Mathematically productive relationships (MPRs): a potentially effective classroom management approach

Positive teacher–student relationships are considered essential for effective classroom management. Thus, focusing efforts to understand better and strengthen teacher–student relationships in order to improve student outcome metrics is prudent. This study examined seven elementary teachers’ descriptions and enactment of mathematically productive relationships (MPRs), a type of teacher-student relationship, and their influence on classroom management practices. Teachers’ descriptions of MPRs incorporated four themes: (a) creating safe, mistake-friendly environments, (b) developing trust and respect between the teacher and the students, (c) establishing and maintaining clear and high expectations, and (d) ensuring that classrooms remain safe spaces. However, these features did not always appear in the teachers’ classrooms. The enactment of MPRs varied across cases, with the participants’ classroom management priorities lying along a continuum from behavioural management to cognitive engagement. Implications of MPRs and the influence of different classroom management practices on students’ learning opportunities are discussed.

Although strong TSRs are a vital component for establishing prosocial, emotional, and behavioural classrooms across disciplines (Jennings & Greenberg, 2009), in mathematics classrooms, the nature of TSRs is particularly important because they influence the type of opportunities in which students must engage and the quality of that engagement. Particularly, mathematics educators and researchers (e.g., El-ahwal & Shahin, 2020;Warshauer, 2015) emphasize the role of productive struggle as essential for effective mathematics problem-solving. The emotional and behavioural engagement that characterizes productive struggle necessitates that teachers establish caring relations with students during mathematical activity in ways that advance mathematical thinking (Hackenberg, 2010). Without the trust and openness engendered through strong TSRs, mathematical caring relations could not be developed, thereby impeding mathematical learning. As such, we consider the type of TSRs needed to promote robust mathematical engagement unique. We refer to these TSRs as mathematically productive relationships (MPRs) (Cross Francis et al., 2019). In this study, we explore how the nature and enactment of MPRs influence elementary mathematics teachers' classroom management.

Building mathematically productive relationships (MPRs)
Teacher-student relationships describe an interpersonal interaction between the teacher and student reflected through both proximal and distal perspectives (McCormick et al., 2013;Pianta, 1999). The proximal perspective captures the interaction at an individual level, and the distal perspective focuses on portraying the environment or culture that reflects TSRs. Studies have revealed that positive proximal TSRs tend to encompass a high level of warmth, trust, and caring (Baker et al., 2008;Pianta, 1999;Roorda et al., 2011). Thus, positive proximal TSRs are an essential aspect of the infrastructure needed to create a supportive distal system for students to develop a strong sense of emotional security (Akdağ & Haser, 2016;Baker et al., 2008), a more satisfying student-student relationship (Hughes & Kwok, 2006), and a stronger connection with school (Birch & Ladd, 1997;Furrer & Skinner, 2003). When TSRs are healthy and positive, students are more willing to engage in learning activities (Kincade et al., 2020;McCormick et al., 2013), such as solving challenging tasks by persevering, taking more mathematical risks, and actively seeking and receiving support (Murray & Greenberg, 2000). Combined, these attributes of TSRs ultimately contribute to higher academic achievement for students (Zhou et al., 2020a(Zhou et al., , 2020bCrosnoe et al., 2004;Fan, 2012;Hattie, 2009;Kincade et al., 2020).
Drawing on the work of Boynton & Boynton (2005) and others (e.g., Davis, 2003;Howes & Ritchie, 2002;Pianta, 1999), the following four actions are considered essential in promoting strong TSRs: (a) communicating positive expectations, (b) developing pride in classroom communities, (c) demonstrating caring, and (d) preventing and reducing teachers' frustration and stress. However, enacting these practices in the mathematics classroom might not support the kind of TSRs critical for developing strong mathematical thinkerswhat we refer to as mathematically productive relationships (MPRs) (Cross Francis et al., 2019). For example, Boynton & Boynton (2005) suggest communicating positive expectations by calling on students equitably as a critical aspect of promoting positive TSRs. However, mathematically effective practices require being selective about student responses showcased for discussion (Smith & Stein, 1998), which might not unfold in ways that are thought to be equitable.
Additionally, mathematics teachers must see struggles as an essential part of the mathematics learning process (Star, 2015) and consistently support student engagement in productive struggle (NCTM, 2014). Unlike other subjects, meaningful mathematical activity is characterized by problem solving, with students being introduced to increasingly challenging and new problems which they have not previously encountered. In this regard, students continuously face cognitively demanding tasks to solve, engendering struggle, and requiring perseverance. Thus, mathematical caring can unfold and look different from general caring (Hackenberg, 2010).
These differences inspired us to conceptualize MPRs as a specific type of TSRs for mathematics classrooms, defined as "healthy teacher-student interactions that foster mathematical caring, and an emotionally safe environment that encourages intellectual vulnerability, problem-solving and academic risk-taking" (Cross Francis et al., 2019, p. 537). As such, common practices for improving TSRs, such as child-led activities, reflective and supportive listening, collaborative problem-solving, peer-assisted learning strategies, and self-regulation (Kincade et al., 2020), are very likely to promote MPRs.

Effective mathematical classroom management
Classroom management includes "the actions teachers take to create an environment that supports and facilitates both academic and social-emotional learning" (Evertson & Weinstein, 2006, p. 4). For teachers, effective classroom management is thought to be the ability to create and sustain an orderly (in a behavioural manner) and productive (in an academic manner) classroom (Evertson & Weinstein, 2006;Henley, 2010;Kwok, 2017;McDonald, 2010;Wong & Wong, 2014;Wubbels, 2011). Kohn (2006) and McCaslin and Good (1998) indicated, in the past and sometimes still alive in today's classrooms, classroom management has a greater emphasis on discipline and controlling students to meet teachers' mechanistic and authoritarian expectations (e.g., Atici, 2007;Bowers & Flinders, 1990;Koehler et al., 2013). This focus on behaviour control has proven to be less effective in reducing off-task, disengaged behavior, and achieving classroom management goals than efforts to stimulate and engage students cognitively. Several studies (e.g., Maguire et al., 2010;Sullivan et al., 2014) have shown that addressing disruptive classroom behaviours tend to be within the teacher's control and is less related to student factors.
Researchers who have focused on developing guiding principles and best practices to support the development of appropriate and effective classroom management behaviours have concluded that "teachers who approach classroom management as a process of establishing and maintaining effective learning environments tend to be more successful than teachers who place more emphasis on their roles as authority figures or disciplinarians" (Brophy, 1988, p. 1). Teachers need to know more about what they want to teach and they should also be able to organize their classrooms and interactions with students in ways that embody a healthy, caring classroom culture where all can thrive (Egeberg et al., 2021).
Earlier conceptualizations of classroom management which focus on student control do not align closely with the current standards for mathematical practice (NCTM, 2014), which require students to actively participate in the development of mathematics through problem-solving, reasoning, arguing, and critiquing. The current standards for mathematical practice (NCTM, 2014), primarily influenced by constructivism (Liu & Chen, 2010;Von Glasersfeld, 1990), positioned students as active mathematical sense makers and knowledge constructors. Accordingly, implementing these practices of teaching mathematics calls for creating a learning climate in which students' thinking, participation, and engagement are foregrounded as central to classroom organization and functioning (Gay, 2013).
Many scholars have advocated constructivist learning environments and labeled them with different terms (Baeten et al., 2016). For example, Struyven et al. (2010) and Loyens & Rikers (2011) used the term student-centered; Alfieri et al. (2011) employed the term discovery-based; Loyens & Rikers (2011) also proposed the term inquiry-based; and Schweder & Raufelder (2021) used the term self-directed to refer to the learning environments inspired by constructivism. However, centering students' thinking (Steffe & Thompson, 2000) is fundamental across these various framings.
Theorists (e.g., Curwin et al., 2018;Glasser, 1999;Kohn, 2006;Kounin, 1970) suggest that focusing more intently on heightening students' cognitive engagement to enhance learning would be more effective in preventing students' misbehaviours (Chandra, 2015). For example, Kohn (2006) argued that "the constructivist critique, which says that a rightanswer focus doesn't help children become good thinkers, also suggests that a right-behaviour focus doesn't help children become good people" (p. xv). His core argument is that classroom management should serve children's needs. Teachers should manage classroom dynamics so that students feel academically empowered, the meaning of misbehaviour is reinterpreted, and students' curiosity is allowed to motivate their learning.
This view aligns with the theoretical assumption of inquiry-based mathematical learning that students are learning agents who should have opportunities to explore mathematics and engage in it in ways that are empowering and stimulate curiosity. Therefore, we argue that effective management in mathematics classrooms will enable students to engage meaningfully in problem-solving, thereby supporting students in regulating behaviour while actively participating in the discourse (Kohn, 2006;Kounin, 1970;Williams, 2008).
According to Glasser's (1999) choice theory, students' behaviour in class is driven by their five basic needs: survival, belonging, freedom, power, and fun. Thus, one of the teacher's essential duties in managing a classroom effectively is to love and care for students (Glasser, 1999), support them to make the right behavioural choices (Glasser, 2001), and make them feel accepted and respected in classrooms (Frey & Wilhite, 2005). The components of strong MPRs are associated with more effective classroom management because they instill pride in the classroom community and belongingness to school (Boynton & Boynton, 2005;Grobler & Wessels, 2020), exhibit mathematical caring (Hackenberg, 2010), trust (Dong et al., 2021;Platz, 2021), and positive expectations (Cross Francis et al., 2019;Boynton & Boynton, 2005;Glasser, 1999;Roorda et al., 2011). By extension, this creates a safe environment that fosters engagement in productive struggle (Warshauer, 2015), openness to making mistakes (Hiebert et al., 1997;Liu et al., in press), problemsolving, and mathematical discussions (Boaler, 2016;Fang, 2010;Smith & Stein, 2018).
Because teachers' beliefs and perceptions tend to influence their actions (Cross Francis et al., 2009Egeberg et al., 2016;Kwok, 2017), teachers' beliefs about MPRs are likely to affect their enactment of MPRs and, by extension, their classroom management (Baker et al., 2008;Kwok, 2017). Given these relationships among constructs, we aimed to explore the following three research questions (

Participants
Participants included seven elementary teachers from three school districts in the midwestern United States (see their demographic information in Table 1). They were involved in a two-year professional development (PD) program that focused on expanding their mathematical knowledge for teaching  (MKT) through a series of professional development activities, including coaching. Each teacher participated in five coaching cycles with the PD team. We coached the participants during five lessons each by providing support before and during the coached lesson. The coaching model is called Holistic Individualized Coaching (HIC) (see Cross Francis et al., 2022 for a full description).

Holistic individualized coaching (HIC)
HIC is designed to integrate knowledge about teachers' MKT, instructional quality, beliefs, identity, and emotions with the goal of providing teachers with comprehensive support as they plan and enact instruction (Cross Francis et al., 2022). The HIC model was designed Pull-out 3-5 Note: *Lia partnered with another fifth-grade teacher to have a flexible classroom structure that varied daily and from lesson to lesson. This ranged from Lia teaching mathematics to all students in a combined format to pulling out students to work in a small group for extra support while the other teacher taught the main lesson Table 1 Demographic data on participants based on the recognition that teachers are learners in the context of professional development; thus, their psychological and affective needs should be given ample attention, in ways similar to their knowledge and pedagogical needs are. HIC involves six steps (see Fig. 1): (i) a multi-dimensional teacher profile that includes information about the teacher's instructional quality (MQI), MKT, beliefs, identity, and emotions; (ii) a pre-coaching conversation about the lesson to be coached; (iii) a content-specific mini teacher profile that includes information about the teacher's MQI, MKT, beliefs, identity, emotions specific to the content of the lesson drawn from the pre-coaching conversation; (iv) pre-lesson support (if needed); (v) coached lesson, which is video recorded; and (vi), the post-coaching conversation that focuses on discussions around video clips from the coached lesson selected by both the teacher and the coach. Each participant engaged in five cycles of HIC. All conversations between the coach and the teacher were audio recorded. The coach did not play an active instructional role during the coached lesson but was present during instruction to video record and provide support mainly if the teacher solicited it. We described this coaching model to indicate that teachers were actively engaged in an MPR with a coach during the study. Although exploration of the ways in which this relationship influences teachers' perceptions of MPRs would be valuable, it was beyond the scope of this study.

Data sources
This study included two data sources -an interview and video recordings of the coached lessons. The semi-structured interviews took place at the end of the year. They were hourlong and focused on asking about teachers' perceptions of positive MPRs and productive struggle. The video recordings of instruction were drawn from five coaching cycles.

Data analysis
To answer RQ1, interviews were transcribed verbatim. All four research team members read one transcript thoroughly and open coded it (Strauss & Corbin, 1990) by applying emergent codes that reflected participants' thoughts about TSRs within the mathematical context. For example, the statement "OK, when I go to Mr. Andrew's class for math, he is going to allow me to make mistakes" was coded "safe environment" as well as "allows mistakes". Then we met to discuss, define the codes generated, and revise those that did not fully capture par- ticipants' meanings. All coders worked collaboratively to modify the codes and definitions until agreements were reached. In some instances, we used in vivo coding because the participants' words reflected the meaning best. This process was iterative across two transcripts to establish a code list. Once the code list was established, members of the research team coded the remaining transcripts independently using the list. After each coder had coded the assigned transcripts, we collectively reviewed the coded transcripts to verify that the codes were applied appropriately. Then, we closely examined the data aligned with the codes to generate themes whose meaning summarized the codes (Charmaz, 2014). We revised the themes until we collectively agreed that they reflected the participants' perspectives about productive MPRs. For example, codes such as "allows mistakes" and "safe environment" were summarized to generate "create a safe, mistake-friendly environment".
To answer RQ2 and RQ3, we scored all the video recordings using the Mathematical Quality of Instruction (MQI) instrument (https://cepr.harvard.edu/mqi) (Hill et al., 2008;LMT, 2006). We identified the dimensions from the MQI instrument related to MPRs and classroom management and used these dimensions as indicators of MPRs that existed in the classroom. Specifically, in alignment with our framing of effective classroom management described in the conceptual framework, we focused on the attributes within the Whole Lesson codes dimension of the MQI. This dimension included 10-whole lesson attributes (attributes for the whole lesson rather than a small proportion of it). The degree to which the attribute was present in the teacher's instruction was scored on a scale from one to five. One indicated that the attribute was not present in the instruction and five indicated that the attribute was very present. From these 10 attributes, we selected three, namely, Tasks and Activities Develop Mathematics, Teachers Use Student Ideas and Teacher Attends to and Remediates Student Difficulty, to determine the nature of teachers' classroom management. We selected two student-related attributes of Students are Engaged and Lesson Contains Common Core Aligned Student Practices to indicate effective classroom management. The scores assigned to these attributes from the Whole Lesson Codes dimension were used to examine how MPRs were enacted and influenced classroom management.
As we defined MPRs as "healthy teacher-student interactions that foster mathematical caring, and an emotionally safe environment that encourages intellectual vulnerability, problem-solving and academic risk-taking", enacting MPRs required teachers: to select meaningful tasks and activities that challenged and developed students' mathematical understandings (assessed by attribute Tasks and Activities Develop Mathematics); to show mathematical caring by valuing and using students' ideas (assessed by attribute Teachers Use Student Ideas); and to encourage intellectual vulnerability and academic risk-taking by treating mistakes as learning opportunities (assessed by attribute Teacher Attends to and Remediates Student Difficulty). As a result, a well-managed classroom can be observed from students' engagement and classroom activities. Thus, we selected the attributes of Students are Engaged, and Lesson Contains Common Core Aligned Student Practices (focused on examining if students engaged in mathematical questioning, reasoning, and communication) to indicate the level of classroom management.
All members of the research team completed MQI training. Two members of the research team scored each participant's video. Both scorers independently watched the participants' instructional videos and used the MQI rubric to score eight-minute segments of the videos and the whole lesson. Disagreements about assigned scores were discussed using the MQI scoring guidelines as a reference until consensus was reached. Scores on all relevant items across dimensions for all participants were then compiled in a table. Finally, we calculated the average MQI scores to indicate teachers' mathematical classroom management (see Table 2). We used teachers' average scores because, first, we were interested in evaluating each teacher's general performance on these attributes and, second, teachers' MQI scores on these attributes varied little across five coaching cycles, which led us to use the average score to represent teachers' general performances on these dimensions.
The following sections describe the themes that emerged from all examined cases concerning their perceptions of MPRs (answers to RQ1) and the average MQI scores (mainly used to answer to RQ2 and RQ3). Based on the MQI scores, we identified the three cases of Lia, Anderson, and Bryan using a diverse case selection method (Seawright & Gerring, 2008) to illustrate the variation of how the teachers' perceptions are reflected in their instruction and how the enacted MPRs impacted their classroom management. Specifically, we selected Lia, a teacher who valued and enacted MPRs and managed her classroom in an orderly and productive manner; Anderson, a teacher who perceived MPRs as important but rarely enacted them and managed his classrooms in an orderly but not effective manner; and, finally, Bryan, a teacher who valued and enacted MPRs, managed his classroom in a productive but not orderly manner. For each case, we looked across their interviews, videos of classroom teaching, and MQI scores to answer RQ2 and RQ3 and narrated our findings.

Findings
Four common themes about the core features of MPRs were uncovered: (a) creating safe, mistake-friendly environments, (b) developing trust and respect between the teacher and student, (c) establishing and maintaining clear and high expectations, and (d) maintaining consistency.

Creating a safe environment
Several teachers noted the importance of building a safe environment where students would not be embarrassed to make mistakes. It is essential to know students holistically and meaningfully, understand how they think and reason, and support them in the mathematics classroom. Andrew stated: What I think is needed to promote high-quality learning in the classroom …where the students know, OK, when I go to Mr. Andrew's class for math, he is going to allow me to make mistakes. He is going to allow me to discuss my answers.
This environment needs to include a culture that promotes high-quality learning where students feel comfortable sharing ideas, asking questions, listening to others, and attempting multiple approaches to solving problems without experiencing ridicule. Another teacher, Willa, also stated: Easing the kids' minds that it's OK to make mistakes. And we talk about that a lotwhatever it is, whether it's behaviour, whether it's math. But, yeah, it's OK to make mistakes and using those mistakes as learning platforms and that everybody is different. And not everybody's going to be in the same place at the same time. And so, it's not about comparing yourself to anybody else. But it is seeing how much you've learned, and how much you've grown, and just trying to make them feel comfortable with themselves and with me-that it's all right. You know? And I tell them all the time-I make all kinds of mistakes, and I'm not expecting perfection from them.
What Willa was trying to emphasize is the value of having multiple brains work in the same classroom. Because everybody is different, it is not realistic for all of them to be always on the same level. By mentioning that she also made mistakes, it is more trustworthy for the children to recognize and accept making mistakes as a natural part of the learning process.

Trust and respect
Building trust in a mathematically productive teacher-student relationship was also emphasized as fundamental to establishing and maintaining MPRs. Lia stated: I think it's that trust between the student and the teacher. To be able to get down to the nitty-gritty, they're going to have to trust you that you're leading them where they need to be taken. And so, I think that the whole trust and the openness. It is going to be parallel to the quantity or the quality. They need to have that relationship and be invested in it.
Lia had mentioned trust in her explanation of building a positive teacher-student relationship, and she saw that as an essential component of MPRs when prompted. She used the word trust to capture a kind of trust that students would have in a leader, in this case, their teacher, and it would lead them to be open. Like Lia, Bryan also highlighted trust as an essential factor for a teacher-student relationship that works well for high-quality mathematics learning: I guess a relationship where the student feels that their teacher trusts them, and they trust their teacher to listen to what they're thinking and set up an environment where it's OK to take risks, and it's OK to get frustrated. And that you know that no matter what happens, the teacher will most likely react in the same way as they did the day before.
Bryan used trust as a mutual component that both the teacher and students have towards each other. So, it is not only gaining students' trust but also making sure that they know that the teacher trusts them and would not change from day to day when things change around the classroom. This also has some indications for consistency, which is the last theme reported in this section.

High expectations
Establishing and maintaining clear and high expectations was also considered crucial in supporting MPRs by our teachers. For example, Janine said: Establishing that safe environment, establishing a personal relationship, and establishing a high relationship where they know their boundaries. They know their expectations. And they know that they are high expectations and that they know that they will be challenged. And they don't-if they choose to play around or not follow the rules of the classroom as far as group work is concerned, as far as getting along is concerned, that there are consequences. And it is-there are consequences for everyone. It doesn't matter who they are.
According to Janine, students would know what they are expected in their mathematics classroom and also know these are high expectations. Having that mindset would bring responsibility to deal with the consequences of their actions, which could sound a little tense, but it also establishes a learning environment where everyone is equal. Both Janine and Andrew were vocal in ensuring that students understood the specifics of the high expectations clearly and that the teachers were there to help them achieve the goals.

Maintaining consistency
Teachers emphasized the importance of maintaining a safe classroom environment. However, this differed from the theme of creating safe, mistake-friendly environments as the focus was on consistency. This meant that each of the aforementioned attributes needed to be applied consistently in the classroom. Bryan, in particular, had strong ideas about this: And I think consistency is really important for kids, especially for kids who don't have this consistency at home, that they know that when you say something that you're going to follow through. So, if you say that you're going to look something up, if they ask you a question, you say you're going to look it up, you actually remember to do that. Or if you say that we're going to work on this today --yeah, just paying attention to what you say and making sure that you follow through with it.
Bryan's idea about being consistent was especially in support of kids who did not have consistency at home. So, it was not only about building a strong MPR but also building a relationship that would support children in other parts of their lives. Table 2 shows that the average scores on selected MQI codes varied both among the features of the MQI dimensions and the teachers. For example, looking across the scores, Lia received the highest scores for all dimensions. Her score was 4.2 for the Lesson Contains Common Core Aligned Student Practices dimension and 4.6 for the Students are Engaged dimension. These scores suggest that Lia could engage her students effectively, resulting in a managed classroom. On the other hand, Andrew gained the lowest scores on the same dimensions (2 and 2.2), which indicates that Andrew was less able to engage his students meaningfully. We selected the highlighted cases in Table 2 to show the range of connections between MPRs and effective classroom management: Lia got the highest MQI scores among the seven cases (managed the classroom orderly and productively); Anderson got the lowest MQI scores (managed the classroom orderly but productively); and Bryan got varied MQI scores (managed the classroom productively but orderly). We describe these cases below.

The case of Lia
Lia's case focuses on an elementary-school teacher who expressed a deep understanding of MPRs and enacted her vision during instruction. As her MQI scores indicate, she managed her classroom in an orderly and productive way -she got the highest scores on all selected dimensions. Related to her understanding of MPRs, she highlighted that one should create a safe and interactive environment for students to experience productive struggle. For instance, she mentioned: When you create that culture where it is safe, and it is comfortable, and they can ask the questions they need to. They can give answers without fearing somebody making fun of them; then they are going to ultimately open their minds a lot more and be willing to work through things…it is not going to maintain those things as strongly as it would if you were struggling to absolutely make total, full sense of it. I think that struggle is absolutely necessary for anybody to get anything in with them. They have to be able to work things. If you are learning things, but you are not struggling through it, your brain's not really working as hard, and it is not going to maintain those things as strongly as it would if you were struggling to absolutely make total, full sense of it.
Her statements indicate that she strongly believes in the importance and necessity of creating a safe and mistake-friendly classroom culture so that the students experience productive struggle. This is important to her because she believes that students learn better and create meaning only if they struggle along the process.
Lia did not only state these principles, but we also observed them in her practice, which is evident from the scores of her MQI analysis. Table 2 shows Lia got the highest score on the dimensions Teacher Uses Student Ideas (4.6) and Teacher Attends to and Remediates Student Difficulty (4.2). This indicates that Lia was able to use students' ideas to develop the mathematics lesson and was attentive to students' difficulties. Of note was that she was effective in remediating these difficulties. Additionally, she was able to respond to the students' misconceptions productively. To accomplish such a task, she had to listen to the students carefully, interpret their comments, and then respond in a manner that would either support the students' contributions or correct their mathematical errors.
For example, in a particular teaching episode, Lia clarified a point about equal partitioning when students had difficulty understanding why the parts had to be equivalent. This occurred during the enactment of a certain activity when students discussed the following question: "How do you know that the green triangle is one-half of the blue rhombus?" using pattern blocks. One of the students responded by stating that two triangles can fit on one rhombus. The teacher then asked students what they noticed about the two triangles fitting on the rhombus. One of the students stated: "If you remove one of the triangles, then you would have one left. Therefore, the triangle that is left on the rhombus represents one-half of the rhombus." Another student observed that the two triangles were equal. The teacher then asked the following question: "Is it necessary for fractions to be equal?" More than one of the students responded by saying "No". The teacher then drew two shapes (one was equally divided into four parts, and the other one was unequally divided into four parts). After that, the teacher then shaded one piece for each of the shapes. She then asked the students if each of the shaded parts represented one-fourth. Initially, some students said "Yes" while some said "No". The teacher then asked the following question: "If I remove the shaded part from both shapes, could we then say that three-fourths is left for both shapes?" A student responded by stating that it would not be three-fourths for the shape at the bottom (one that had been divided unequally). He also justified his answer by saying that, because four parts for that shape were not equally divided, that part did not represent one-fourth.
This example captures Lia's typical interaction with the students by showing the ways in which her perspectives on MPRs were reflected in her practices. First, we observed the case in which Lia was able to engage students in conversations around an error made in completing the task. Lia did not state that there was a mistake; instead, she engaged the students in discussions through questioning that encouraged them to rethink their responses. Students were engaged, thinking through the questions that she posed and responding with justification, and it was reflected by the high scores on dimensions of Students are Engaged (4.6) and Lesson Contains Common Core Aligned Student Practices (4.2). This interaction shows that Lia had established a "culture where it is safe, and it is comfortable" and where students were able to "ask the questions they need to" and "give answers without fearing somebody making fun of them." As she asked questions and probed, we observed students "open their minds … [and were] willing to work through things". Lia set high expectations for the level of thinking she expected by engaging students in mathematically-productive work and conversation, as indicated by the MQI scores and excerpts from the video-recorded lesson. She engendered a safe and supportive classroom culture and interacted with students in trusting ways that supported them in fulfilling those expectations.

The case of Andrew
Andrew's case shows an elementary teacher who understood the importance of MPRs, but it was not reflected in his teaching practice. While he maintained physical order in the classroom, he struggled to manage the class effectively through mathematical activity. Similar to Lia, Andrew also believed that creating safe, mistake-friendly environments was necessary for effective math instruction. He described such MPRs: What I think is needed to promote high-quality learning in the classroom …where the students know, OK, when I go to Andrew's class for math, he is going to allow me to make mistakes. He is going to allow me to discuss my answers.
According to Andrew, to enable high-quality learning, students should feel safe to make mistakes and have opportunities to discuss their mathematical thinking. However, his comments did not align well with his teaching practice. Analyzing his teaching videos across five coaching cycles, we found that he tended to use procedure-oriented tasks, and his instruction was mainly based on lecturing students. He usually attended to students' mathematical mistakes but did not address them for deep sense making (he scored 3 points on the dimension Teacher Attends to and Remediates Student Difficulty). He seldom used their ideas or created spaces to express their understanding. This approach was reflected in his low MQI scores on Teacher Uses Student Ideas (MQI score: 2.2). For example, in a teaching episode when the students were asked to convert 44 quarts to gallons, one student mentioned that you should divide 44 by 11. Instead of giving the student ample time to provide a rationale for this particular step in solving the problem or asking another student to assist him in remediating this error, the teacher told the student that the step was not meaningful and then moved on with his instruction. Similar situations frequently occurred in Andrew's classes. By not engaging the student in exploring his error, Andrew communicated that the student's contribution was not valuable, showing that his classroom was not a safe and mistake-friendly environment for students' mathematical learning.
We observed that Andrew was more concerned about disciplining students than implementing meaningful tasks or activities during the coaching cycles. He expected his students to be seated quietly and to listen to him carefully. Despite his ability to identify worthwhile mathematics tasks, Andrew appeared uncomfortable with the more open-ended tasks, which required substantial teacher-student interactions. He stated that he was fearful of "losing control of the class" and preferred to offer students a list of tasks to practise and then check the answers. In this regard, he got the lowest average score (2.8) of the five coaching cycles in the dimension Tasks and Activities Develop Mathematics, which indicated that, although Andrew's maintained control in his classes, the low-level MPR, the lack of discussion, and procedural activities did not allow students to engage with the content meaningfully.
Andrew's students had few opportunities to express their ideas, ask questions, and show their reasoning, which was reflected by a low MQI score of 2.0 in the dimension Lesson Contains Common Core Aligned Student Practices. (As mentioned earlier, this dimension assessed the volume of students' mathematical practice in classrooms.) Correspondingly, as a general outcome, students' engagement in his classroom was also relatively low (a score of 3.6 in the Students Are Engaged dimension). Although most students were seated quietly without any misbehaviour, opportunities for mathematical productivity were low. Andrew advocated a classroom environment that aligns with the features of MPRs, but he organized his classroom in ways that foregrounded physical order and control but minimized students' intellectual engagement. This instructional approach was intentionally stemming from his fear of losing control of the class. In this regard, Andrew's case was an example of a teacher who is struggling with conceptualizing classroom management in ways that primarily serve students' needs and support students in feeling empowered academically. Maintaining order and control while foregrounding students' thinking was a significant challenge, potentially jeopardizing students' opportunities to learn.

The case of Bryan
Different from Andrew's and Lia's cases, Bryan had strong ideas about MPRs. Bryan valued personal trust (e.g., "show a genuine interest in the kids' life") and tended to build academic trust by believing that "they [students] can do things well" as well as "elevating their voice." Bryan's perception of trust is aligned with his perception of teaching mathematics effectively. He valued students' mathematical thinking and challenged students based on their thinking: I interact with students, always put their thinking first, and then with their thinking, I try to create something that pushes them towards a new, higher level of understanding of a topic.
Bryan also had a favorable view of students' struggles in his classroom, stating that "I think a teacher has to have a lot of flexibility in their mind about the different ways that kids are first expressing when they struggle, and that you interpret that as an expression of engagement, rather than misbehaviour". Bryan was not only able to see struggle "as an expression of engagement", but also believed that different students had different ways of engaging by creating an all-ideas-are-valued classroom environment: And then set up an environment where you know that all ideas are valued, which sometimes requires kids who share all the time to listen to someone else. And then kids who don't like to share as much, you might find ways for them to engage as well.
In alignment with Bryan's clear and positive perception of MPR and his attention to building trust and rapport with his students, we also observed that students in Bryan's classroom were highly engaged in the lesson (MQI score: 4.6 on the Students are Engaged dimension). However, while he managed the classroom productively, it would not be considered orderly. Nevertheless, students communicated well about mathematics, provided explanations and reasoning, and asked questions about the task at hand. These were also evident in Bryan's high scores in the dimensions Activities and Tasks Develop Mathematics (MQI score: 4) and Teacher Use Student Ideas (MQI score: 4.4). For example, Bryan invited students to present their solutions on the whiteboard in the fourth coached lesson. Having students share their ideas openly aligned with his statements about building academic trust by believing that "they [students] can do things well" and "elevating their voice". In so doing, he demonstrated that their contributions were worthwhile and that their thinking was valued. He posed another question to expand one student's thinking and interacted with students to "push them towards a new, higher level of understanding of a topic". He created space for students to struggle and solve problems independently.
On the other hand, while he managed the classroom productively, it would not be considered orderly. For instance, different from Lia's classroom, where students followed formal classroom norms, in Bryan's classroom, we could see that students in the videos selected their seats based on their preference and assumed a range of postures. Students were also free to move around the classroom at will. They spoke freely and often loudly and used the whiteboard frequently based on their needs. Bryan did not often object to or reprimand students about these behaviours, reflecting Bryan's idea that different students have different ways to engage in the learning process. Aligned with more current notions of classroom management (e.g., Kohn, 2006), Bryan tried to interpret students' expressions of struggle to determine what they needed. Also, Bryan's students had emotional and behavioural challenges. Thus, to foreground their thinking, he might have considered it necessary to balance both physical and intellectual freedom.

Discussion and implications
The teachers' descriptions of MPRs aligned with some of the core features of building and sustaining positive teacher-student relationships (e.g., Boynton & Boynton, 2005;Davis, 2003;Kincade, 2020) and the critical elements of productive classrooms described in the literature (e.g., Hiebert et al., 1997). Notably, teachers discussed the need to create safe, judgment-free spaces for learning-centered relationships grounded in trust and respect. Although the three teachers selected stated that an emotionally-safe, risk-supporting environment is essential for building positive MPRs, Lia and Bryan integrated this perspective into their teaching practice, while Andrew did not. By examining teachers' views about MPRs in detail, we observed that although their statements aligned with the features of MPRs, the depth and rationale underlying them were different. For example, Andrew was able to point out his expectation of creating a mistake-friendly culture, but he did not articulate the genesis or rationale for this thinking. The analysis of his teaching indicated that he did not enact these ideas; instead, his classroom did not welcome mistakes or emerging conceptions. On the other hand, the other two participants, especially Bryan, showed a deep understanding of why creating a safe environment is essential and how to build such a culture in their daily practice. Our analysis showed that both Bryan and Lia tried to develop such a culture in their classrooms so that students would feel free to make mistakes and engage in productive struggles in the process of learning.
We also observed that, in classes where students adhered to traditional classroom norms, this did not necessarily mean the teacher exercises mathematically-effective classroom management skills (e.g., Andrew's case). At the same time, it is possible to manage a mathematics classroom productively without having a traditional physical "orderly" classroom (e.g., Bryan's case). For example, in a video, we observed a student providing insightful contributions to developing the solution to a problem while sitting on top of a trash can. These situations might be more likely to happen in a classroom that has a small number of students (Bryan's classes ranged between 5 and 7 students). Yet, we still question the rationale of the common belief that is held by most teachers and administrators implicitly -all students should sit at their desks quietly and follow their teachers' commands for learning to take place. Therefore, these three cases indicate that orderly and physically-controlled classrooms (Evertson & Weinstein, 2006) might not always create mathematically-productive classrooms. Lia's case is an example of a classroom with high-quality instruction with students primarily adhering to traditional classroom behavioural norms; however, Anderson's and Bryan's cases demonstrate alternative ways in which lessons might unfold. Both cases show realistic situations between behavioural management and intellectual engagement as an approach to classroom management. Further examination of the nature of teachers' beliefs about classroom management and how they influence the ways in which they manage their classrooms provide meaningful insight for determining the features of effective classroom management in mathematics classrooms (e.g., Kwok, 2017).
Our findings have several implications for instruction and organizing learning environments. First, rather than considering "order and control" as preconditions of effective classroom management, a well-managed classroom involves optimizing students' cognitive and emotional engagement (Henley, 2010;Korpershoek et al., 2016;McDonald, 2010), empowering students academically and focusing on interpreting students' struggles in ways that allow attending to their needs (Kohn, 2006). This approach better facilitates positive student outcomes, as we observed in Lia's and Bryan's classrooms. When translated into the mathematics classroom, this involves providing students opportunities to share their thinking and hear the thoughts of others, with the teachers listening to students and using their ideas to drive instruction (e.g., In Lia's case and Bryan's case). This aligns with the work of Chandra (2015) who argued: "If lessons are precise and completed at a steady and continuous pace, students will have little time to misbehave or get into conflicts" (p.13). This suggests that, if teachers want to engage and minimize disruptive behaviour, they should focus on implementing teaching strategies to foster a classroom environment where students are involved throughout the lesson. These perspectives align with that of mathematics educators (Hiebert et al., 1997;NCTM, 2014;NRC, 2001) and educational researchers more broadly (e.g., Baeten et al., 2016;Loyens & Rikers, 2011;Schweder & Raufelder, 2021;Struyven et al., 2010), who advocate student-centered instructional approaches which prioritize students' thinking, voice, and empowerment.
Second, we observed that believing that MPRs are essential for high-quality instruction and effective classroom management is necessary but not sufficient for these principles to be enacted in the classroom, which aligns with Kwok's (2017) findings. As has been documented elsewhere (Cross Francis et al., 2009, Egeberg et al., 2016, we observed that there could be contrasting beliefs that become foregrounded in the classroom context that impede the enactment of MPRs. For example, Bryan foregrounded control and order over creating a classroom accepting of mistakes and open discussion. In addition to advocating the development of student-centered beliefs about classroom management -thus shifting conceptualization of classroom management from emphasizing different disciplinary mechanisms (Kohn, 2006) to supporting order -teachers might need support in implementing instructional strategies (e.g., design activities and tasks to develop mathematics and use students' ideas Liu & Jacobson, 2022) that promote students' meaningful mathematical engagement (e.g., offer students classroom participation opportunities, creating a safe environment). Both Lia's and Bryan's cases support that this could be more effective for classroom management (Emmer & Sabornie, 2014). For some teachers, like Bryan, both shifting beliefs and instructional support could be prudent.
Such shifts might not be easy because they require that teachers, administrators, and students reconsider what classroom management entails and what it means to learn and be actively engaged in the learning process. This is particularly important for administrators who provide support for and/or evaluate teachers. Bryan feared losing control of his class, which could reflect the school culture or what is highly valued in his school community. It is therefore essential for administrators and those who provide support for teachers to understand the role that MPRs play in students' mathematical engagement, hold student-centered perspectives on classroom management, and be able to identify and scaffold relevant instructional practices for teachers. Backgrounding a behaviour control perspective and foregrounding one focused on cognitive engagement demand teacher educators and administrators ardently work to continue to support teachers in teaching boldly and ambitiously.
Additionally, teachers need time to make such shifts. Reflecting on our interactions with teachers during the five coaching cycles, we created a space for teachers to experience MPRs, and we were able to support teachers in strengthening MPRs to some extent. Participating in HIC encouraged teachers to focus attention, increased their ability to notice (cf., Jacobs et al., 2010) students' thinking, and created opportunities to effectively utilize students' ideas and mistakes (Cross Francis et al., 2022). However, observed improvements in MPRs were gradual and sometimes unstable, which posed difficulty for measurement across the five coaching cycles. Notably, the effect of coaching on advancing MPRs seemed to be more effective for those teachers who held more student-centered beliefs (e.g., Bryan's case). As has been widely discussed in the literature (e.g., Pajares, 1992), attending to teachers' beliefs, in this case, beliefs about teacher-student relationships and classroom management, could be the critical first step to creating safe and empowering learning environments for students.