis of classroom data, Nachlieli and Tabach (2019) have
shown the opportunities for learning generated by an interplay between activity that ena-
bles students to engage in rituals and activity that requires students to engage in explorative
routines. Deritualisation draws attention to the mathematical efforts students exert when
they are working not only with investigative tasks but also with sensemaking linked to
more ritual tasks. However, what are the teaching moves that can scaffold a transformation
from rituals to explorations?
Furthermore, as we discuss in the next section, teachers adapt lessons whilst they are
underway, which affects the learning opportunities. What does this mean in terms of rituals
and explorative routines?
2.2 Teachers’ lesson adaptations
Many researchers outside of the commognitive tradition have recognised difficult balances
in the classroom, for instance between sustaining student engagement and keeping the task
cognitively demanding (Hofmann & Mercer, 2016; Warshauer, 2015), between teachers’
orientations to or conceptions of teaching and learning mathematics, goals and knowledge
(Schoenfeld, 2010; Wilhelm, 2014) or between learning standard algorithms and learning
to participate in mathematical discourses (Lampert, 1990).
Tasks that are meant to engage students in cognitively demanding activities often
evolve into less-demanding cognitive activity, not the least as a consequence of teacher
actions (Henningsen & Stein, 1997; Stein etal., 1996). Teachers may reposition their direct
instruction from the whole class to groups of students instead (Baxter & Williams, 2010),
or take the lead in the problem solving or investigative work (Hofmann & Mercer, 2016).
Cognitive load reduction can also be a result of a focus on correct answers, classroom
dynamics, lack of or too much time—even both at the same time for different students—
unclear tasks or tasks with unclear expectations, tasks not appropriate for students’ previ-
ous knowledge, students’ lack of motivation or willingness or students not held account-
able for their work (Henningsen & Stein, 1997; Warshauer, 2015).
Teachers’ adaptations during a lesson may result from considering curricular goals, stu-
dents’ engagement and their mathematical activity (Gallagher etal., 2020; Jacobs et al.,
2010), though decision making can vary with culture (Yang etal., 2019) as well as with
knowledge, beliefs and experience (Gallagher etal., 2020). In a review of 19 articles on
adaptive teaching in mathematics, Gallagher etal. (2020) identified the following respon-
sive teacher actions: helping students connect to the key mathematical concepts, changing
inequity in social structures, engaging students discursively, providing feedback, choos-
ing tasks or deciding not to adapt the teaching. In contrast to the studies on how teachers
reduce the cognitive demand of tasks, the teacher actions summarised by Gallagher etal.
(2020) are infrequently mentioned as reducing the learning opportunities.
An analysis of the nature of teachers’ moves grounded in a discursive/commognitive
perspective may provide additional insights into the varied opportunities for learning
that adaptations of teaching generate. As our focus is on how the teaching moves affect
learning opportunities, we build onto previous work by Nachlieli and Tabach (2019) on
52 I. M. Christiansen et al.
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1 3
opportunities to learn, as well as on the aspects of deritualisation (Lavie etal., 2019). It is
to these theoretical conceptualisations we now turn.
3 Theoretical framework
In this section, we briefly summarise Nachlieli and Tabach’s (2019) notions of ritual-
enabling and exploration-requiring OTLs, and the concept of deritualisation (Lavie etal.,
2019; Sfard, 2016). The combination of these concepts allows us a first configuration of
some hybrids of ritual-enabling and exploration-requiring OTLs.
3.1 Ritual‑enabling andexploration‑requiring OTLs
Nachlieli and Tabach (2019) have built on the notions of ritual and exploration to focus on
the role of teaching. They propose two related concepts, ritual-enabling and exploration-
requiring OTLs. A ritual-enabling OTL refers to “teachers’ actions that provide students
with tasks that could be successfully performed by rigid application of a procedure that had
been previously learned” (Nachlieli & Tabach, 2019, p. 257). This is contrasted with an
exploration-requiring OTL, defined as “teachers’ actions that provide students with tasks
that could not be successfully solved by performing a ritual” (p. 257). Since the same activ-
ity may be a ritual to one student and an exploration to another, the use of “enabling” indi-
cates that the opportunity offered to the students makes it possible—but not required—to
complete the task ritualistically. Many tasks can be exploration-enabling, but only when
the task is not likely to be solved by any students by means of a ritual is it considered
exploration-requiring. This terminology alone suggests that there are tasks which are both
Table 1 Methodological lens of Nachlieli and Tabach (2019, p. 258)
Teaching (OTLs)
Ritual-enabling Exploration-requiring
1.
Initia-
tion
What is the question
that the teacher poses
(raises)?
How do you proceed? What do you want to achieve?
2.
Proce-
dure
How is the procedure of
the routine determined
(specifically with
respect to the flexibility
the teacher allows her
students)?
Students are expected to
apply a rigid procedure
that was previously
performed by others
in similar situations.
They are not expected
to make independent
decisions
Students are expected to choose from
alternative procedures. They are
expected to make independent deci-
sions
3. Clo-
sure
What type of answer does
the teacher expect?
A final answer. If reason-
ing is provided, it
details the steps of the
applied procedure
Stating the new narrative produced. If
reasoning is provided, it details the
mathematics reasoning involved
4. (Clo-
sure)
Who determines the end
conditions (to indicate
the task has ended)?
The teacher The student (based on mathematical
reasoning)
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ritual- and exploration-enabling, and that the nature of an OTL varies with time, in rela-
tion to previous engagements with tasks of the same type. To operationalise the concepts
for application in analysis of classroom data, Nachlieli and Tabach worked from the three
subsets of routines previously mentioned, to construct a methodological lens, captured in
Table1.
Analysing several lessons for OTLs, Nachlieli and Tabach (2019) find that ritual-ena-
bling and exploration-requiring OTLs can be embedded or nested within each other. They
decide “that the type of an OTL would be that of the external OTL as it constrains the stu-
dents’ opportunities to learn that could be offered by the internal routines” (2019, p. 263).
However, there is no reason why the relationship between ritual-enabling and exploration-
requiring OTLs could not also be, for instance, one of transforming one type of OTL into
another through lesson adaptation. As deritualisation concerns a transformation of rituals
into explorative routines, it seemed relevant to consider the opportunities to engage in der-
itualisation generated in the teaching.
3.2 Opportunities toengage inderitualisation
Deritualisation is a gradual change from a ritual to an exploration when a student’s rou-
tine develops from being process-oriented to becoming outcome-oriented (Lavie etal.,
2019). For deritualisation to happen, students must participate in the discourse whilst
also exerting effort to shift attention to the outcomes of routines rather than how to per-
form them (Nachlieli & Tabach, 2022; Sfard, 2020). Deritualisation is characterised by
increased flexibility, bondedness, applicability, performer agentivity, objectification and
substantiability (Lavie etal., 2019). If the circularity proposed by Viirman and Nardi
(2019) in moving between rituals and explorative routines is reflected in the classroom,
some teaching moves may be fostering deritualisation and others, ritualisation. Indeed,
Heyd-Metzuyanim etal. (2022) have suggested characteristics of learner-enacted ritual
and explorative routines corresponding to aspects of deritualisation, whilst Österling
(2022) has proposed an operationalisation of teachers’ deritualisation moves.
This paper explores an analysis of deritualisation moves in three classroom obser-
vations where the teacher altered the initial student task. Treating each of the six—at
the time of writing—characteristics of deritualisation as a separate dimension generates
a six-dimensional space. For the sake of brevity, we focus on the two deritualisation
moves that varied the most in our data—openings for performer agentivity and sub-
stantiability. This constitutes a two-dimensional plane within the generated space. This
process allows us to point to different relations and hybrid spaces alongside exploration-
requiring and ritual-enabling OTLs—but we do not claim to fill the space.
Increased agentivity implies the student being able to make more decisions indepen-
dently of external authorities (Heyd-Metzuyanim etal., 2022; Lavie etal., 2019). Sub-
stantiability increases when the student is increasingly able to assess their own perfor-
mance and substantiate the relevance or correctness thereof (op. cit.). By distinguishing
the extent to which a lesson offers opportunities for students to engage in aspects of
deritualisation—or ritualisation—we suggest that it is possible to nuance descriptions
of adaptations of teaching and their effect on learning opportunities. The pairing of
Nachieli and Tabach’s and Lavie etal.’s work helps us understand what may be at stake
from a teacher perspective. However, this pairing requires revisiting the operationalisa-
tion of the concepts.
54 I. M. Christiansen et al.
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In the last column in Table1, student agentivity seems central to criteria 2 and 4 for
exploration-requiring OTL. However, the focus appears to be more on the task situa-
tions—as in formulations about what is expected from students regarding procedures—
than on how the teacher may invite participation as the lesson evolves. To identify
opportunities for agentivity reflected in the teacher’s discourse, we utilise the idea of
overtures made by the teacher for student participation, as suggested by Sfard (2016).
Following Sfard, teacher utterances prompting students to explain or engage were coded
as overtures (openings) for student participation, however, only when students were
given time and room for participation. Closed questions as well as open questions which
the teachers answered themselves or for which students were given no time or opportu-
nity were coded as limited openings for student participation.
When it comes to what students are expected to produce (criterion 3 in Table1),
the distinction between requesting “a final answer” and a “new narrative” may not
be easily observable, given that providing an answer to a sum may be a new nar-
rative for some students (Sfard, 2008, p. 134, 223ff). More easily discernible is the
nature of the reasoning or substantiations requested from students by the teacher. In
Table1, the distinction made is between referring to steps in a previously performed
rigid procedure and detailing mathematical reasoning supposedly related to students’
“independent decisions.” As we understand this, it is in line with Sfard (2016), who
distinguishes relying on arguments of others or using own mathematical arguments in
substantiations:
[O]ne has to inquire about the sources of mathematical narratives and of their
endorsement. In explorative discourse, only those narratives are endorsed that can be
logically deduced from stories already endorsed (considered as true). In contrast, the
ritualization would often express itself in the problem-solver’s frequent recourse to
memory or to authority. (p. 7–8)
On this basis, we operationalised the opportunities to engage in substantiability accord-
ing to the sources of narratives and substantiations (others’ or own steps in a previously
learned procedure or developed in response to the task) which students were encouraged—
and given opportunity—to generate. Figure 1 illustrates these dimensions and the sug-
gested names for the OTLs generated by variations in the dimensions.
As can be seen, two of the options—ritual-enabling and exploration-requiring
OTLs—are taken directly from Nachlieli and Tabach (2019). The two other options
have some characteristics of an explorative routine and some of a ritual, and there-
fore we have chosen to refer to them as hybrids. We anticipate that a teacher who
gives students opportunity to develop their own substantiation without inviting agen-
tivity is closely guiding the exploration; therefore, we named it a guided-exploration-
enabling1 OTL, which we refer to as guided OTL for short. As for the combination
using others’ substantiation and inviting student agentivity, we expect the teacher to
encourage students to participate (for instance formulate narratives or develop argu-
ments during group work), but to value a specific way to address these open ques-
tions, recreating modelled substantiations, hence recreated-exploration-requiring
OTL, which we refer to as recreated OTL for short.
1 We have retained the use of enabling here, and later of requiring, to emphasise our focus on teacher
actions. Whether learners respond as desired to these OTLs is a different question.
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3.3 Research questions
We now have a conceptualisation which may help distinguish the adaptations that the lessons
undergo, using two dimensions: teacher openings for student agentivity and sources of encour-
aged substantiations.
We can therefore specify our research questions as follows:
1. To what extent do hybrid OTLs—characterised by teacher moves in terms of invitations
for student agentivity and encouraged substantiations—provide meaningful distinctions
of lessons?
2. What learning is enabled in such hybrids of rituals and explorative OTLs?
4 Methodology
In this section, we briefly describe the cases and the reasons for their selection, followed by
a presentation of our methods of analysis.
4.1 The cases andtheir selection
To obtain a range of lessons to analyse, we took one high school lesson from existing mate-
rials collected in the Swedish TRACE project, which focuses on the teaching of novice
Infrequent overtures
for learner agenvity
Frequent overtures
for learner agenvity
Learners’ own
substanaons
encouraged
Others’
substanaons
encouraged
Recreated-
exploraon-
requiring OTL
Ritual-
enabling OTL
Exploraon-
requiring OTL
Guided-
exploraon-
enabling OTL
Fig. 1 Dimensions in the plane intersecting the OTL space, with ritual-enabling OTL and exploration-
requiring OTL at opposite extremes
56 I. M. Christiansen et al.
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1 3
mathematics teachers; one Grade 4 lesson collected in the Canadian MathéRéaliser project,
which focuses on the use of manipulatives in the teaching and learning and the US3 Grade 8
lesson from the TIMSS Video Study. The latter is the lesson analysed in detail by Nachlieli
and Tabach (2019). The two former lessons were both characterised by containing one or
more iterations of an explorative routine which was then ritualised to various degrees as the
lesson progressed, and the parts of the lessons discussed here are those where such adapta-
tions took place. As argued in Sect.5.1, even the US3 lessons had elements of such adapta-
tions. The ritualisations, however, took different forms, which is another reason for selecting
these cases. Thus, the selection of cases was purposive, in line with our focus of adaptations
of exploration-requiring OTLs towards increasingly ritual-enabling OTLs. At the same time,
we aimed for differences in grade level, level of teacher experience, country and content, so
as to apply our analytical distinctions to as varied data as reasonably possible. An overview
of the cases is given in Table2.
4.2 Method ofanalysis
Our analysis took the above operationalisation of invited student agentivity and encour-
aged substantiations as its starting point. We detail the three phases of our analysis below.
For Julian’s and Sven’s lesson, the researchers had chosen a lesson to transcribe in which the
aforementioned adaptation took place. Native-speaking researchers were responsible for the
coding of the Swedish and Canadian lesson (in French), though substantial parts of each tran-
scription were translated into English to enable a joint interpretation.
4.2.1 Phase 1—identification ofOTL episodes
Following Nachlieli and Tabach (2019), we identified opportunities to engage routines in the
classroom by identifying initiations of new tasks by the teacher. Subsequently, we determined
the phases of working with procedures on the task and the closing. A section of a lesson
from initiation to closing was then considered one unit of analysis. As found by Nachlieli and
Tabach (2019), these OTL episodes could be embedded in one another.
4.2.2 Phase 2—codification
In each of the identified OTL episodes, we coded all teacher utterances for invited student
agentivity and substantiations, as described above. Table3 shows examples.
These codings allow us to identify shifts in opportunities for agentivity or substantiability
in the lessons. For instance, Julian changed, at some point during the lesson (see Sect.5.2),
from always encouraging students’ substantiations to encouraging use of a previously demon-
strated procedure two-thirds of the time.
Table 2 The three cases
US3 lesson Julian’s lesson Sven’s lesson
Country USA Canada Sweden
Grade level 8 4 11
Teacher’s experience level Experienced Experienced Recent graduate
Content Rules for exponents Fractions (part-whole relation) Proving statements about geo-
metrical figures
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4.2.3 Phase 3—Synthesis ofeach lesson
At this stage of the analysis, we synthesised the coding from phase 2 within each episode
determined in phase 1. The analysis of the changes in deritualisation-encouraging moves is
presented in the next section.
5 Analysis ofthelessons
This section has three parts. In each part, we discuss the ways in which the respective les-
son unfolded in terms of changes in calls for substantiation and student agentivity. In the
following section, we interpret and discuss the results of our analysis.
5.1 Lesson US3
In their 2019 paper, Nachlieli and Tabach shared their analysis of lesson US3 from the
TIMSS 1999 Video Study website and concluded that it consisted of embedded explora-
tion and ritual routines. They summarised the first 20min of the lesson as an exploration-
requiring OTL (routine 1) embedding three exploration-requiring OTLs, one for each of
three rules for exponents. The first of these—routine 1.1—concerned arriving at the gen-
eral rule
a
m
×a
n
=a(
m
+
n
)
and is the one we focus on here.2
When we considered lines 111–173 of the transcript, we saw modelling preceding explo-
ration-requiring OTL 1.1, and the teacher deciding when the task ended, which did not fit
the criteria (2) and (4) for an exploration-requiring OTL, according to Nachlieli and Tabach
(2019) (see also Table1). As we are precisely interested in hybrid OTLs, we coded for agen-
tivity and substantibility. Table4 shows the stages of the first 15min and 40s of the lesson.
Table 3 Operationalisation of key concepts with illustrative examples
Concept Operationalisation Categorisa-
tions
Examples
Agentivity Invitations for student
agency
Encouraged “See if you see a pattern developing on your own.”
(US3)
Limited “Can you make a geometrical figure [with the
manipulatives], stick the pieces together?”
(Julian)
Substanti-
ability
What are the sources
of narratives and
substantiations that
students are encour-
aged to produce?
Others’
arguments
or steps in
a known or
previously
demon-
strated
procedure
“And if you need to expand it out to get the answer
like I did, that’s what I’d like you to do.” (US3)
Students’
own argu-
ments
“For the first challenge, show us how you did
it.”(Julian)
2 The lesson with both video and transcription is available at http:// www. timss video. com/ us3- expon ents#
tabs-2
58 I. M. Christiansen et al.
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1 3
Table 4 The first part of lesson US3
Time Focus Openings for student
agentivity
Encouraged substantiations
00:00–03:42 Classroom management,
etc
N/A N/A
03:43–06:48 Introduction to the topic
“exponents.” Review
of the concept of
exponential expressions.
Introduction and exem-
plification of exponen-
tial growth
Generally closed ques-
tions
None
06:49–07:08 Initiation of routine 1:
finding out about three
rules for multiplying
exponents
None None
07:09–09:26 Teacher goes through
three numerical exam-
ples corresponding to
each of the three rules:
22×23
;
(2
3
)3;(2
x
)2
. In
each case, she expands
the expressions before
rewriting
Closed questions except
one
None
09:27–10:00 Initiation of routine 1.1
(in groups to find a rule
for
an×am
), and of
routine 1.1.1 (to expand,
multiply, and write as
one exponential expres-
sion the three examples:
a2×a4
;
a2×a
;
a3×a×a4
)
Limited time to work.
The teacher asks one
open question
Followed by a request to use
the previously demonstrated
procedure
10:01–11:30 Whilst students are work-
ing on the three exam-
ples, the teacher gives
instructions concerning
which procedure to use
Only a brief time for
students’ work
Repeated instruction to use
the teacher’s demonstrated
approach to substantiation
11:31–13:06 Return to routine 1.1 as
students are asked to
discuss patterns and
formulate a rule for
an×am
Encouraged through
open questions, but
there are fewer than
90s for the discussion
None
13:07–15:19 Students are asked to
share their proposed
rules, and, during this,
one example is revisited
Begins with open
questions about
which rules students
proposed but turns
to closed questions
around specific com-
ponents
None
15:20–15:39 Teacher summarises
the rule in words and
reinforces that it only
applies when the bases
are the same
None except for one
closed question
None
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Invitations of students’ participation mainly occur in the part of the lesson where students are
invited to share their proposed rules. The students’ individual work takes less than a minute, the
group work around 90s and the whole-class discussion around 2min, a large part of which is
taken up by addressing an incorrect suggestion from one student. Such short time spans for gen-
erating a new narrative and sharing substantiations indicate that very little or no time was spent
deciding on a procedure and weighing substantiations. Substantiation is only encouraged when
the teacher suggests following her demonstrated procedure to answer the three exemplary tasks
as a basis for developing the rule. Together, the teacher’s demonstrations (07:09–09:26) and her
insistence on following her example (10:01–11:30) steer the students towards a particular rule.
It is on this basis that we suggest that the situation around the generalising task (1.1)
should indeed be considered a hybrid rather than an exploration-requiring OTL. The only
time substantiation was invited is when the teacher instructs students to replicate her
demonstrated approach (we have considered this a substantiation both because the dem-
onstrated procedure is used to justify the addition of exponents, and because in the com-
mognitive framework following the steps of a procedure is considered a substantiation, as
also indicated in Table1 for step 3 of the ritual-enabling OTL). Therefore, we have charac-
terised this part of the lesson as a recreated-exploration-requiring OTL. This characterisa-
tion does not mean that we reject the importance of what goes on in this classroom or the
appropriateness of the teacher’s actions, on the contrary. In this particular case, demon-
strating a mathematical substantiation and facilitating students’ recreation thereof enable
the lesson to culminate in the formulation of three rules abstracted from examples worked
through with a given procedure. We return to the point of strengths and weaknesses of the
hybrid OTLs in the discussion.
5.2 Julian’s lesson onrelating fractions andwholes
Julian (pseudonym) teaches 9–10-year-old students in Canada. He is an elementary teacher
with more than 20years of experience. In the analysed lesson, the students worked on a
fraction task which Julian had developed together with researchers from MathéRéaliser to
study the use of manipulatives in numerical contexts. The task consisted in finding a frac-
tion starting from a given different fraction of the same unit. The students already knew
what a fraction was in relation to a whole and how to identify and represent fractions.
However, they had not previously engaged in “reconstituting” a whole from a given frac-
tion. To work on the task, students received a bag with pattern blocks: six green triangles,
three blue diamonds, two red trapezia and three yellow hexagons (Fig.2). The lesson epi-
sodes and their coding are summarised in Table5.
Fig. 2 Kit of pattern blocks
60 I. M. Christiansen et al.
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Table 5 Julian’s lesson
Time Focus Openings for student
agentivity
Encouraged substantiations
07:57–11:23 Julian makes sure
students have their
manipulatives and
presents the first two
tasks
He announces task 1:
Given that the green
triangle represents a
twelfth, he asks the
students to “make up
a third of the same
whole” (i.e., generate a
new narrative). He also
presents task 2: Given
that the blue diamond
represents a twelfth,
“make up a third of the
same whole”
Encouraged through the
very open tasks
None encouraged explicitly, but the
task was given to students working
in pairs; therefore, they may have
been expected to engage substantia-
tions between them
11:24–18:20 Students work in pairs
for 7min, mainly on
task 1. They are invited
to use manipulatives as
they wish
As above. Students are
given time to work
As above
18:21–23:45 Julian asks a student
(Lou) to come to
the board to explain
and substantiate the
group’s procedure.
Magnetic pattern
blocks were available,
and the student utilised
these, generating a
whole out of two
yellow hexagons, and
arriving at the answer
of four green triangles
representing a third
During the student’s
explanation, Julian
asks for further
justifications and asks
questions scaffolding
the student’s explana-
tion. Also, he asks for
a specific way to use
the manipulatives
Varied Both students’ own (mathematics)
and Julian’s (way of using the
manipulatives)
23:46–24:48 When the student ends
her explanation, Julian
asks for other ways to
solve the task. Two
other students present
mathematically similar
solutions
Encouraged Others’ and Julian’s
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During Lou’s presentation at the board (18:21–23:45), Julian uses both open and closed
questions to guide the student to refer to mathematical objects in her own discourse, and
to make unspoken aspects of Lou’s substantiation explicit—as in asking the student: “Can
you just show what 12 twelfths is going to look like?” and “why are there 12 twelfths?” In
addition, the teacher instructs the student to use the manipulatives in ways not specified by
the original task, asking the student, “can you make a geometrical figure”, or instructing
her to “stick the pieces together.” The task as well as this interaction generally has the char-
acteristics of an exploration-requiring OTL.
Task 2 is similar to task 1, except that this time, the blue diamond represents a twelfth.
Students must again decide on a figure to represent one-third of the same whole. A student
(Stephanie) is invited to the board to present her solution. It was the interaction between
Stephanie and Julian which we first noticed as a hybrid OTL. It proceeded as follows:
Transcript 1 Part of Julian’s lesson
Table 5 (continued)
Time Focus Openings for student
agentivity
Encouraged substantiations
24:49–27:19 Julian asks another
student (Stephanie) to
explain how to solve
task 2. See Transcript
1 below
Varied Encouraged to use the same substan-
tiation as the previous student
27:20–32:05 Students are asked to
perform a subtask.
They must represent
the whole, knowing the
blue piece represents a
twelfth
Task invites some
reasoning. Teacher’s
questions are closed
No calls for substantiations
32:06–34:34 Three students present
their solution to the
subtask on the board.
The first works on the
wrong whole (two
hexagons from task
1) and the other two
provide two different
ways of constituting
the “whole”
Mainly closed questions,
but different methods
allowed
Mainly students’ own
34:35–37:00 A new student presents
the rest of the solution
to task 2 (starting from
the subtask) on the
board
Mainly closed questions Invites replicated substantiations
37:01–38:48 Julian asks for other
ways to solve task 2. A
student (Lou from task
1) explains almost the
same procedure
Mainly closed Invites replicated substantiations or
procedures
38:48–49:01 Question 3 (not included
for analysis)
Mainly closed questions,
but time to work
Mainly replicated substantiations or
procedures
62 I. M. Christiansen et al.
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Speaker Verbal Action
Stephanie Well, it’s, like, there Points to the magnetic pattern blocks on the
board from the first task
well uh… The third of 12 is 4 so you had to
do that
Points to three blue diamonds
but here we didn’t have enough pieces, so we
took two triangles
Combines two green triangles to make a fourth
diamond
Julian But I don’t really understand what you are
saying
Stephanie Well, if we know, from the same principle
with the triangles…
Points to the four green triangles on the board
from task 1
It’s just that here… with the diamonds… Points to the blue diamonds
We needed four [diamonds], but we only
had three [in their kit], so we used that
instead…
Points to two green triangles
because it’s equivalent to that Points to one blue diamond
Julian But where is the whole?
Instead of using the same procedure as established in the first task—first constructing
a whole of twelve twelfths and from there a third—Stephanie uses what was established
in task 1, namely that a third is four times a twelfth. However, the teacher asked questions
to guide Stephanie to follow the same procedure as previously by first constructing the
whole. She struggles to do so, whereupon the teacher asks the whole class to work in pairs
and construct a whole from the blue diamond representing a twelfth. As a second student
comes to the board, Julian shifts to invite the student to use the particular procedure, rather
than inquire as to the group’s substantiations.
The lesson is clearly initiated as an exploration-requiring OTL. The teacher guides
Lou’s presentation of the group’s work at the board with both open and closed ques-
tions and encourages the student’s own substantiations of the mathematics. However,
when Stephanie introduces a shortcut to the answer of the second task, Julian guides
her towards using the same procedure and thereby the same substantiation as the one
previously demonstrated by Lou. This part of the lesson is best characterised as a rec-
reated OTL. That Julian did not only expect but also wanted an explanation that goes
via the construction of a whole is evident from the introduction of a subtask that fol-
lows the interaction with Stephanie. It constitutes a turning point in the lesson, as the
remaining parts are more akin to recreated OTLs and even ritual-enabling OTLs with
the teacher asking mainly closed questions and encouraging students to use the already
demonstrated procedure as substantiation of their answers.
As with lesson US3, this adaptation of an exploration-requiring OTL into a hybrid form
with stronger replicated or recreated elements serves a purpose. Stephanie’s shortcut is math-
ematically valid, but not all her classmates may have deritualised Lou’s method to the extent
that they can skip steps—as their activity on the task introduced by Julian shows. Indeed, the
flexibility of the “shortcut” may be limited to tasks with the starting fraction of one-twelfth.
By restricting the students to Lou’s method, Julian may draw on his experience to ensure that
all students are given the opportunity to deritualise the method rather than adapting the result
of one student’s deritualisation.
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5.3 Sven’s lesson onproving geometric statements
Sven is a Swedish teacher in upper secondary school. In the 60-min lesson discussed here,
the class was working on proving statements about geometrical figures using theorems
such as the exterior angle theorem.3 After a short recap of the previous lesson, the class
worked on three tasks. The episode used here covers the first of these tasks and can be
summarised and coded as in Table6.
The task for this part of the lesson was to show that x + y + z = 360° where x,
y and z are exterior angles of a triangle (Fig.3). The class worked under Sven’s
leadership. He initiated the task by displaying it on the whiteboard and gave stu-
dents time to read and think. A student said she saw that the three angles together
would make a circle, but she did not know how to prove it. Sven marked the inte-
rior angles with w, u and v and gave the students additional time to work. One
student said that u + v + w = 180, another that v + u = x.
Sven confirmed the use of the exterior angle theorem and showed the students
how to use the theorem on all three exterior angles but dismissed additional questions
on why x = v + u. After indicating an addition of the three expressions, he gave the
students time to think. No suggestions emerged so Sven proceeded to add the three
expressions, resulting in the equality x + y + z = u + v + w + u + v + w. There was some
joint reasoning about what was to be proved before Sven circled each of the two sums
u + v + w. Sven closed the task by saying, “180 + 180 = 360, so, x + y + z = 360°.”
The initiation of the task indicates that Sven wanted the solving of the task to be an
exploration, or at least a generation of the substantiation behind the given narrative. He
gave students a lot of time to think. However, this episode changed to being strongly led
by the teacher, who made decisions and demonstrated what to do through asking students
narrower questions along the way. The questions concerned choosing from alternative pro-
cedures and making some decisions, but Sven provided nearly all the key clues and so
reduced, or even took away, students’ agentivity. Sven, noticing that the statement was now
proven, closed the task.
Through asking more closed questions, Sven determined the direction of
the substantiation. In doing so, he did not legitimise the choice of the exterior
angle theorem, nor the procedure of adding expressions—these suggestions were
offered as hints, encouraging the students to construct the remaining pieces of
the argument. Whilst the coding indicates an oscillation between open and closed
questions, there are continued invitations for students’ substantiation inter-
spersed with hints. For this reason, we have characterised the lesson as moving
from an exploration-requiring to a guided OTL. Still, Sven’s lesson illustrates
the difficulty in characterising the OTL in a lesson with fluctuating degrees of
deritualisation.
As with the previous lessons, this adaptation of an exploration-requiring OTL into a
hybrid form with stronger teacher guidance is meaningful. Very simply put, it serves no
purpose to require an exploration from the students if they are either not willing or not able
to execute it.
3 The exterior angle theorem states that the measure of an exterior angle of a triangle equals the sum of the
opposite and non-adjacent interior angles.
64 I. M. Christiansen et al.
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1 3
Table 6 Part of Sven’s lesson
Time Focus Openings for stu-
dent agentivity
Encouraged substantiations
10:10–
13:22
The teacher introduces the task:
Exterior angles are marked
outside of a triangle. “Show that
x + y + z = 360°”
Open questions
and time for
students to read
and think
Encourages students to make their own substantiation
13:22–
14:31
The teacher marks the interior
angles u, v and wOpen questions
and time for stu-
dents to work
Encourages students to make their own substantiation
14:31–
18:00
A student says that v + u = x. The
teacher confirms her use of the
exterior angle theorem. The
teacher asks what could be said
about the other exterior angles
and gives the students time
to think. He writes students’
answers on the whiteboard:
x = v + u, y = w + v, z = w + u
Time for students
to think
Support the student’s substantiation and encourage
students to use it again on the other angles
18:00–
22:04
The teacher indicates adding the
three expressions. One student
follows; others are silent.
One student says, “I don’t
understand anything.” The first
student tries to explain how the
substitution is done. A student
asks, “how do you know that
x = v + u?” The teacher tells
the student that this will not be
revisited
Closed questions.
Dismisses a
student’s ques-
tion on previous
knowledge
Encourages the students to explain the current deriva-
tion to each other
22:04–
25:00
The teacher tries to move on, ask-
ing, “if we accept these, what
should we do now?” A student
says that x + y + z = 360°. The
teacher answers with some
reasoning together with the
students about what must be
proved. He then adds the three
expressions, resulting in the
equality x + y + z = u + v + w +
u + v + w, and circles each of
the two sums u + v + w. After
some reasoning together with
the students, he closes the task
by saying that “180 + 180 = 360,
so, x + y + z = 360°”
Open and closed
questions, but
less time for stu-
dents to think
Encourages students to come up with the proof, but
provides some parts of the reasoning
25:00–
29:40
The teacher comments on the
students’ good questions about
what is to be shown and to be
used. He asks if there are any
questions, but after explaining
the first part of the proof again,
he dismisses the questions
Open and closed
questions
Encourages students to look back into their substan-
tiation
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6 Interpretation anddiscussion
In this section, we provide our interpretation of what we have learned about the hybrids of
ritual-requiring and exploration-enabling OTLs spanned by degree of invitation of student
agentivity and source of substantiations, as well as about the characteristics of the adapta-
tions of OTLs throughout a lesson. This constitutes a return to our research questions.
In our engagement with Nachlieli and Tabach’s (2019) concepts of ritual-enabling and
exploration-requiring OTLs, we applied operationalisation of deritualisation to further the
analysis of lesson transcripts which were not clearly one or the other. For the purpose of
this paper, this resulted in a two-dimensional model—a cross section of the space of pos-
sible variations of deritualisation moves—which generated the additional hypothetical
notions of recreated-exploration-requiring and guided-exploration-enabling OTLs.
6.1 Adaptations ofexploration‑requiring OTLs—introducing two hybrid forms
The analysis of the three lessons indicated that the proposed hybrid OTLs were non-empty
categories, since we saw different manifestations of both. Figure4 illustrates the hybrids
manifesting in the three lessons.
Guided OTLs are characterised by the students having opportunities to engage in sub-
stantiating but with less agentivity. Two manifestations of such OTLs were presented here.
There were commonalities and differences between Julian’s and Sven’s cases. The common
aspects of the lessons were that the students were first introduced to a task and had a lot of
time on their own to work on the task or to think about a way to solve it. However, the two
teachers’ guidance differed. In Julian’s case, the guidance came after the students’ inde-
pendent work, in a part of the lesson in which students were invited to share their work.
The first student shared her procedure and Julian’s questions scaffolded her substantiation.
We can even hypothesise that the purpose of such focus was to use the student’s procedure
and substantiation as a teaching strategy, drawing attention to a particular key mathemati-
cal idea (e.g., Asami-Johansson etal., 2020; Ceron, 2019; Kazemi & Hintz, 2014). How-
ever, the second learner’s strategy introduces a different key idea, which appears not to be
what Julian wanted to focus on. In Sven’s case, the guidance (questions and hints) came
during the students’ work on the task and scaffolded the construction of a proof. The ques-
tions at the beginning of the lesson sequence invited student agentivity but this changed.
However, the content of the task was to generate a substantiation, and the students were
Fig. 3 The triangle with exterior (left) and both exterior and interior angles marked (right)
66 I. M. Christiansen et al.
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1 3
encouraged to develop their own. In both cases, we interpret the teachers’ actions as deritu-
alisation-encouraging moves. By “narrowing” the options, the teachers make it possible for
students to engage in some degree of deritualisation but with scaffolding. This is a very dif-
ferent situation from the teacher restricting both students’ agentivity and the development
of their own substantiations.
Recreated OTLs are characterised by students’ agentivity but with limited oppor-
tunity to produce their own substantiations. This might sound paradoxical, but rather
it highlights the constitutive role the teacher plays in students generating routines.
Indeed, we observed teachers inviting students’ participation, but giving little value
to alternative procedures, hence eliminating the need for substantiations. Common
and different aspects were observed in Julian and the US3 lesson. In the US3 lesson,
the students were given a procedure to mimic and generalise; however, the teacher
presented task 1.1 as a rule to develop. By working this way, the teacher passed some
agentivity onto the students, but without inviting them to substantiate their work. In
Julian’s case, the first procedure and its substantiation came from one student. When
another student was encouraged to present her work, her procedure and its substan-
tiation were not valued by the teacher. Thereby, the teacher implicitly informed the
other students that what is expected is the first procedure and the substantiation that
goes with it. Whilst an important focus is put on substantiation, the culture of the
classroom that is constituted by such actions instead gives students the message that
they must replicate what has been done and substantiate the way previous claims had
been justified. However, it is plausible that it is exactly this insistence on developing
familiarity with a method and its substantiation—which in itself could be considered
a ritualisation—that makes deritualisation accessible for the majority of the students.
Fig. 4 The theoretical space with the progression of the three lessons indicated by coloured arrows
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The way to a desired practice is often not through engaging in the full practice from
the first time it is encountered. It suggests, with Warshauer (2015), that there is a
continuum of reasonable teacher responses to situations, and that lessons may fruit-
fully move back and forth between ritualisation and deritualisation, between more or
less cognitive demand.
6.2 Hybrid opportunities tolearn
Each case demonstrates adaptations of OTLs during a lesson (Fig.4), but what underlies
them?
Unlike the two lessons from our own data, lesson US3 adheres to the plan indicated
by the teacher’s introduction and the distributed worksheet. Through tight guidance, the
teacher may not engage the learners in such classroom activity as has been suggested to
increase cognitive demand (Amador & Carter, 2018; Hofmann & Mercer, 2016; Olawoyin
etal., 2021), but yet appears to ensure that almost the entire class comes to accept the pro-
posed narrative and find it justified. Furthermore, not just one but three narratives—rules
for exponents—are produced over the course of one lesson. In addition, a meta-routine
of exploring examples as a stepping stone for proposing generalisations has been demon-
strated (something which the current analysis does not capture). A curriculum goal has
been achieved, and perhaps no child left behind. As also argued by Nachlieli and Tabach
(2019), the use of a ritual-enabling OTL at the beginning of the lesson is what enables
the students to complete the worksheet tasks swiftly and reach the intended generalisa-
tion almost effortlessly. In our view, it is too tightly controlled by the teacher to qualify as
an exploration-requiring OTL, but at the same time it is the requirement to use an exist-
ing procedure which likely enables students to make the generalisation. This illustrates the
point made in relation to previous research, namely that reducing the cognitive load or
working with closed tasks still offers substantial and relevant opportunities to learn (see
also Lavie etal., 2019).
Sven is confronted with students who likely find the task too difficult—as indicated
in their statements about not understanding “anything” or what to do. Sven does not
abandon the goal of facilitating students’ substantiations, but he takes over some of the
agentivity through asking more closed questions. In doing so (whilst allowing students
time to think), he still requires active participation from students and keeps open the
possibility for them to be engaged in exploration or deritualisation. As mentioned by
Sfard (2008, referring to Gee, 1989, p. 7): “one gains access to a discourse ‘through
scaffolded and supported interaction with people who have already mastered the Dis-
course’” (p. 282).
Finally, Julian is confronted with a mathematically elegant solution which builds
on a previous solution. However, as the succeeding interaction with students around
the introduced subtask indicates (see Table5, time slot 27:20–32:05), it is reasonable
for Julian to assume that many students in the class may struggle to follow Steph-
anie’s argumentation. Through closed questions, he tries to direct Stephanie towards
the more elaborate and generalisable but mathematically less sophisticated solution,
and when that fails, he restricts the acceptable substantiation to the one already dem-
onstrated. In this way, he consolidates the introduced procedure whilst strengthening
the connection between fractions of the same whole. We argue that it is this seem-
ingly restricted space for students’ engagement which lays the foundation for future
fraction work for the majority of students, again showing the relevance of varied
68 I. M. Christiansen et al.
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1 3
cognitive demand. According to commognitive analyses, students’ persistent partici-
pation in mathematical talk, even when this kind of communication is for them but a
discourse-for-others, seems to be an inevitable stage in learning mathematics (e.g.,
Lavie etal., 2019; Sfard, 2008).
All teachers are part of complex systems, with objectives for lessons and long-term out-
comes, with limitations and opportunities. Teachers must follow the curriculum (e.g., Gal-
lagher etal., 2020). The most common way to do so is to treat lessons as parts of a whole,
where some content closure must be reached in each lesson. Teachers are responsible not
just for the students’ acquisition of a prescribed content, but also for their development as
human beings, individuals and citizens. They work in classrooms with a diversity of stu-
dents, which requires them to balance considerations towards the classroom community
and individual students simultaneously. Within such a view, the moves made by the teach-
ers are driven by the teachers’ commitment to many masters and their attempt to balance
the tensions this commitment generates. Whilst teachers have agentivity and can provide
substantiations for their choices, this is always framed by the activity system in which the
decision is made. The teacher’s activity may well be the didactically and circumstantially
best practice.
7 Conclusion
This article was driven by the desire to better understand how and why intended mathemat-
ics explorations changed towards rituals during a lesson. To do so, we offered a way to
address the interplay between ritual and exploration from a teacher’s perspective. Many
researchers perceive ritual and exploration routines as two extremes. To theorise an elabo-
ration of the space in-between these, we combined the OTL concepts from Nachlieli and
Tabach (2019) and deritualisation from Lavie etal. (2019). By summoning two aspects
of deritualisation—students’ agentivity and substantiation—we were able to propose two
hybrid OTLs: recreated-exploration-requiring and guided-exploration-enabling OTLs.
Analysing teachers’ discourse in the interaction with students allowed us to identify actual
forms of these hybrid OTLs and illustrate lesson adaptations.
The guided and recreated OTLs offer learners opportunities to learn even if tasks are
closed and/or cognitive demand reduced. They are ways of being inclusive—not all stu-
dents can make it through a full exploration in the time-limited lessons and large, diverse
classrooms, and these hybrids may generate opportunities for gradual deritualisation. To
understand constructive ways of using the hybrid OTLs, more work on teachers’ choices
around these is needed. This may suggest ways of bringing such approaches to work in
teacher education as a means to reduce the oft-experienced theory–practice gap. In future
work, we explore how a novice teacher struggles with balancing the different obligations
of her teaching when she tries to expand her teaching repertoire towards more exploration-
enabling learning opportunities (Christiansen & Corriveau, forthcoming).
Acknowledgements Thank you to Julian and Sven for allowing us access to their classrooms and thoughts,
and to our colleagues for comments on earlier drafts of this paper.
Funding Open access funding provided by Stockholm University. This study is part of the TRACE project
supported by the Swedish National Research Foundation, under project/grant number 017–03614. It also
utilises data from the Canadian MathéRéaliser project supported by the Social Sciences and Humanities
Research Council of Canada, under the grant number 430–2016-00332.
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