Keywords

1 Introduction

The focus of this chapter is on interactive mathematics teacher activities (Type C). That is, the activities in which teachers engage in the presence of students. Researchers are often interested in innovative pedagogies aimed at enhancing the teaching and learning of mathematics. Studies, therefore, typically investigate classrooms in which teachers are participating in an intervention aimed at influencing their practice in ways deemed desirable by researchers or are attempting some kind of atypical practice that aligns with contemporary views of effective mathematics teaching. Fewer studies consider the nature of normative mathematics teaching practice. Those that do are necessarily large scale and provide less rich data than is usual for studies of atypical practice. They are, however, important for system level understanding and as a context in which to consider innovative practice.

Manizade, Moore and Beswick (this volume) describe eight epistemological perspectives that formed a framework for examining research on teacher’s activities, such as lesson planning, reflecting on teaching, and assessing students, when not in the presence of students (i.e., Type D variables). Aspects of Type D that occur prior to teaching are intended to inform what happens when teachers interact directly with students (Type C). Nevertheless, we know that many factors intervene to ensure that there is rarely a direct translation from plan to practice by constraining the interactive activities that are feasible or desirable. They include Type I variables such as system and school policies and priorities, resources available to schools and choices about their allocation, resources of the families that schools serve, and cultural considerations; and Type E variables, specifically the skills, knowledge, dispositions, and accumulated experience that individual teachers bring to their task.

Apparent disjunctions between what teacher’s say they believe about teaching (an aspect of Type E), including their epistemological perspective, and hence plan to do, (Type D), and what in fact happens in their classrooms (Type C) gave rise to studies pre-dating the focus of this review that highlighted apparent discrepancies between beliefs and practice (e.g., Frykholm, 1999; Sosniak et al., 1991). An important development in recent decades has been a growing consensus that teachers are reasonable when they state and enact their beliefs (Leatham, 2006) with more than a dozen ways in which apparent discrepancies can be reasonably explained having been documented (Liljedahl, 2008). In addition, Beswick (2003) highlighted the influence of the differing contexts in which teachers typically talk about their beliefs and then enact them. This certainly applies to the contexts in which teachers plan for interacting with students (Type D) and then implement those plans (Type C). Each of the epistemological paradigms identified by Manizade, Moore and Beswick (this volume) allow for a degree of contingency; that is, the teacher needs to respond to the ways in which students respond to teaching. Indeed contingency, defined as involving deviating from the plan, responding to student’s ideas, and making use of unplanned opportunities, is a dimension of the Knowledge Quartet that was developed based on observations of teacher’s practice and presented as a framework for observing mathematics teaching (Rowland & Turner, 2007). Speer (2005) argued that apparent discrepancies between teacher’s espoused and enacted beliefs are likely artefacts of the research methods employed, specifically a failure to consider data from practice as well as teacher reports via surveys or interviews when attempting to infer their beliefs. Care also needs to be taken to ensure that there are shared understandings between teachers and researchers of the meanings of words and interpretations of events (Beswick, 2005; Schoenfeld, 2003) which in turn are influenced by the researcher’s beliefs. In the case of large-scale studies, choices about the scales and items included reflect what the test designers assume to be desirable practices (Eriksson et al., 2019).

Consistent with this, research interest in particular interactive teaching practices has followed developments in theoretical understandings of mathematics teaching and learning and the epistemological perspectives from which practices have been examined. As noted by Manizade, Moore and Beswick (this volume), these are not always explicitly stated but can be inferred with varying degrees of confidence from reports of studies. In this chapter we review what we know about teacher behaviours in typical mathematics classrooms and discuss the range of less widespread pedagogical approaches that are evident in the literature. In both cases we make links to underpinning epistemological perspectives as described by Manizade, Moore and Beswick pointing to how these appear to have both influenced the Type C variables that have been of interest and that may relate to practices observed, although we recognise the difficulties inherent in making such connections. Large scale surveys, for example, necessarily rely on teacher’s self-reports rather than on direct observations. Desimone et al. (2005) cited research showing that although self-reports are acceptably reliable and valid measures of the content taught and the teaching strategies that are emphasised (e.g., Mullens & Gayler, 1999, as cited by Desimone et al. 2005), they may not be well-suited to measuring aspects of practice such as teacher-student interaction (e.g., Mullens & Kaspryzyk, 1999, as cited by Desimone et al., 2005). According to Eriksson et al. (2019), there was almost no connection between teacher’s responses to items on an “instruction to Engage Students in Learning Scale” and student’s achievement, leading them to recommend relying instead upon student reports about what happens in their classrooms. Studies that involve direct observations of teaching provide more certainty about actual classroom events but are necessarily smaller in scale and present their own challenges for researchers who seek to go beyond reporting what teachers and students do and say to make inferences about their intent and motivations.

We also examine what has been found about the impacts of technology on what happens in classrooms (real or virtual) in which teachers and students interact, and the theoretical lenses that informed the work. We conclude with reflections on aspects of classroom practice that have been less or un-scrutinized, but which warrant attention in future studies.

2 Our Approach

We conducted an organically evolving search of research articles discussing teacher’s interactions with students in mathematics classrooms. We began with a search of relevant databases of the high-ranking mathematics education journals identified by Williams and Leatham (2017). The databases searched were ERIC, ProQuest Education, Informit A + Education, OECD Library, EBSCO Education Source and JSTOR. We began with a small number of relevant articles which were searched for relevant keywords (e.g., classroom environments, teacher-student interaction, teacher behaviors). Further researchers engaged in the field were identified and we focused specifically on articles that discussed teacher’s actions in classrooms, and identified the different perspectives, methods, recommendations, and issues raised. We restricted the sample to publications dating from 2000 and conducted further searches by prominent authors in the field and keywords (e.g., mathematics pedagogy, mathematics teaching, classroom practice). We refer to older literature when it is important to framing more recent trends and identifying their progress over slightly longer timeframes.

The matched articles were transferred into an Excel spreadsheet in which they were categorised by title; author; date of publication; type of data (direct observation, indirect, other); type of activity (e.g., problem solving); theoretical approach; and whether it concerned existing practice or practice connected with an intervention. In addition, we found articles more broadly related to the topic, such as when a particular issue, e.g., conceptual understanding of fractions, was examined with an intervention impacting teacher’s actions. Those articles only tangentially related were excluded from the core analyses but were discussed in author meetings thus informing our discussion in this chapter. For normative practices we also referred to reports of large-scale international surveys.

The chapter is structured in two broad sections; the first describing the development of research about widely practiced teacher student interactions, and the second exploring studies that have considered teacher’s behaviors with students in particular projects or in response to specific interventions. For the former, normative practices, we rely on large scale assessments of mathematics teaching and learning whereas for the atypical practices described in specific studies we refer to research reports available in the mathematics education literature.

3 Normative Mathematics Teaching Practices

In this section we survey what is known about what typically happens in mathematics classrooms. We rely primarily on the large-scale international surveys, Trends in International Mathematics and Science Study (TIMSS) that assess mathematics achievement at Years 4 and 8 in participating countries. The first TIMSS was conducted in 1995 but we confine our attention to those in the past two decades, beginning with TIMSS 2003. The Programme for International Student Assessment surveys (PISA) similarly provide insights into the classroom activities that constitute mathematics learning for 15-year-olds in participating countries. We begin with a brief overview of TIMSS and PISA before highlighting changes in the classroom activity that successive iterations of these surveys have revealed.

Country participation in TIMSS has steadily increased over the years reflecting increased interest at government and education system level in the performance of their students relative to those in other countries. In 2003, 46 countries participated from the continents of Africa, Asia, Australia, Europe, North America, Oceania, and South America (Gonzales et al., 2004). By 2019 participation had risen to 64 countries, representing a broad range of geographic, demographic, and economic diversity (Mullis et al., 2020). Although the focus of this book is on Western countries, TIMSS is relevant because it provides an international overview of mathematics education, allowing comparisons among countries and the identification of distinctive characteristics of mathematics teaching and curriculum objectives in particular countries of interest.

PISA, undertaken by the Organisation for Economic Co-operation and Development (OECD) assesses how well 15-year-old students can apply the knowledge and skills they have learned in the areas of reading, mathematics and science to real-life problems and situations. Seventy-nine countries participated in PISA in 2018 (Schleicher, 2019), which was the seventh cycle of the international assessment since the programme was launched in 2000. Each assessment focuses on one of the three subjects and provides a summary assessment of the other two. So, while Mathematics has been assessed by PISA once every 3 years since 2000, the mathematics domain was the main area of focus only in 2003 and 2012. Mathematics will again be the major domain assessed in 2022.Footnote 1

We focus on aspects of TIMSS and PISA that relate most directly to what teachers do, or are able to do, in the presence of students. In terms of constraints on teacher’s activity with students, resources including teacher’s expertise and time for mathematics teaching, are especially salient and hence considered here.

3.1 Resources for Teaching Mathematics

Hopper et al. (2017) explained how the TIMSS Context Questionnaire gathers data about two types of resources that affect the teaching of mathematics. These are Type I variables, beyond the direct control of the teacher but that, nevertheless, provide constraints and affordances for what teachers are able to do in their interactions with students (Type C). The first are general resources such as school infrastructure (e.g., buildings, and grounds, heating and lighting, classroom space), teaching supplies, and the availability of technology. The second resource type is specific to mathematics including such things as particular software, calculators, and instructional materials. Data are also gathered on the difficulty or otherwise of finding well-qualified mathematics teachers, and on the rates of attainment of tertiary discipline and pedagogical study deemed necessary for teaching mathematics that teachers have undertaken. While acknowledged as crude proxy for knowledge for teaching mathematics, the extent to which mathematics teachers have undertaken such studies contributes to Type E variables that inform and constrain Type D and hence Type C activities.

The amount of time that teachers are able to spend with their students constrains the kinds of activities in which they can engage. Lack of time is frequently cited by teachers as an obstacle to implementing innovative practices in their mathematics classrooms (e.g., Livy et al., 2021). There was a significant variation in the amount of mathematics instructional time across the 64 countries surveyed in TIMSS 2019. On average, the fourth-grade students received 154 h of mathematics instruction per year, which equated to approximately 17% of total instructional time. The average number of hours received by eighth grade students was 17 h less than in fourth grade (137 h or 13% of the total) (Mullis et al., 2020). The increase in the amount of mathematics instructional time since 2003 is noteworthy. Although the sample size in TIMSS 2003 was considerably smaller (19 countries at fourth grade and 35 countries at eighth grade) the data point to a smaller time allocation: on average, fourth-grade students in 2003 received 149 h of mathematics instruction per year, which equated to approximately 16% of total instructional time. The average number of hours received by eighth grade students in 2003 was 26 h less than in fourth grade (123 h or 12% of the total) (Mullis et al., 2004). The reduction in hours dedicated to mathematics instruction from Year 4 to Year 8 likely reflects the broader range of subject areas taught at eighth grade (Mullis et al., 2020).

3.2 Instructional Practices

Rožman and Klieme (2017) identified three major international trends in education based on contemporary educational policy and discourse. These were: an increased interest in regular assessment of student progress; greater advocacy of student-centred pedagogies; and promotion of reasoning and problem-solving rather than the development of computational and procedural skills as the goals of mathematics teaching. They investigated four cycles of TIMSS (1995–1999–2003–2007) at eighth grade across 18 countries. Only slight evidence of increased use of testing was found across TIMSS assessments from 1995 to 2007 (Rožman & Klieme, 2017). In relation to the second trend—greater advocacy of student-centred pedagogies – there was some evidence that associated pedagogical approaches, such as making connections between mathematics and student’s daily lives and working in groups had increased in several countries, most particularly in East Asia. In relation to the third trend—the promotion of reasoning and problem-solving rather than the development of computational and procedural skills as the goals of mathematics teaching—contrary to expectations, there was an increased practice of computational skills, with a particular emphasis in Central and Eastern Europe. Despite an initial increase in the frequency of problem solving, there was a decrease from 2003–2007.

In the 2003 and 2007 TIMSS studies, Year 8 students were asked about instructional practices in their classrooms considered relevant to instructional quality (Eriksson et al., 2019). In their discussion Eriksson et al. (2019) focused on three items, namely: (1) we listen to the teacher give lecture-style presentations, (2) we relate what we are learning in mathematics to our daily lives and, (3) we memorise formulas and procedures. As Eriksson et al. (2019) pointed out there is no consensus as to the optimal frequency with which any of these practices should occur. The frequency of lecturing, for example, that might be considered beneficial depends upon what the teacher is aiming to achieve, that is their goals for teaching. As explained by Manizade, Moore and Beswick, a teacher adopting a behaviorist perspective is likely to be concerned with helping students to perform flawlessly the steps of a procedure to obtain correct answers to a class of mathematical problems. In this case telling students clearly the steps that need to be followed is likely to be effective. In contrast, from other perspectives such as social constructivism, where the goals of teaching relate to the quality of interactions among students and building subjective knowledge, much less frequent use of lecture style presentations would be deemed desirable.

TIMSS 2015 data indicated positive associations between instructional clarity and student achievement (Hooper et al., 2017) as did TIMSS 2019 (Mullis et al., 2020) which used updated scales to further explore this trend. Students at fourth grade in 2019 reported clearer instruction than did students in eighth grade: Most students in fourth grade (95%) reported moderate to high clarity of instruction compared with only 46% of students in eighth grade.

TIMSS 2019, like TIMSS surveys since 1995, collected data on instructional practices and strategies. For mathematics these concerned how often students; worked on problems on their own, explained their answers in class, and decided on their own strategies for solving problems (Hooper et al., 2017). Just as the theoretical perspectives that teachers bring to their work influence the goals they have for their teaching (Manizade, Moore & Beswick) and hence the instructional practices that they are likely to adopt, the choice of items included in TIMSS studies reflect the theoretical perspectives, and their concomitant goals and practices, that are of interest to the test designers, influenced by theoretical developments and recent research on approaches to teaching mathematics. The three items listed from TIMSS 2019 suggest interest in the extent to which problem solving and reasoning, and collaborative or individual working, are fostered in mathematics classrooms. These are consistent with problem solving and social constructivist perspectives on mathematics teaching. Researchers have, across successive iterations of TIMSS, explored associations between particular instructional practices and mathematics achievement. As Eriksson et al. (2019) pointed out the results of these studies do not always support theoretical assumptions about what constitutes instructional quality. They suggest that instructional practices should only be considered characteristic of quality teaching if they are found empirically to support student achievement.

TIMSS video studies were conducted in 1995 and 1999. The 1995 study involved a total of 231 mathematics lessons in the United States (81 lessons), Germany (100 lessons), and Japan (50 lessons), while in the 1999 a total of 638 mathematics lessons were video recorded across the seven participating countries: Australia, Czech Republic, Hong Kong SAR, Japan, the Netherlands, Switzerland, and the US (Neubrand, 2006). Video studies offer an opportunity for teachers (and student) behaviors to be studied repeatedly from different theoretical standpoints, and to address different questions about what is happening in those classrooms. Researchers have been interested in such things as how teachers structure their lessons, the clarity of instruction, interruptions, and how homework is treated. For example, Neubrand (2006) re-analysed 22 lessons from each of the three participating countries in the 1995 study to explore the number and types of tasks that teachers offered their students in the three countries. The 1999 lessons have also been examined in terms of lesson structure, mathematical content, and instructional practices, and to discern differences in mathematics classroom activity in different countries. Hiebert et al., (2003) observed that while there were some similar features in the relatively higher achieving countries, there were also distinct differences. For example, eighth-grade lessons in all participating countries included both whole-class work and individual/small group work. However, lessons in Australia, the Netherlands and Switzerland allocated more time, on average, to students working individually or in small groups. Another finding of note was that across all of the participating countries, at least 80% of lesson time in eighth grade, on average, was dedicated to solving mathematics problems. But there was considerable variation in respect to drawing the relationships between mathematics problems and real-life situations ranging from only 9% of problems per lesson in Japan to 42% of problems per lesson in the Netherlands. Regarding computers, relatively few eighth-grade lessons in the participating countries made use of them. However, 91% of eighth-grade lessons in the Netherlands used calculators; a percentage much higher than in the other countries which ranged from 31 to 56% of lessons (except in Japan where no reliable estimate could be reported due to their infrequent use). In summary, Hiebert et al., reported that ‘no single method of teaching eighth-grade mathematics was observed in all the relatively higher achieving countries participating in this study’ (2003, p. 15).

Eligible mathematics teachers and students in a representative sample of 150 PISA participating schools in eight countries (Australia, Finland, Latvia, Mexico, Portugal, Romania, Singapore, and Spain) responded to the OECD’s Teaching and Learning International Surveys (TALIS) on classroom practice (OECD, 2017). Teachers and students were asked to rate teacher’s use of eight classroom practices. These practices were clustered according to three broad teaching strategies: structuring practices, student-oriented practices, and enhanced learning activities. Structuring practices entailed the explicit specification of learning goals; student practice until all students have understood the content; and a summary presentation by the teacher of recently learned subject matter. Student-oriented practices were the differentiation of the work for students with learning difficulties or the ability to progress more quickly than their peers, and groupwork that allows students to devise a collective solution to a problem or task. Enhanced learning activities comprised students undertaking projects of at least one week’s duration, an expectation that students explain their thinking, and encouragement to seek multiple ways to solve problems (OECD, 2017).

Both teachers and students reported that almost all mathematics teachers across participating countries used clear and structured teaching practices; specifically, explicitly stating learning goals; allowing students to practice until they understand the content; and providing summaries of recently learned content. The teacher’s use of enhanced learning activities was also commonly reported by both teachers and students, suggesting strong encouragement of students to solve problems in more than one way, and a high expectation that students explain their thinking on complex problems. The use of project work lasting at least one week was less frequent. While used less often than the other two practices, most teachers and over half of students confirmed the use of student-oriented practices, i.e., giving different work to students according to their level of understanding, or the use of small groups for students to come up with joint solutions.

Structuring practices were the most frequently used teaching practices in mathematics classrooms, according to both teachers and students. According to the authors, “Since they (structured practices) aim to deliver an orderly and clear lesson, they could be seen as the necessary foundation for the development of any other practice. This would explain why they are so predominant in the teaching strategies implemented by teachers” (OECD, 2017, p. 7). Nevertheless, “classroom instruction time is a scarce resource, and an overemphasis on structuring practices could limit teachers in their use of other potentially more innovative strategies, such as enhanced learning activities and student-oriented strategies” (OECD, 2017, p. 7).

3.3 Teacher’s Use of Technology

The growing presence of digital learning technologies has brought new opportunities and challenges for mathematics teachers. An array of mobile devices, application software and other online technologies have transformed the landscape of mathematics classrooms providing myriad pedagogical opportunities, notably in relation to problem-solving, experimentation and collaboration. Based on the most recent TALIS Vincent-Lancrin et al. (2019) noted that changes in the use of ICT in mathematics lessons has been a major driver of pedagogical innovation in mathematics classrooms, along with professional development of mathematics teachers through peer learning. However, the challenge for teachers to equip themselves with the requisite skills to effectively use new technologies and engage in higher-order pedagogical tasks is significant. An observation made by Handal and Campbell et al. in 2012 still has currency a decade on:

In the case of online tools, there is a vast range of technologies available, but do teachers feel that they know how to find them and use them once located? A range of dynamic geometry software (e.g., Geometer’s Sketchpad) and computer algebra software is available. These tools have a steep learning curve and teachers need to be able to model these technologies for students for use in the classroom. (2012, p. 394)

A corollary of a digitally-rich classroom is a shift in the role of the teacher and hence what they do in their direct interactions with students. This is particularly discernible in the context of the ‘flipped classroom’ where instructive videos typically replace ‘traditional’ homework tasks to allow more focused teaching in class time (Muir, 2020). In such circumstances where recorded teaching is made available to students to engage with in their own time, the teacher and each student are effectively interacting, albeit in a uni-directional way, asynchronously. Medley (1987) did not envisage interactions of this kind, but they have become increasingly common as technology has evolved and as circumstances have demanded the use of distance learning. Teacher behaviors as they engage in virtual asynchronous teaching are an aspect of Type C that warrants research. The content that is presented and whether or not it is presented in a way that elicits student-centered interactions depends on the theoretical perspective adopted by the teacher.

TIMSS 2019 investigated three areas relating to the use of technology: computer access for instruction; technology to support learning; and tests delivered on digital devices. Teachers were asked about availability of computers during mathematics lessons and the types of access i.e., whether each student has a computer, the classroom has shared computers, and/or the school sometimes gives access to computers. Teachers reported similar levels of access to computers at fourth and eighth grades (39% and 37% respectively), but there was variation in the level of access to computers across countries as well as in the types of access. The type of access most frequently reported for both fourth and eighth grades was that the school has computers that the class can sometimes use (29% and 28% respectively). Average student achievement was associated with access to computers at both grades, not surprisingly given that access to computers would be related to socio-economic advantage (Mullis et al., 2020).

TIMMS 2019 also investigated the frequency with which teachers used computer activities to support learning in mathematics. Around two-thirds of students in both fourth and eighth grades were in classes in which their teacher reported that they “never or almost never” do computer activities to support learning (67% and 68% respectively). Average student achievement was lowest for students in these classes, with a 15 point-average difference at fourth level and an 18 point-average difference at eighth level (Mullis et al., 2020). The way in which teachers administer tests, and specifically whether they use computers or tablets for this purpose was also examined with eight grade students reporting the lowest occurrence of digitally delivered tests having the highest achievement.

4 Atypical Mathematics Teaching Practices

In this section, we consider studies that have addressed practices that have been less common in mathematics classrooms. We discuss the topics that have attracted researcher’s attention when it comes to teacher’s efforts to implement non-traditional practices and discuss aspects that have been most influential in shaping teacher’s activity in mathematics classrooms in the last two decades.

Since 2000, smaller scale studies have emphasized the examination of pedagogical approaches based on constructivism, with many studies having involved examining the implementation and impact of particular practices. Teacher-student interactions have sometimes been observed directly, but artifacts such as teacher’s lesson plans (Type D), have also been reviewed, and teacher actions inferred from them. Artifacts of this kind provide indirect insight into what teachers do in their classrooms but need to be interpreted carefully because of their indirectness. There are, for example, many reasons for which a lesson may not be implemented as planned. Small scale studies have focused on broad pedagogical approaches or perspectives (e.g., project-based learning, culturally responsive teaching), aspects of teacher’s practices (e.g., questioning, types of listening), the organization of teaching and learning (e.g., flipped classrooms), and classroom environments. In the sections that follow we describe findings from these studies according to themes identified from the foci of the studies.

4.1 Pedagogical Approaches

Boaler (e.g., Boaler, 2001) has made extensive contributions to research on teacher’s use of student-centered approaches complemented by practical work providing resources (underpinned perhaps by a social constructivist, cognitive learning theory, or structuralist perspective (Manizade, Moore, & Beswick)) and instructions for teachers to inform their classroom activity. In Boaler’s work, the concept of rights of the learners, that include such things as the right to be heard, make mistakes and be confused, requiring a degree of sensitivity from teacher’s side (Kalinec-Craig, 2017) features as something that should guide teacher’s interactions with students. What one considers to be the rights of a learner depends in part upon teacher’s perspective on mathematics teaching and hence what the goal of teaching is. From a situated learning theory or social constructivist perspective it would be quite natural to allow students to voice their thinking whereas a teacher approaching their task from a behaviorist perspective might see this as detracting from the effectiveness of teaching aimed at the perfect performance of procedures. From this perspective, affording students rights necessarily constrains the actions available to and appropriate for teachers as they interact with students.

Fewer studies have considered how the student-centered approaches proposed are understood by teachers, or how they are translated in classrooms. Silver et al. (2009) analysed portfolio entries submitted by teachers. In the entries, teachers proposed lesson plans with pedagogical features to support the development of students’ understanding. They found that teachers were not able to systematically embed innovative pedagogical approaches in their best practice submissions. While this study shed light on the degree of teacher’s adaptation to some atypical practices, the study did not address the question of how each innovative, student-centred approach was understood by teachers; that is how the teachers defined and hence might enact the atypical practices they were proposing in their entries.

The research literature suggests that student-centered interactions and teacher’s role in those interactions have been thoroughly researched and are well understood. Nevertheless, large-scale studies such as TIMSS and PISA suggest these approaches are not widely used. Reasons for the limited spread of student-centered approaches has been the subject of considerable speculation. For example, Buschman (2004) pointed to a “blame game”, described as teachers commonly arguing that good activities don’t exist and ‘blaming’ the supply of activities, as an explanation and canvassed many of the features of the debate about the uptake of atypical practices in which researchers in the field, have participated. These include: generic definitions of the approach in question (problem-solving as a loose term that refers to enhanced understanding, student centeredness and shifting the teaching from drilling to supporting genuine ideas); the realization that such practices have not been fully entertained by teachers, implying that the suggested practices would work as expected should the teachers only learn the way to acquire what is suggested to them; and providing informed, but not thoroughly evidence-based speculations about the situation.

Approaches that were innovative but not student-centered were hard to find in the body of research conducted in the last two decades, suggesting that perspectives that underpin teaching with features that could be characterised as student centred (e.g., social constructivism, structuralism, problem solving, culturally relevant pedagogy, and project and problem-based learning), are the lenses through which researchers have envisioned effective mathematics teaching. We struggled to find studies that examined innovative teacher-centered approaches and did not find studies taking a fresh perspective on behaviorist approaches.

There is, however, a body of research on cognitive load theory (Paas et al., 2004; Sweller, 2011), that has investigated teaching practices and techniques that reduce unnecessary load on students’ working memory. Such an approach is, if not teacher-centred, at least teacher-led, and often considered as an opposing approach to student-led problem solving, inquiry-based learning or ‘discovery learning’ (Paas, 2004, p. 6), although the intention of the approach is not to avoid mental challenges, but to question the external interruptions that may appear in student-centred, inquiry-based or collaborative problem-solving settings. Best practices to reduce (unnecessary) cognitive load have been developed and delivered through laboratory studies, as well as within training programs for teachers (Van Merrienboer & Sweller, 2005). One can find comparative studies testing the effects of reduced cognitive load on student’s learning (e.g., on geometry in Reis et al., 2012; the use of spreadsheets and sequencing in Clarke et al., 2005) but how teachers have applied those practices in their mathematics classrooms and the extent to which laboratory-based findings can be reproduced in classroom contexts seems less known.

The pedagogical approaches discussed above have their roots in ideas presented in earlier decades. For example, “a quasi-empirical” approach to mathematics teaching was proposed by Lerman (1990). In that approach, teachers were encouraged to take mathematical misconceptions as hypotheses (as a source of something productive) and investigate the conditions under which they might or might not work (and why). Similarly, Ball and colleagues (e.g., Ball & Bass, 2000) have contributed to the general understanding of student-centered, constructivist pedagogies. Schoenfeld (e.g., 1992) has been influential in elaborating and building understanding of problem-solving as a means of teaching mathematics. Influential elaborations such as these have likely contributed to student-centred, inquiry-based approaches becoming dominant in the small-scale intervention studies. Comparison of these studies with typical mathematics teaching practices discerned from large scale studies, along with studies that suggest many teachers may have deeply ingrained views aligned with a behaviorist perspective on teaching (Schoenfeld, 2018) offers an explanation for the limited traction that student centred teaching has achieved. Not only do theoretical perspectives constrain the behaviors of teachers in their interactions with students, but they also constrain the kinds of questions researchers ask, the way studies are designed, and the questions that remain unanswered.

In the next section we discuss approaches in mathematics classrooms, namely, the practices in, and organisation of, the environment of a mathematics classroom.

4.2 Aspects of Teacher’s Practices

Burkhardt (2006) reviewed the benefits and the spread of teaching modelling in the mathematics classrooms, concluding that the approach is only moderately used despite the opportunities it affords for student learning. Boaler (2001) described research in which modeling was a practice that had made a difference in student’s learning in an investigation contrasting mathematics teaching in two schools. She concluded that teachers needed to change their practices to allow students to develop transferable problem-solving skills.

The concept of robust understanding was introduced by Schoenfeld et al. (2020) along with a framework for teaching in ways that support the development of student’s robust understanding of mathematics (Schoenfeld, 2018). He described activities derived from three teacher’s lessons and analysed them in terms of the framework. The three teachers differed in the aspects of the framework that they emphasized. Each was able to address some aspects but struggled in others. In general, teachers seemed to struggle to shift from pedagogies that develop procedural knowledge to facilitating more connected understanding, and to build on student’s thinking, making sure everyone had access to opportunities to develop their agency (Schoenfeld, 2018). Similarly, Buschman (2004), noted that teachers often miss opportunities to build on student’s ideas, and speculated that there is a need for more examples of the desired practice, more collaboration among teachers, and greater acceptance of making mistakes while adapting to new practices.

Others such as Conner et al. (2014) have examined ways in which teachers can support argumentation, while Handal and Bobis (2004) considered thematically structured teaching. Sullivan et al. (2003) investigated context-based teaching and Shahrill (2013), conducted a review of teacher’s questioning, focusing on what makes questioning effective, rather than on what teachers are actually doing in relation to questioning.

A particular practice, “instructing between the desks” was investigated as part of the cross-cultural Lexicon project by Clarke and colleagues (e.g., Dong et al., 2015). In this project, aspects of teachers’ practices were labelled in order to provide a vocabulary to make it easier for researchers and teachers to address the various aspects of teachers’ conscious and unconscious actions in mathematics classrooms. Clarke and his team were able to identify significant cultural differences in the ways in which teachers facilitate students’ learning. For example, instructing between the desks seems more casually and less systematically applied in many Western countries, but rigorously practiced as “Kikan-shido” in some cultures (O’Keefe et al., 2006). Linguistic aspects of mathematics teaching have also been addressed by Sfard (2021), who elaborated on the role of language in the mathematics learning process.

4.3 The Organization of Teaching and Learning

Flipped classrooms have attracted considerable attention from mathematics education researchers during the last two decades. The enactment of a flipped classroom relies on technology, as the learner needs to acquire some of the content through digital resources independently. The need for independence on the part of the learner has been suggested to require self-determination (Deci & Ryan, 2012) from the learner’s side and being well informed of appropriate resources from the teacher’s side (Muir, 2020; Muir & Geiger, 2016). Muir (2020) observed a teacher implementing a flipped classroom approach and concluded that with careful preparation, the teacher was able to support all aspects of her students’ self-determination (competence, autonomy, and relatedness), while also helping students to develop their conceptual and procedural knowledge.

In addition to the well-studied flipped classroom approach, we found several case studies of community engagement. Many of these studies were reported in conference proceedings, but there were also a few such cases documented in journal articles. For example, Leonard and Evans (2008) described an intervention in which teachers worked closely with local churches in urban settings to adapt practices from community building. The aim was to address social justice and improve cultural responsiveness. Leonard’s and Evan’s (ibid.) study serves as an alternative example of what teachers (with or without a research-connection) could engage with in order to widen their perceptions of what is possible to support mathematics learning, as well as to better meet the needs of their students as individuals with varying backgrounds.

4.4 Classroom Environments

Research studies are typically based on researchers’ initiatives inspired by their beliefs about what constitutes good mathematics teaching. Teachers may adopt the new practice during an intervention, but reports of what happened before and after these interventions are rare.

Some researchers have made extensive efforts in creating resources to help teachers apply recommended ideas independently of participation in a project. Liljedahl (2019), for example, has suggested tangible changes in the classroom environment. His concept of “thinking classrooms” includes the use of vertical surfaces as a mean to support student argumentation. Working in small groups and documenting the mathematical work on vertical boards that everyone can see has attracted attention (as evidenced in teacher groups in social media) but is hard to find evidence of precisely how these practices have been adopted or the extent of their adoption.

Research literature is written and initiated by researchers, and when teachers share their ideas (for example, in professional journals or on social media), the accounts are mostly anecdotal. One of the authors of this chapter considers herself “an insider” in relation to what we can infer of teachers’ attention to educational ideas in social media. Having her own media to spread research-based resources for teachers to use, she has learned that even if there is a “hype” from the teacher’s side about a new practice, the real change may remain undone or only partially implemented. As Buschman (2004) explained, it is hard work for a teacher, who most likely has never been experienced alternative methods as a learner or observed them being used by colleagues, to adopt them, no matter how much value they might see in doing so.

In the digital era, online resources are also available for teachers to use for a range of purposes (e.g., as enriching the activities, outsourced feedback, creating excitement). Handal et al. (2013) reviewed more than one hundred mobile applications designed for mathematics learning. They categorized applications using three main clusters: explorative, productivity and instructive tools. It was noted that teachers should understand an application’s instructional value when deciding which to use as some are of little instructional value. They recommended a “watchful but enthusiastic eye” (p. 126) on new mobile learning developments in mathematics teaching.

Other examples of the ubiquity of digital resources vary from general organisation of teaching such as hybrid learning environments (Cribbs & Linder, 2013), to specific techniques, such as teaching with embodied learning technologies (Flood et al., 2020), or applications of known learning theories, such as cognitive load theory, in digital settings (Pass & Sweller, 2005). In an overview of the impact of the Internet on mathematics classrooms Engelbrecht et al. (2020) discussed the new meanings for old constructs such as ‘tool’, ‘resources’ or ‘learning setting’. These new meanings, introduced in mathematics classrooms in the digital era include using Massive Open Online Courses and blended approaches (referred to as Principles of design), technologies in online contexts supporting social interaction and construction of knowledge, and online tools and resources (traditional resources in a digital form, as well as new conceptualisations of what is perceived as a mathematical activity).

In sum, the mathematics classroom as a physical environment has begun to be transformed along with the expansion of the digital world (Engelbrecht et al., 2020). Teachers teaching mathematics are no longer restricted to being the key source, let alone the sole source, of mathematical knowledge. What is more, Engelbrecht et al., (ibid.) discussed the Internet Era transforming the traditional teacher led push approach to mathematics teaching into a student led pull approach, increasing student engagement and agency. Again, the ways in which teachers have reacted to these recent opportunities is less documented (Clark-Wilson et al., 2020) but appears to vary from not using technology, supporting student’s use of technology, through to deliberately eliciting student thinking with and through technology.

Finally, COVID-19 pandemic has accelerated the adoption of technologies in mathematics classrooms, and the impacts are yet to be fully identified. Some insights about impacts of digital technologies in mathematics education during the COVID-19 pandemic were discussed by Borba (2021). The sudden move to online classrooms around the world required teachers to react quickly and with minimal preparation. There is an urgent need to study how the mathematics learning process looks, and specifically what teachers do as they interact with students in new online settings on such a massive scale. The impacts of COVID-19 might have included a decrease in equity as a result of differing access to technologies according to student’s socio-economic background (e.g., using a phone to attend the mathematics class instead of a computer) (Clark-Wilson, 2020). The pandemic necessitated all teachers of mathematics engaging with technologies to teach. Studies of teachers’ activities with students will continue to need to include conceptions of mathematics classrooms that transcend physical boundaries.

5 Implications and Conclusions

The research considered in this chapter is far from exhaustive. Rather we surveyed a broad range of literature to identify the kinds of research being undertaken relating to teacher’s interactive classroom behaviors, and the extent to which promoted practices are used beyond specific studies.

We distinguished between normative mathematics teaching practice and atypical mathematics teaching practices. Large scale studies such as TIMSS and PISA provide insight, albeit indirect, into what happens in the majority of mathematics classrooms. It seems that, in contrast with the student-centered approaches that have dominated mathematics education literature in recent decades, behaviorist approaches remain prevalent. Researcher’s beliefs about, or theoretical perspective on, mathematics teaching inform and constrain their research (its design, conduct and reporting) just as teacher’s theoretical perspectives in either the pre-active (Type D) or interactive phase of teaching (Type C) limit the actions that they perform in their classrooms. The mismatch between the teacher behaviors that researchers advocate and the pedagogies that students most commonly report experiencing raise the longstanding issue of how teacher’s practice can be influenced in ways deemed desirable. Researchers’ interests in particular perspectives on teaching mathematics seem also to have limited research on the practices that most commonly occur in mathematics classrooms. A better understanding of these practices, including the reasons for which teachers adopt and often stick with them, and the variations in context and the practices themselves that affect their efficacy would be valuable in its own right as well to inform efforts to influence teacher’s interactive classroom activity.

In some classrooms technology has had a profound impact on pedagogical possibilities and has led to new ways of structuring teaching such as flipped classrooms. There has been recognition that in a digital world interactions between teacher and students can be both virtual and asynchronous. This development extends Medley’s conception of Type C variables research on teacher behaviors during online synchronous or asynchronous teaching. It problematizes what it in fact means to be in the presence of students.

It is apparent that researchers bring their own theoretical perspectives and beliefs to their work, just as teachers do. The theoretical perspectives described by Manizade, Moore, and Beswick apply equally well to interactive teacher behaviors (Type C) and to pre- and post-active teacher behaviors (Type D). Our review has also highlighted the relative dearth, beyond large scale studies, of research on normative interactions in mathematics classrooms.