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

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.


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 mathematics as a discourse with a particular terminology and visuals, accepted statementsreferred 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 implication 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 applicability 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 adaptations counter to their initial aim, we undertook a further analysis to enable a description of different 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 deritualisation from Lavie et al. (2019). The latter concept enables us to identify changes in opportunities to learn between ritual and explorative routines. Thereafter, we explicate the methodological 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.

Earlier work
The focus of this article is to inform the unpacking of situations when teachers' invitations 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 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.

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 mathematical 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 motivation, 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" (Schoen & 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 consider teaching moves, from the perspective of which openings they generate for students. 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 students 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. Others 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 mathematical 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, passim) 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 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 enables 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?

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 studentsunclear tasks or tasks with unclear expectations, tasks not appropriate for students' previous knowledge, students' lack of motivation or willingness or students not held accountable for their work (Henningsen & Stein, 1997;Warshauer, 2015).
Teachers' adaptations during a lesson may result from considering curricular goals, students' 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 responsive teacher actions: helping students connect to the key mathematical concepts, changing inequity in social structures, engaging students discursively, providing feedback, choosing 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 opportunities to learn, as well as on the aspects of deritualisation (Lavie et al., 2019). It is to these theoretical conceptualisations we now turn.

Theoretical framework
In this section, we briefly summarise Nachlieli and Tabach's (2019) notions of ritualenabling 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. 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 explorationrequiring 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 activity may be a ritual to one student and an exploration to another, the use of "enabling" indicates 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 Who determines the end conditions (to indicate the task has ended)?

Ritual-enabling and exploration-requiring OTLs
The teacher The student (based on mathematical reasoning) ritual-and exploration-enabling, and that the nature of an OTL varies with time, in relation 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-enabling 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 students' 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 explorationrequiring 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 deritualisation generated in the teaching.

Opportunities to engage in deritualisation
Deritualisation is a gradual change from a ritual to an exploration when a student's routine 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 perform 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 observations 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 substantiability. This constitutes a two-dimensional plane within the generated space. This process allows us to point to different relations and hybrid spaces alongside explorationrequiring and ritual-enabling OTLs-but we do not claim to fill the space.
Increased agentivity implies the student being able to make more decisions independently of external authorities (Heyd-Metzuyanim et al., 2022;Lavie et al., 2019). Substantiability increases when the student is increasingly able to assess their own performance 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 operationalisation of the concepts.
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 situations-as in formulations about what is expected from students regarding proceduresthan 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 opportunity 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 narrative 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 according 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 encouragedand given opportunity-to generate. Figure 1 illustrates these dimensions and the suggested 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 therefore 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 agentivity is closely guiding the exploration; therefore, we named it a guided-explorationenabling 1 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 arguments during group work), but to value a specific way to address these open questions, recreating modelled substantiations, hence recreated-exploration-requiring OTL, which we refer to as recreated OTL for short.

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 encouraged 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?

Methodology
In this section, we briefly describe the cases and the reasons for their selection, followed by a presentation of our methods of analysis.

The cases and their selection
To obtain a range of lessons to analyse, we took one high school lesson from existing materials collected in the Swedish TRACE project, which focuses on the teaching of novice  Fig. 1 Dimensions in the plane intersecting the OTL space, with ritual-enabling OTL and explorationrequiring OTL at opposite extremes 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 adaptations took place. As argued in Sect. 5.1, even the US3 lessons had elements of such adaptations. 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.

Method of analysis
Our analysis took the above operationalisation of invited student agentivity and encouraged 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 transcription were translated into English to enable a joint interpretation.

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.

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 demonstrated procedure two-thirds of the time.

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.

Analysis of the lessons
This section has three parts. In each part, we discuss the ways in which the respective lesson 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.

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 exploration and ritual routines. They summarised the first 20 min of the lesson as an explorationrequiring 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 general 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 exploration-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 agentivity and substantibility. Table 4 shows the stages of the first 15 min and 40 s of the lesson.  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 generating 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 demonstrated procedure is used to justify the addition of exponents, and because in the commognitive 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 characterised this part of the lesson as a recreated-exploration-requiring OTL. This characterisation 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, demonstrating 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.

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 fraction 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 fraction. 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 episodes and their coding are summarised in Table 5.  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 substantiations 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 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 characteristics 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 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 questions 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 recreated 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 follows 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 mathematically 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.

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 students 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 interior 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 procedures 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 interspersed 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. 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 question on previous knowledge Encourages the students to explain the current derivation to each other 22:04-25:00 The teacher tries to move on, asking, "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 students 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 substantiation

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 adaptations 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 possible variations of deritualisation moves-which generated the additional hypothetical notions of recreated-exploration-requiring and guided-exploration-enabling OTLs.

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 substantiating 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' independent 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 mathematical idea (e.g., Asami-Johansson et al., 2020;Ceron, 2019;Kazemi & Hintz, 2014). However, 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 questions 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 encouraged to develop their own. In both cases, we interpret the teachers' actions as deritualisation-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 different 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 opportunity 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 substantiation 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. 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 fruitfully move back and forth between ritualisation and deritualisation, between more or less cognitive demand.

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 proposed 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 demonstrated (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 generalisation 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 existing 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: "one gains access to a discourse 'through scaffolded and supported interaction with people who have already mastered the Discourse'" (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 Stephanie'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 demonstrated. In this way, he consolidates the introduced procedure whilst strengthening the connection between fractions of the same whole. We argue that it is this seemingly restricted space for students' engagement which lays the foundation for future fraction work for the majority of students, again showing the relevance of varied cognitive demand. According to commognitive analyses, students' persistent participation 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 outcomes, with limitations and opportunities. Teachers must follow the curriculum (e.g., Gallagher 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 students, which requires them to balance considerations towards the classroom community and individual students simultaneously. Within such a view, the moves made by the teachers 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.

Conclusion
This article was driven by the desire to better understand how and why intended mathematics 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 elaboration 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 students 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 explorationenabling learning opportunities (Christiansen & Corriveau, forthcoming).
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