Keywords

In this chapter, we take the position that there is a two-way street between theory for teaching and teaching for theory. Our goal is to articulate the linkages between these two dynamic processes through a particular conceptualization of professional knowledge for teaching. Within the context of this conceptualization of professional knowledge for teaching, we address a set of questions about theory and teaching: (1) What is a theory (of teaching)? (2) What should it contain and why? (3) Can such a theory accommodate differences across subject matters and student populations taught? If so, how? If not, why? (4) Do we already have a theory or theories on teaching? If so, which are they? (5) In the future, in what ways might it be possible, if at all, to create a (more comprehensive) theory of teaching? We draw on the lens of Confucian learning as well as examples from Chinese and U.S. mathematics education to elaborate on understanding, assessing, and accumulating professional knowledge for teaching. In addition, we describe how our conceptualization of theories of teaching relies on the creation, evolution, and sharing of artifacts that embody the active processes of theorizing and teaching. The examples that we share are based on research involving Chinese and U.S. students and teachers that some of the authors have been heavily involved in.

This chapter is structured to address each of the above questions in a separate section. The exception is Question 4, for which our response is integrated throughout the paper in the form of examples, such as in our discussion of professional knowledge for teaching and our discussion of effective mathematics teaching.

1 What Is a Theory (of Teaching)?

Speaking very generally, a theory of teaching is a system of ideas that helps to explain the mechanisms of teaching and its effects on students’ learning. This allows for a broad range of things to count as a theory of teaching, from theories that are narrowly focused and tailored to explain teaching phenomena in particular contexts to theories that explain much wider categories of teaching phenomena across many different contexts. Nevertheless, for every theory of teaching, we do expect that the system of ideas includes some explanatory principle that broadens the field’s understanding of teaching in general, and we also expect that it can help inform the actions and decisions of teachers in classrooms on some level (ranging from a general principle to specific guidance). At the general level, a theory of teaching could promote ideas about what kind of teaching or what ways to teach would be best and most effective. An example of this general level might be the idea that the more teachers know about students’ thinking, the better they can teach those students. Although this is a principle that may be broadly applicable, at this level of generality, the idea may not necessarily be directly related to teachers’ teaching; the generality means that the specific implications for practice are not sharply in focus. A narrower theory of teaching might tighten the focus to a particular context or a specific phenomenon happening in a classroom. At this level, the theory would speak more directly to teachers’ planning and to what they actually choose to do in their classrooms.

The connection (or lack of connection) between theories of teaching and the actual practice of teaching has been a topic of interest and concern to scholars in mathematics education and education more broadly for many decades. Silver and Herbst (2007) took a “practice-oriented perspective” centered around an active approach to theory development and use in mathematics education (p. 39). They acknowledged that theory, especially as it relates to mathematics education, takes many forms that have different uses according to the situation at hand. They found commonalities across these various forms and presented a framework aimed at decreasing the divide between those for whom theory is not theoretical enough (e.g., researchers) and, conversely, those for whom it is not practical enough (e.g., teachers). They noted the growing importance of theory in the field and the growing number of studies in mathematics education that include theoretical considerations as well as growing diversity among the theories drawn upon, ranging among psychological, sociological, sociocultural, linguistic, and others. In fact, they claimed that “a theoretical perspective (as opposed to a practical perspective) currently dominates the process of scholarly publication in mathematics education” (Silver & Herbst, 2007, p. 43). The framework they presented places theory and theory making at the center of the triangle of relationships between problems, practices, and research, with theory acting as the mediator for the bidirectional relationships between all of the vertices of the triangle.

Although Silver and Herbst (2007) described the role of theory in mathematics education in general, we focus this chapter specifically on teaching. Moreover, we propose that it is most important to focus not on the entire category of theories of teaching but on the critical subset that we describe below—theories for teaching that are specifically intended to inform and improve teaching by enriching our (teachers’ and researchers’) collective understanding of some aspect of teaching, how and why that aspect matters, and how it might be leveraged to create richer learning opportunities for students. We choose to focus our gaze on these kinds of theories because they tend to take seriously the challenge of harmonizing the perspectives of teachers and researchers in a way that can ultimately contribute to a usable base of practical professional knowledge (the idea of building a professional knowledge base is also taken up by Hiebert & Stigler, this volume, and Herbst & Chazan, this volume). In addition, we posit that it is not feasible to discuss theories for teaching without also discussing its dual, what we refer to as teaching for theory. More than simply advocating for greater connections between theory and practice, we believe it is necessary for the field to consider both of these together because although they appear to be fundamentally different constructs, they are unified in a perpetual cycle of interaction, development, and evolution—a two-way street. Indeed, from an Asian perspective, the greater connections between theory and practice as a two-way street are viewed as part of the identity of Asian mathematics education (Leung, 2001). Thus, for the remainder of this chapter, when we refer to a “theory of teaching,” our conceptualization of this term should be interpreted not as the more general case but as a specific instance of the unification of both theory for teaching and teaching for theory.

2 A Two-Way Street: Theory for Teaching and Teaching for Theory

By a theory for teaching, we mean a theory that is designed to provide guidance for creating more and better learning opportunities for students. The choice of the word “for” is deliberate; it is meant to highlight this purpose of the theory. Theories for teaching can build on a variety of foundations such as theories of learning and major components of the practice of teaching. For example, learning can be conceptualized as both an individual and a social process (Cai, 2003; Cobb, 1994a). Correspondingly, theories for teaching would need to address how to create the kinds of learning opportunities in which students are able to construct their own knowledge as well as address how to create a social environment for learning, establish classroom norms, and reach shared understanding (Cobb, 1994b). Other elements of the classroom experience might motivate theories for teaching that address the nature of instructional tasks or the characteristics of classroom discourse (e.g., Hiebert & Wearne, 1993; Stein & Lane, 1996) because these are also fundamental to how teachers organize classroom teaching to promote students’ learning. Each of these theories is fundamentally aimed at improving teaching by better understanding some aspect of teaching and showing how leveraging that aspect can change teaching for the better. In essence, theories for teaching can provide essential information about what is worth trying, what is unlikely to work, and whether an instructional design is based on theoretically sound principles because such research is deliberately framed to guide or shape teaching (Burkhardt & Schoenfeld, 2020).

Similarly, we use “for” in the expression “teaching for theory” to signify that we are focused on a kind of teaching that is for a particular purpose in addition to the immediate and usual purpose of helping students learn. If theory for teaching refers to the theories which are drawn upon to teach well so that students learn, then teaching for theory refers to teaching that is deliberately designed to generate, elaborate, and test theory so that the field learns. In educational research, teaching experiments are a well-known mechanism for using teaching to help generate, develop, and articulate theory (Steffe & Thompson, 2000). However, the usual image of teaching experiments is not the only kind of teaching for theory that can be invoked here. Rather, we include in this category a wide range of activities in which teaching produces professional knowledge that accumulates and gives new insights into the problems of practice, including the iterative design and implementation cycles of design-based research (Cobb et al., 2017) and the “rapid prototyping followed by iterative refinement cycles in increasingly realistic circumstances” of the engineering research approach (Burkhardt & Schoenfeld, 2020). Of course, not every theory in education is built through this kind of deliberate accumulation of professional knowledge through teaching, but we explicitly highlight this mechanism for two reasons. First, our perspective is informed by the emphasis on codependency between practitioners and researchers among some Asian cultures (e.g., Fan et al., 2004). Second, even when a theory of teaching is constructed through other means, it must still at some point survive contact with actual practice. Teaching for theory can provide critical feedback that defines or constrains the validity and applicability (or generality) of a given theory.

Given our definitions of theory for teaching and teaching for theory, there are two natural corollaries. The first is that these two things are not static objects but instead dynamic, evolving processes. Theory for teaching is, perhaps, more properly expressed as “theorizing for teaching”—engaging in a constant process of evaluating and reevaluating assumptions and connections to refine our understanding of teaching. And, teaching for theory is an ever-iterating process of shaping teaching so that it provides new data, new hypotheses, and new ways to teach that enrich the theorizing. Ultimately, theories for teaching must have practical implications for teaching, but they must also evolve in response to teaching for theory.

For example, in the era of the COVID-19 pandemic, the model for teaching has had to completely change in many places. As Engelbrecht et al. (2020) acknowledged, COVID-19 has drastically changed teaching and learning as we know it; thus, some of the preexisting theories for teaching no longer apply. In this environment, teaching for theory takes on an important role in stimulating the generation of new theories about the best ways to teach through online or virtual learning. In addition to the logistical hurdles of obtaining equipment and access and learning new technologies, the importance of digital communication and the home environment (including physical, social, and family resources) has never been more front and center as classrooms have switched to blended and online learning environments. And, the potential for educational technologies to facilitate student-centered learning is preeminent. Adapted theories for learning will be required to account for these radical changes in teaching and learning. For example, in their examination of how students expand their mathematical knowledge through their collaborative creation of mathematics videos, Oechsler and Borba (2020) demonstrated how digital technology can fundamentally change not only the educational problems that are to be addressed in the learning of mathematics (the problems of practice; Cai et al., 2018a) but also the roles of teachers and students (e.g., expanding students’ responsibility for their own learning).

It is reasonable to ask why it is important to highlight theories for teaching within the larger set of theories of teaching (taken in the broader sense). Our perspective comes from foregrounding the practical aspect of teaching—it is an applied science that involves doing. Thus, theories related to teaching must guide practice in teaching and not simply involve the generation of theories purely for human beings’ curiosity without any implications or realization:

Teachers’ expertise can play a leading role in identifying and formulating important problems of practice. Teachers directly interact with students around mathematics, and they are well positioned to raise red flags when those interactions consistently go awry or fail to produce the desired outcomes. Because teachers are necessarily focused locally, what they see is framed by their students, their lesson, their curriculum, their classroom, and their school. Moreover, teachers’ conceptions and beliefs about mathematics, teaching, and learning influence how they perceive and identify instructional problems. (Cai et al., 2018a, p. 515)

Included among the theories for teaching are a number of theories that are concerned with the kinds of knowledge that are needed for teaching. Many of these theories have stemmed from Shulman’s (1987) seven categories: subject-matter content knowledge; general pedagogical knowledge; curriculum knowledge; pedagogical content knowledge; knowledge of learners and their characteristics; knowledge of educational contexts; and knowledge of “educational ends, purposes, and values, and their philosophical and historical grounds” (p. 8). The field has seen extensive development of these constructs, such as Mathematical Knowledge for Teaching (MKT; Hill et al., 2008), which includes several aspects of both subject matter knowledge and pedagogical content knowledge; extensions of MKT to geometry (e.g., Herbst & Kosko, 2014; Mohr-Schroeder et al., 2017) and algebra (e.g., McCrory et al., 2012); and teachers’ general pedagogical knowledge (e.g., Döhrmann et al., 2012; Tatto et al., 2012), which includes knowledge like classroom management techniques.

In a series of editorials, Cai et al. (2020) discussed professional knowledge for teaching from several angles centered around the divide between isolated teaching practice by individuals in unique contexts and practice that is built on shared knowledge gained from collective profession. As models of teaching and learning change, as in the case of the response to COVID-19, teachers must be able to adapt their professional knowledge for teaching on a rapid, iterative basis. Cai et al. (2018b) proposed how the use of a professional knowledge base storing lessons and instructional adaptations that are aggregated over time and that involve teacher–researcher partnerships could have direct implications for developing professional learning. Cai et al. (2020) discussed how researchers must work to supplement and build teachers’ specific, lesson-level professional knowledge to create learning opportunities for students as well as how to share this knowledge and make it accessible. Regardless of the form they take, theories for teaching must be very practical, useful, and accessible for teachers.

3 Components and Generality of a Theory of Teaching

In this section, we continue this line of reasoning by considering what components a theory of teaching should contain as well as how such theories can accommodate differences across subject matters and student populations. As a reminder, we are using the term “theory of teaching” to refer specifically to the pairing of theory for teaching and teaching for theory. That is, the theories of teaching that we are concerned with are specifically those that reflect the combination of both sides of the two-way street.

3.1 Components of a Theory of Teaching

As noted above, we take a broad view of what counts as theory with respect to its grain size. Whether a theory of teaching is broad and concerned with a widespread teaching phenomenon or narrowly focused on a local problem of practice, it must address some aspect of teaching, big or small. Moreover, to improve the quality of learning opportunities for students, a theory of teaching must exist in a form that supports teachers to think through, evaluate, and translate into actions the ideas about teaching that the theory comprises. As Biesta (this volume) argues, the knowledge generated by science “can never tell teachers what they should do, but can at most inform their judgments” (p. 273). Thus, a theory of teaching that builds on a continual process of formulating and testing hypotheses in actual practice must exist in a form that teachers can make sense of and draw on to craft instruction. Indeed, without the ability to provide such support for teachers to engage with the ideas of the theory, a theory of teaching cannot easily benefit from the process of teaching for theory because it becomes difficult to frame useful hypotheses related to the theory that can be informed by carefully planned teaching experiences. This means that a theory of teaching should provide a framework that teachers can use as they think through principles of the best ways to teach in a given situation. In addition, it has to have some operational aspects that address the practical translation of principles into actions. These two components, a framework for thinking about teaching and an operational side, are both needed for a theory of teaching to provide guidance for teaching.

As an example, consider the dimensions for examining the effectiveness of mathematics instruction (Cai, 2003). These dimensions address three critical aspects of effective classroom instruction: (1) students’ learning goals, (2) instructional tasks (both as set up by teachers and as implemented in the classroom), and (3) classroom discourse. The role of teachers is to select and develop tasks that are likely to foster students’ development of understanding and mastering procedures in a way that also promotes their development of abilities to solve problems, to reason, and to communicate mathematically. We examine each of these dimensions in greater detail to highlight how this theory includes both a framework for supporting teachers as they think about their teaching and an operational aspect.

3.1.1 Learning Goals for Students

It is assumed that effective teaching is related to the goal of high achievement for all students (National Academy of Education [NAE], 1999). Effective teaching requires that teachers understand what students know and need to learn and what challenges and supports their learning (National Council of Teachers of Mathematics [NCTM], 2000). What teachers do in the classroom depends on the nature of their learning goals for their students, and there are important connections between goals for learning and teaching practices that affect students’ abilities to accomplish these goals (Bransford et al., 2000). In particular, the learning goals teachers set influence both their planning for each lesson as well as how they make in-the-moment decisions to address the unexpected to guide students toward the learning goals. As Schoenfeld (this volume) puts it, teaching involves knowledge-based decision making in complex social contexts, and this decision-making process depends fundamentally on teachers’ goals (as well as the resources available). So, when planning a lesson, teachers must take into account the learning goals as well as the knowledge and experiences their students bring with them. They use that information, along with their curricular resources, to choose appropriate instructional tasks that can help their students build on their existing knowledge to achieve the learning goals. Thus, the nature of the learning goals has a large influence on the shape of the lesson and on the mathematics that students have the opportunity to learn. In addition, when actively engaged in teaching, teachers frequently make in-the-moment instructional decisions in response to what students are doing, especially when something unexpected arises. These in-the-moment decisions, which include such choices as how to respond to students’ questions, how to react to or make use of students’ responses and mathematical work, and when to provide additional guidance or additional encouragement to persevere, serve to shape the ongoing enactment of the lesson so that it continues to orient students towards the learning goals.

Given the role of learning goals in shaping lesson planning and enactment, it follows that teaching benefits from teachers setting productive learning goals—ones that result in learning opportunities that encourage students to develop conceptual understanding, mathematical reasoning, and positive relationships with mathematics. Setting productive learning goals requires teachers to draw on their knowledge of mathematics, the curriculum, their students as learners, and pedagogical strategies. In addition, teachers’ beliefs about mathematics and conceptions about teaching mathematics also factor into teachers’ decisions about learning goals for their students. Thus, on one level, the inclusion of learning goals as an aspect of effective classroom instruction provides a general guide to teachers that it is important to carefully consider the kinds of learning goals they explicitly set for their students.

On an operational level, how might the learning goal aspect of effective classroom instruction address teachers’ actual day-to-day practice? One example comes from a comparative study of U.S. and Chinese students’ problem-solving abilities. Although we know that in mathematics it is important for students to have basic algorithmic knowledge to solve many kinds of problems, this does not ensure that they have the conceptual knowledge to solve nonroutine or novel problems (Cai, 2000; Hatano & Inagaki, 1998; Steen, 1999; Sternberg, 1999). In one of a series of studies examining U.S. and Chinese sixth-grade students’ mathematical problem solving and problem posing (Cai, 2001), four types of tasks were used: multiple-choice tasks measuring basic computation skills, 18 multiple-choice tasks measuring simple problem-solving skills, process-constrained tasks measuring complex problem-solving skills, and process-open tasks measuring complex problem-solving skills. Process-constrained tasks refer to problems that can be solved by executing a “standard algorithm.” In contrast, process-open tasks are problems that usually cannot be solved by an algorithm and more typically require novel exploration of the problem situation. Furthermore, a process-open task usually lends itself to a variety of acceptable solutions.

The Chinese students in the study scored significantly higher on average than the U.S. students on the computation tasks, the simple problem-solving tasks, and the process-constrained tasks. However, the U.S. students scored significantly higher on average than the Chinese students on the process-open tasks. Indeed, on average, the U.S. students scored highest on the process-open tasks and lowest on the computation tasks, whereas the Chinese students scored highest on the computation tasks and lowest on the process-open tasks. Reported 20 years ago, these results reflected then-prevalent characteristics of teaching in the United States and China (in particular), specifically with respect to cultural differences in teachers’ beliefs about the relationships between developing basic skills and higher order thinking skills in mathematics (Fan et al., 2004) and the kinds of learning goals teachers set for their students.

Twenty years later, data gathered from the same Chinese schools using the same tasks reflect a major shift in learning goals for students in China; students’ learning goals now include explicit attention to process-open complex problem solving. Since 2001, teaching in Chinese schools has thus shifted to include a focus on process-open tasks so that these tasks are now a specific part of mathematics teaching and built into teachers’ day-to-day lessons. Comparing the performance of current Chinese students to their predecessors, we see relatively similar performance on computation, simple word problem solving, and process-constrained complex problem solving (from 88% to 82%, from 77% to 70%, and from 75% to 78%, respectively) and a sharp increase in performance on process-open complex problem solving (from 57% to 75%) that exceeds even the earlier U.S. students’ performance on those tasks. Clearly, the operationalization of this evolution in students’ mathematical learning goals has come with a parallel evolution in students’ learning.

3.1.2 Instructional Tasks

As we noted above, teachers choose instructional tasks to create opportunities for students to move towards the desired learning goals. Instructional tasks provide the intellectual environments for students’ learning and the development of their mathematical thinking. Broadly, instructional tasks include such things as projects, questions, problems, constructions, applications, and exercises in which students engage. Doyle (1988) argued that tasks with different cognitive demands are likely to induce different kinds of learning. Indeed, tasks influence students’ attention to particular aspects of content and the ways they process information. In particular, instructional tasks that are truly problematic for students have the potential to promote their conceptual understanding, foster their ability to reason and communicate mathematically, and capture their interest and curiosity (NCTM, 1991). It is recommended that students in classrooms be exposed to truly problematic tasks so that mathematical sense-making is practiced (NCTM, 1991, 2000). Thus, a framework for thinking about the characteristics and impact of instructional tasks on students’ learning can be helpful for teachers to be sensitive to the nature of the tasks they use and to differentiate between tasks that will or will not help their students to achieve the learning goals. For example, Stein and Lane (1996) highlighted the importance of the level of cognitive demand that an instructional task supports. They classified tasks into four increasingly demanding categories of cognitive demand: memorization, procedures without connections, procedures with connections, and doing mathematics. Tasks with higher levels of cognitive demand can support students to engage in higher level thinking and problem solving (Cai, 2014). Thus, as an aspect of effective teaching, the nature of instructional tasks is a key dimension for teachers to attend to.

Operationally, teachers must have ways to decide which instructional tasks to select or what tasks to develop in order to meet the specific learning goals of a lesson. For example, Lappan and Phillips (1998) proposed a set of characteristics that could be used to evaluate whether a problem was worthwhile for students to engage with:

  • The problem has important, useful mathematics embedded in it.

  • Students can approach the problem in multiple ways using different solution strategies.

  • The problem has various solutions or allows different decisions or positions to be taken and defended.

  • The problem encourages student engagement and discourse.

  • The problem requires higher level thinking and problem solving.

  • The problem contributes to the conceptual development of students.

  • The problem connects to other important mathematical ideas.

  • The problem promotes the skillful use of mathematics.

  • The problem provides an opportunity to practice important skills.

  • The problem creates an opportunity for the teacher to assess what his or her students are learning and where they are experiencing difficulty.

Although textbooks can be a useful resource for selecting worthwhile instructional tasks, teachers can use criteria such as these to evaluate the suitability of problems for supporting effective teaching. In addition, teachers can draw on these criteria to develop additional worthwhile and interesting mathematical tasks by modifying problems from textbooks.

In our recent work on mathematical problem posing, we have begun to work with teachers to revise or develop problem-posing tasks to teach mathematics (Cai & Hwang, 2021a, 2023). By “problem-posing tasks,” we refer to instructional tasks that engage students in generating new problems and questions based on given situations (including mathematical expressions or diagrams) or changing (i.e., reformulating) existing problems (Cai & Hwang, 2023; Silver, 1994). As we have argued in Cai and Hwang (2023), because problem-posing tasks are cognitively demanding, such tasks can engage students in productive struggle with challenging mathematics so as to maximize their learning opportunities. Although problem-posing activities are cognitively demanding tasks, they are also adaptable to students’ abilities and thus can increase students’ access such that students with different levels of understanding can still participate and pose potentially productive problems based on their own sense-making.

3.1.3 Mathematical Discourse

Worthwhile instructional tasks and rigorous goals alone do not guarantee effective teaching and students’ learning. Even the most worthwhile tasks that have been designed to help students move towards important learning goals may fail to play out in the classroom as intended. For example, Stein and Lane (1996) found that only about 50% of the tasks that were set up to require students to apply procedures with meaningful connections were implemented effectively. A key factor lies in the choices that teachers make when organizing mathematical discourse in their classrooms, including choices like how long to wait for students to respond. Therefore, in the classroom, students’ actual opportunities to learn depend not only on the type of mathematical tasks that teachers present but also on the kind of discourse that teachers orchestrate to implement the tasks in support of the learning goals (Cazden, 1986). More generally, discourse refers to the ways of representing, thinking, talking, and agreeing and disagreeing that teachers and students use to engage in instructional tasks. Considerable theoretical and empirical evidence exists supporting the connection between classroom discourse and student learning. The theoretical support comes from both constructivist and sociocultural perspectives of learning (e.g., Cobb, 1994a; Hatano, 1993). As students explain and justify their thinking and challenge the explanations of their peers and teachers, they are also engaging in clarification of their own thinking and becoming owners of “knowing” (Lampert, 1990). Indeed, patterns of discourse in classrooms can serve both to position students as knowers and doers of mathematics as well as to establish classroom norms (Wagner & Herbel-Eisenmann, 2009). The empirical evidence supporting the positive relationships between teachers asking high-order questions and students’ learning can be found in Hiebert and Wearne (1993) and in Redfield and Rousearu (1981). Thus, if teachers do not orchestrate discourse effectively, it is possible that students will miss many learning opportunities.

Given the potential power of well-orchestrated classroom discourse and the relative lack of such discourse in many classrooms (Spillane & Zeuli, 1999; Stigler & Hiebert, 1999), it is particularly important to provide an operational aspect of this dimension of the theory to support teachers’ efforts. An operationalization of this can be seen in the work of the Mathematics Discourse in Secondary Classrooms (MDISC) project (Herbel-Eisenmann et al., 2013). In developing professional development experiences for teachers around discourse, MDISC described six “teacher discourse moves” (Herbel-Eisenmann et al., 2013, p. 183): waiting, inviting student participation, revoicing, asking students to revoice, probing a student’s thinking, and creating opportunities to engage with another’s reasoning. They provide transcripts illustrating the various moves as materials for teacher discussion as they think about how they can intentionally use powerful mathematics classroom discourse. In addition, MDISC describes interpretive lenses to help teachers notice and interpret the productivity and power of discourse patterns that they observe.

3.2 Generality of a Theory of Teaching

One quality of a theory that is typically prized is its generality—the breadth of the set of conditions to which it applies. In the context of our discussion of theories of teaching, the relevant question regarding generality is whether a theory of teaching can accommodate differences across subject matters and student populations. Again, because our definition of a theory of teaching admits both theories with broad scope and much more narrowly focused theories, the answer to this question depends on the particular theory.

Some theories of teaching are focused broadly, and by their nature they span different subject matters, grade levels, and even cultural contexts. For example, a theory that claims that teaching needs to build on students’ prior knowledge and thinking to be effective in teachers’ design and delivery of lessons is a very general theory of teaching that can apply to students in different subject areas, grade levels, and cultures. As another example, consider higher order thinking skills in mathematics. According to Resnick (1987), higher order thinking incorporates the following:

  1. 1.

    Is non algorithmic. That is, the path of action is not fully specified in advance.

  2. 2.

    Tends to be complex. The total path is not “visible” (mentally speaking) from any single vantage point.

  3. 3.

    Often yields multiple solutions, each with costs and benefits, rather than unique solutions.

  4. 4.

    Involves nuanced judgment and interpretation.

  5. 5.

    Involves the application of multiple criteria, which sometimes conflict with one another.

  6. 6.

    Often involves uncertainty, not everything that bears on the task at hand is known.

  7. 7.

    Involves self-regulation of the thinking process.

  8. 8.

    Involves imposing meaning, finding structure in apparent disorder.

  9. 9.

    Is effortful, a considerable mental work involved in the kinds of elaborations and judgments required.

This list clearly shows that higher order thinking skills involve the abilities necessary to think flexibly to make sound decisions in complex and uncertain problem situations. Resnick’s list does not include the ability to collaborate with others, but being able to work together with others is also one of the characteristics of having higher order thinking skills (Chi, 2009). Through students’ collaborative work, they can think together about ideas and problems as well as challenge each other’s ideas and ask for clarification and further explanation. The theory of higher order thinking can be used as a way to analyze and improve teaching so that students engage with higher quality learning opportunities. For example, curriculum developers could make use of the characteristics of higher order thinking to guide instructional task design so that tasks are likely to foster these higher order thinking skills. Or, teachers may draw on the characteristics to evaluate their own instructional decisions and to guide how they choose tasks, launch them in class, and organize discourse around them. In these senses, the theory of higher order thinking can act as a theory for teaching as we have described above. Moreover, the theory is broad enough that it has been applied to different subject matters, grade levels, content areas, and cultures (e.g., mathematics education (Stein & Lane, 1996), science education (Barak et al., 2007), and with students at various academic levels (Zohar & Dori, 2003)).

On the other end of the spectrum are theories of teaching that are very narrowly focused on a particular teaching phenomenon in a specific context. By nature, these kinds of theories for teaching are not apparently very general. They do not necessarily accommodate differences across subject matters and student populations. Here, however, the idea of generality comes from the ongoing evolution of these theories for teaching through teaching for theory. Although a particular theory may have arisen from addressing a specific local problem of practice, many such problems exist across contexts and classrooms. Indeed, teachers commonly face many problems of practice that are closely related to one another; such problems are found repeatedly in many places. Examples of common problems of practice abound in mathematics teaching and learning, including difficulties teaching students about adding and subtracting fractions, developing students’ understanding of triangle congruence theorems and their use, and dealing with common student errors related to place value in multidigit multiplication. Extremely small-grained theories of teaching may involve something as limited as one teacher’s hypothesis that a particular instructional task or approach will help his or her students realize that a key point of the triangle congruence theorems is that they each identify a minimal amount of information needed to determine a particular triangle. By implementing this task or approach to test the hypothesis, the teacher engages in a local instance of teaching for theory. But, because this problem exists in many contexts, other teachers may also try such an approach with their students. Their experiences, that is, their own teaching for theory, help to define the boundaries of generality of the original teacher’s hypothesis—under what conditions it holds or does not hold. Moreover, adaptations of the approach may expand the theory to cover more contexts. Thus, there is the potential to link small-grain-size theories for teaching to each other by the common aspects of the problems they address. A local theory of teaching may be expanded by other teachers teaching for theory in different (though possibly similar) contexts who attempt to apply the local theory to their own classrooms and contexts. In this way, the boundaries of local theories can be iteratively mapped so that, as Kyriakides et al. (this volume) emphasize, we can better know for whom and under what conditions a theory is useful.

4 Theories of Teaching: East Meets West

Given the examples we have discussed above, it seems that theories of teaching span a wide range of grain sizes and attend to many different aspects of teaching. One key aspect of teaching that we have not yet specifically addressed is the fact that teaching is a cultural practice: It is a practice embedded within a larger cultural milieu. Teaching is thus also shaped by cultural expectations, and, consequently, theories of teaching may naturally end up reflecting the cultural practices of the context in which they are conceived, used, and refined. Indeed, the influence of culture on theories of teaching can be striking; it can shape, for example, conceptions of effective instruction.

As an illustration of this influence, consider the following story recounted by Howard Gardner, a distinguished professor and scholar from Harvard University. In the spring of 1987, Gardner was visiting China to study arts education in kindergartens and elementary schools. During the visit, he, his wife, and his son (Benjamin) stayed in the Jinling Hotel in Nanjing. The key to their hotel room was attached to a large plastic block that made noise when it was shaken. Benjamin loved to carry the key chain around, shaking it vigorously. He also liked to try to place the key into the slot. Because Benjamin was very young, it was a challenge to correctly orient the key into the slot. However, Benjamin seemed to enjoy the sound it made when the key banged against the slot, and he also loved this exploratory activity. Because Gardner and his wife were not in a hurry at the time, they allowed Benjamin to have a good time. But they soon observed an intriguing phenomenon. Any Chinese attendant nearby would come to watch Benjamin. At one point, an attendant noticed Benjamin’s lack of initial success in placing the key into the slot, so she would hold onto Benjamin’s hand and directly help Benjamin insert the key. Then she smiled at Gardner or his wife, as if having done a favor for them and awaiting a “thank you.” Interestingly, neither Gardner nor his wife appreciated the intervention of the attendant since what mattered to them was that Benjamin was having a good time exploring. Later Gardner realized that this incident pointed to important differences in the educational and artistic practices between the United States and China. After studying Chinese education in general and arts education in particular, the world-renowned scholar wrote: “Some of my most entrenched beliefs about education and human development had been challenged by my observations in Chinese classrooms” (Gardner, 1989, pp. vi).

Gardner’s story provides an insight into the cultural practices of education in China and how they differed from his Western expectations. The attendant’s reaction to Benjamin reflects a set of expected behaviors rooted in a Confucian cultural perspective. The Confucian model of education is “centered on the teaching and learning of ren, the benevolent relationship among human beings” (Chan et al., 2017, p. 21). By this notion, all individuals regardless of background can engage in the pursuit of advancing personal and moral character through diligent practice and harnessing their unique potential (Chan et al., 2017). Li (2003) found that Chinese students’ conceptions of learning suggested a “person orientation” in which knowledge is a part of the person’s life and a cognitive, social, and moral process of seeking rather than an externally existing object or absolute truth. These are in line with a Confucian philosophy of learning, which is based on the lifelong process of seeking toward self-improvement, available to anyone who sets their intentions on this path. The teacher’s role in this process is to serve as a model who exemplifies the process of and commitment to learning and to guide students on their individual paths to enlightenment (Tan, 2017), just as the attendant in Gardner’s story sought to guide Benjamin. Indeed, Confucian teaching and learning are two sides of the same coin. Many of these ideas are reflected in Chuang’s (2012) study of Western and Confucian-influenced graduate students’ educational philosophies, with the Confucian-influenced students’ philosophies reflecting the notion of lifelong learning; the goal of self-cultivation to achieve personal virtue and collective harmony; and the approaches of observation, listening, questioning, memorization, experience, and deference to the teacher.

Confucianism emphasizes the role of environment and practice on peoples’ development of skills and knowledge. Through learning and practice, people develop different ways of being and thinking, and every source of observation is a potential teacher, whether it be books, peers, and so on. Central to this process is the role of critical thinking in the acquisition of knowledge and skills as well as their application to real life and localized problems to verify that what is learned is accurately reflected in subsequent observations. Thus, it is a learning based on meaning and synthesizing multiple ideas and perspectives.

This characterization of learning in a Confucian perspective may seem at odds with the example of Chinese teaching and learning of mathematics in the past as being focused on computation and process-constrained tasks rather than the kinds of open-ended tasks that would benefit from a focus on meaning and synthesis. However, the evolution over decades to a model of teaching in Chinese schools that also focuses on process-open tasks reflects the two-way street between theory for teaching and teaching for theory. In fact, as we indicated before, Chinese students were able to perform better than U.S. students even on process-open tasks in recent years.

In a comparison of U.S. and Chinese teachers’ instructional methods, Cai et al. (2014) found that Chinese teachers focused more on addressing student thinking and challenging students in fostering deep synthesis between interconnected mathematical ideas and conceptual structures. Similarly, Cai and Wang (2010) found that, compared to U.S. teachers, Chinese teachers emphasized connecting different conceptual ideas to foster students’ mathematical understanding.

According to Cai and Wang (2010),

For Confucius, knowledge and truth should be acquired by learning from authority figures/masters (e.g., a teacher) rather than being generated by the learners themselves. In teaching and learning, the Confucian tradition emphasizes teacher’s authority and students’ hard work. (p. 284)

Thus, the role of the teacher in the learning process is that of a mentor or disciple who serves as a model and resource for their students. On a survey of U.S. teachers’ reactions to Confucian teaching philosophies and methodologies, Chan et al. (2017) identified these philosophies as follows: promoting character education, teaching students from all backgrounds, improving teacher knowledge and skills, perseverance in teaching, and teaching with no reservations in sharing personal experiences. These philosophies translated into different methodologies, including: providing differentiated instruction potentials, stimulating student learning, teaching students by role modelling, and teaching with a step-by-step approach.

Notably, despite the orientation towards learning from authority figures, the Confucian theory of learning is actually conducive to a student-centered approach to teaching. Teachers are encouraged to guide all individuals regardless of background in their educational pursuits and to know their students well so they can adjust their methods according to the individual needs of the learner (Chan et al., 2017). The teaching methods, resources, and approach are customized to maximize each individual learner’s self-cultivation process; through observing each student’s learning status and characteristics, the teacher can provide a personalized response that best fosters their educational attainments (Tan, 2017). Rather than merely dictating educational content, the teaching process depends on guiding the students to play an active role in their own learning through appropriate prompts that facilitate students’ reflection and critical thinking skills (Tan, 2017). Reflection, then, is a key component of both the teaching and learning process: Both the teachers and the learners rely on a regular process of reflection to see where the learner’s current stage of knowledge is and where it has gaps or conflicts. Tan (2017) mentioned two specific techniques that exemplify this process: the questioning technique, whereby teachers engage students in questions and prompts, and peer learning, whereby students are encouraged to discuss among themselves in pairs or groups to sort out their understanding of the content.

5 Artifacts That Embody and Bridge Theory for Teaching and Teaching for Theory

Thus far, we have described the relationship between theory for teaching and teaching for theory as a two-way street. That is, these two constructs exist in a reciprocal relationship in which each is a driving force that stimulates progress in the other. In practice, however, it can be difficult to establish this kind of pairing of theory for teaching and teaching for theory. Theorizing for the purpose (at least in part) of informing the decisions of practice is an ongoing, dynamic act; similarly, teaching for the purpose (again, at least in part) of informing the growth of theory is active. Capturing what is happening in both of these activities so that they may mutually support each other requires a third element—a way of embodying them and making them tangible and accessible to the teachers and researchers who are engaged in the processes.

Elsewhere, we have discussed the need for artifacts—tangible products—that can store professional knowledge and that can form the foundation of a knowledge base for the profession (Cai et al., 2018b). Such artifacts are a way to give a physical reality to the dual processes of theory for teaching and teaching for theory; they act as “carriers” that facilitate the storing, sharing and growth of professional knowledge. Figure 8.1 shows our conception of how such an artifact works. The strip includes both theory for teaching and teaching for theory, apparently on opposite sides. However, this is a Mobius strip, and the two apparent sides are, in fact, the same side, flowing in an infinite cycle. The artifact, then, serves as an embodiment of both processes simultaneously, capturing their mutual development, interaction, and influence. This conception of an artifact bears some similarity to what Burkhardt and Schoenfeld (2020) have described as “replicable materials” to support implementation that would integrate (or embody) a set of engineering principles, including being “grounded in robust aspects of theory from prior research,” “flexibility … that affords adaptation to the range of contexts across the intended user community,” and “continued refinement on the basis of post-implementation feedback ‘from the field’” (p. 8). Similarly, Hiebert and Stigler (this volume) discuss how lesson plans may serve as an artifact that records, preserves, and shares information across classrooms while remaining at a grain size that is amenable to the work of teaching.

Fig. 8.1
An illustration of a curved ribbon in the shape of the infinity symbol has two texts, theory for teaching on the left and teaching for theory on the right. A text in front of the structure reads artifact, for example teaching case.

A diagram illustrating the relationship between theory for teaching and teaching for theory as embodied in a tangible artifact. The Mobius strip represents the artifact. It is the medium in which theory for teaching and teaching for theory exist and interact. The two are not on opposite sides of the artifact (there is only one side) but rather flow continuously into one another. Teaching cases (considered as dynamic, evolving objects) are a specific example of such an artifact

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Below, we will elaborate on the idea of artifacts by describing one possible form that such artifacts can take—continuously developed teaching cases produced by teacher–researcher partnerships—as an example that is currently embedded in an Eastern culture of teaching. We will then consider the more general question of what features an artifact might need to have to fulfill this function, whether in an Eastern or Western cultural context. Finally, we will look to the future and suggest some considerations that we believe the field will need to focus on to make theory for teaching and teaching for theory feasible, if not commonplace, mechanisms for improving the quality of teaching and learning.

5.1 Teaching Cases in Chinese Mathematics Education

In China, there is a multitude of lesson plans that have been developed by award-winning teachers. This type of lesson plan is often produced from a focus lesson that is developed through the work of a teacher research group and incrementally improved upon until the lesson is ready for others to examine and use as a model. Thus, the development and refinement of individual lessons through the work of a teacher research group is a normal part of the work of teachers in Chinese mathematics education. Indeed, this kind of work has strong parallels with activities such as lesson study, a process that has been a longstanding part of teacher learning and professional development in Japan (Becker & Shimada, 1997) and which has been studied as a potential avenue for teacher professional development in other countries (e.g., Lewis & Perry, 2017). Those related activities also embody connections between research for teaching and teaching for research, but here we will focus on the specific example of China to illustrate our argument. In particular, we argue that the process of refining focus lessons can be further developed and built upon systematically to create the kinds of artifacts that would capture theory for teaching and teaching for theory.

A longitudinal research project based in a school district in Hangzhou, China, aimed to develop elementary and middle school teachers’ ability to use mathematical problem posing to teach mathematics (Cai & Hwang, 2021a; Zhang & Cai, 2021). As part of the project, the teachers participated in professional development workshops in which they learned about mathematical problem posing and how it can be used to teach mathematics, and they designed mathematics lessons in which problem posing was used as an instructional tool. In addition to the workshops, a central element of the project was a collaboration between the participating teachers, teacher researchers, and teams within each of the teachers’ schools to develop problem-posing teaching cases based on lessons (and entire units of lessons) that the teachers designed.

In this project, the initial conception of teaching cases draws both on the typical Chinese form of teaching cases as a way to share professional knowledge and on Western conceptions of case-based education (e.g., Smith & Friel, 2015; Stein et al., 2000). Moreover, the teaching cases being developed are more than simply a collection of lesson plans or a single report on a lesson and its implementation. Indeed, the teaching cases are dynamic objects that grow as lessons evolve and that, once shared, may continue to grow through adaptations from others (as well as the originators). When a teaching case is published, what is shared with the reader is an instantaneous snapshot of one part of the full, dynamic teaching case. To embody the support that theory for teaching and teaching for theory offer to each other, these teaching cases include multiple components. The first component explains the mathematical learning goals for the lesson, including a description of what it means to understand the content topic. In addition, this component includes a mathematical analysis that situates the content within the mathematical framework of the curriculum. The second component is a cognitive analysis of the learning goals and content, focusing on potential difficulties for students and the prior understanding and knowledge students need to succeed in the lesson. The third component is a description of the major components of the lesson, broken down by instructional task (mainly problem-posing tasks, but not all tasks are necessarily problem-posing tasks). This includes a rationale for each problem-posing task that explains the purpose of the task and what students should take away from it. In addition, the description includes details on implementation, including potential student responses (e.g., posed problems), ideas about how the teacher could deal with those responses, and specific reflections from experiences with implementing the lesson. The fourth component is an overall reflection and summary of how the lesson fostered students’ mathematical understanding and what other teachers might want to pay attention to when using the lesson. The teaching cases (and all four components) are iteratively and continuously improved as the lessons (and units) are repeatedly implemented so that they embody the best of what the teachers and researchers learn as they work towards refining the lessons and units.

The teaching cases are dynamic physical artifacts that store professional knowledge that comes from both theory for teaching and teaching for theory. Theory for teaching informs the elaboration of the mathematical learning goals, including helping to define what it means to understand the mathematical content in the lesson. In addition, theory provides useful perspectives for the cognitive analysis of the learning goals and content, such as specifying necessary prior knowledge and understanding that students will need to take advantage of the learning opportunities in the lesson (e.g., by drawing on a learning trajectory) and identifying common misconceptions that students may develop. Moreover, theory for teaching provides explanatory power (possibly specific to the context) for reflecting on how the lesson fostered students’ mathematical understanding. Thus, theory for teaching is embedded in the teaching cases through multiple components as well as in the design of the lesson itself. At the same time, the teaching cases embody what is being learned through teaching for theory. Each time the lesson is implemented, there are opportunities to test small, local hypotheses about how attributes of tasks or instruction may influence students’ learning in the particular context. Through teaching the lesson, teachers accumulate additional professional knowledge such as how students respond to tasks, what kinds of conceptions (productive or counterproductive) that students generate, and what teaching moves best make use of students’ responses to move the class towards the learning goal. Again, the teaching case provides a dynamic, tangible resource that can help store this knowledge gained from teaching for theory and, in turn, allow teachers and researchers to use that knowledge to extend theory for teaching.

Because they serve as a tangible carrier of theory for teaching and teaching for theory, the teaching cases are also natural mechanisms for sharing and disseminating professional knowledge beyond the immediate context in which they were created. Many of them have been disseminated widely through practitioner-focused journals in China, although a published teaching case is, as noted above, only a snapshot of the full, dynamic teaching case artifact. These journals reach teachers throughout China. The articles in these journals are typically lesson focused with analysis of real teaching so that teachers who read the articles will be able to visualize what the lessons look like in practice. Moreover, the teaching cases have also provided the foundation for further development of theory in research-focused journals. For example, the development process that led to one teaching case has also led to the further development of theory for teaching—specifically, an analysis of the factors that are critical for implementing new pedagogical approaches (Cai & Hwang, 2021b).

5.2 Features of Artifacts That Embody and Bridge Theory for Teaching and Teaching for Theory

Thus far, we have used the example of teaching cases in the context of Chinese mathematics education to illustrate how an artifact may serve as a tangible representation of professional knowledge by embodying the dynamic between theory for teaching and teaching for theory. Although teaching is a cultural activity, and the teaching cases described above are certainly rooted in the norms of Chinese mathematics education (Huang & Bao, 2006), we suggest that this dynamic can exist across different cultures and thus can similarly be embodied through an appropriate artifact. The teaching case is only one example of such an artifact. This prompts the question of what features an artifact must have to suitably embody and bridge theory for teaching and teaching for theory so that it can represent the ongoing growth of professional knowledge. In this section, we propose three characteristics that can exist across cultures and which seem to be necessary for such an artifact.

The first characteristic is that the artifact must be able to include both the operational details of teaching and the principles that guide those details in ways that are interpretable by both teachers and researchers without extensive translation. To be useful to teachers, the ways of teaching that are captured in the artifact need to be accessible and directly applicable to teachers’ practical work. As with the task-by-task descriptions in the teaching cases and the information about students’ responses, an artifact that supports teaching for theory needs to paint a clear picture of the procedural details of teaching—what teachers can do in their own classrooms to create the desired learning opportunities. However, those details and procedures are not arbitrary. They are guided by principles—the theory that motivates the choice of actions. Clear explanations of how and why particular actions should produce the desired learning opportunities enable both teachers and researchers to make informed hypotheses that they can test through teaching (i.e., teaching for theory).

A second critical characteristic is that the artifact must be able to evolve over time. As we noted, both theory for teaching and teaching for theory are dynamic processes, not static objects. Thus, an artifact that embodies them and their relationship cannot be static either. For observers of lessons and for the teachers themselves, once an activity is implemented, a lesson is taught, or a unit is completed, there must be a way to capture what is learned from that teaching, both the practical knowledge and the theoretical advances, and to revise the artifact so that it carries the history of learning. Without this feature, the artifact cannot support the further development of either theory for teaching or teaching for theory.

The third necessary characteristic is that the artifact must be sharable. Expertise and experience that is entirely bound up in a local context does not ultimately contribute to the wider knowledge of the profession. But there are simply too many possible problems of practice to solve to count on every local context to individually address every problem. Sharing the work of building professional knowledge and solving the problems of practice allows the profession to make shared progress. However, because the artifact is the embodiment of the dynamic processes of theory for teaching and teaching for theory, requiring that the artifact be sharable means that both of those processes must also be designed to be sharable in some sense. Of course, there are local aspects of the theory for teaching and teaching for theory that are necessarily rooted in the context in which the theory and practice were developed. However, by including in the artifact information about what aspects of the local context seemed to be important for the success of instruction and what aspects were not so important, it is possible to allow others who use the artifact to generate their own hypotheses about what will work in their own local context. Some aspects may be universal, such as a lesson needing to have a clear way for the teacher to understand the students’ thinking during the teaching process so that the teacher can make adjustments based on different students, countries, or textbooks.

6 Future Directions for Research: Spiralling Up the Two-Way Street

In the future, in what ways might it be possible to create a more “comprehensive” theory of teaching? Given a system oriented towards artifacts that embody theory for teaching and teaching for theory, what would it mean for theories of teaching to evolve to be more comprehensive? Following the characterization we have given of theories of teaching, we take it to mean that a theory of teaching grows in generality to accommodate differences between subject matter, grade levels, and cultural aspects and grows in connection to other theories of teaching. Growing in generality means that although a theory should span these different areas, we have to keep in mind the specific character and requirements of each of them. For example, the level of higher order thinking between elementary and secondary students is not the same, but the theory of using higher order thinking should still be adjusted to fit the needs of the students. Growing in connectedness means that we should strive to find commonalities and parallel ideas across theories of teaching. For example, despite the seeming lack of overlap between Confucian and Western modes of learning, there may be areas of connection. Zhao (2013) identified four areas of overlap between Confucian concepts and other theories of education such as those based on Dewey and Freire, suggesting areas for cross-cultural integration of theories: “mutual learning, integration of theory and practice, importance of reflection in teaching and learning, and democratic purpose of education” (p. 9). Moreover:

Despite the differences between Confucius and critical educators, due to vastly different social contexts, there exists a strong resemblance between the two in terms of integration of theory and practice, reflective teaching and learning, teachers as learners (mutual learning) and social transformation. (Zhao, 2013, p. 23)

Ultimately, although we believe that the theory of teaching can become more comprehensive, we continue to stress that there is a two-way street. Thus, theory keeps evolving along with teaching, and we do not anticipate there will ever be an end-all, be-all comprehensive theory for teaching. Rather, as teaching and theory co-evolve, we anticipate continuous improvements in both.

In describing the role that an artifact can play in supporting the mutually reinforcing activities of theory for teaching and teaching for theory, we have drawn on ideas similar to those of others who have highlighted the potential role of artifacts or instructional products to act as a central focus for the work of educational improvement (e.g., Cai & Hwang, 2021b; Huang & Bao, 2006; Lewis & Tsuchida, 1999; Morris & Hiebert, 2011; Rothkopf, 2009). However, much more work is still needed to address critical questions about how such artifacts can be conceptualized, developed, and used more broadly. Our aim here is to call the field’s attention to these questions. We suggest four specific directions where further work is needed: (a) conceptualizing the construct of artifacts more precisely and in greater detail; (b) understanding the mechanisms by which partnerships between teachers and researchers can work productively; (c) exploring the wider impacts on instruction and students’ learning when researchers and teachers engage with artifacts that embody theory for teaching and teaching for theory; and (d) understanding how artifacts such as the ones we have described fit, practically speaking, into the complex ecosystem of existing curriculum materials, guidelines, and resources.

Although we have highlighted the example of teaching cases in China and discussed features that potential artifacts must have to successfully embody the dynamic relationship between theory for teaching and teaching for theory, we do not claim that we have fully conceptualized or characterized the artifact as a construct. We have merely sketched an outline of how to make the two-way street a productive reality. We believe that there remains much work to better define the essential elements of artifacts that serve this purpose in and across many different contexts. What other features are necessary characteristics? For example, how should such artifacts embody learning over time—the steady accumulation of professional knowledge without losing “institutional memory”—in a way that still allows for sharing that learning across contexts? In other words, what does the artifact have to be like to reach and connect a broader set of researchers and teachers engaging in theory for teaching and teaching for theory? The teaching cases described here begin to move in that direction through publication in practitioner-focused journals, but this is still a somewhat haphazard way of broadening the base of professional knowledge. Not every teacher who needs to will encounter the relevant teaching case for their situation.

Because the dynamic relationship between theory for teaching and teaching for theory is based on the assumption of close collaboration between teachers and researchers, the mechanisms for such partnerships also need to be better understood (Kilpatrick, 1981). What are the characteristics of productive partnerships, and what are the conditions needed to support their work? Cai et al. (2018a, 2019) have described how the roles of researchers and teachers might need to be reconceptualized and how alternative research pathways might be needed for the work of teacher–researcher partnerships to be fully developed. Fundamental changes to incentive structures and institutional norms could encourage the productivity and longevity of such partnerships. Ultimately, the potential of teacher–researcher partnerships to improve instruction and students’ learning may depend on attending to many factors, including cognitive, affective, and structural considerations (Cai & Hwang, 2021a, b).

Indeed, another area for future work is to understand and measure the potential of this kind of work for improving teaching and learning. How will collaboration around artifacts actually effect change? To what degree will incremental accumulation of professional knowledge improve the kinds of instructional decisions that teachers and researchers can make when planning and implementing instruction? Because change (and improvement) is likely to be incremental and slow, it is likely that longitudinal studies will be needed to analyze how collaborative work around artifacts—that is, engaging in the two-way street of theory for teaching and teaching for theory—actually affects how teachers teach and how and what students learn.

Finally, any attempt to embody theory for teaching and teaching for theory in an artifact will intersect with existing elements of curriculum. There is an abundance of curriculum materials, including textbooks, teachers’ guides, supplemental resources, and online resources. How will a shared artifact that both includes curriculum (e.g., by documenting lessons) and embodies a great deal of additional work around curriculum fit into this landscape? What are the practical considerations for teachers who wish to engage with these artifacts in addition to or alongside their existing curricular resources? If the field is to pursue theories of teaching that continuously evolve through artifacts that embody the two-way street of theory for teaching and teaching for theory, these and many other operational aspects of engaging in this kind of work will need to be systematically explored.