Learning metaphors and classroom enactments: Understanding webs of association and their entailments for school mathematics

Abstract

Oriented by the observation that cross-cultural comparisons of mathematics teaching have had little impact on the evolution of classroom practice, we develop the thesis that approaches to school mathematics are rooted in metaphors and sustained by associations that are language specific. We examined distinct webs of meaning for the words “learn” and “teach” in three languages: English, Czech, and Japanese. By applying a tool designed for analyzing bodily classroom (inter)activity, we aim to demonstrate that the meanings of these words are evident in practices that are readily observed in video recordings of mathematics lessons conducted in these languages. We suggest that efforts toward “transcreation” rather than “translation” are better fitted to hopes of interpreting and informing classroom practices across languages.

Cross-cultural comparisons of mathematics teaching have been prominent in the mathematics education research literature for decades, owing in no small part to the video-based studies associated with the Trends in International Mathematics and Science Study (TIMSS; Stigler et al., 2000) and the Learner's Perspective Study (LPS; Clarke et al., 2006). Most such projects have been concerned with comparing classroom methods and emphases from different parts of the world. Although one of their consistent conclusions has been that mathematics classroom practices are culturally specific, based on our work with practicing teachers over the past quarter century, we suspect that these studies are most often received within the teaching profession as recommendations to adopt or abandon specific classroom practices.

However, it is unclear whether many resulting insights into effective teaching and learning have been broadly embraced or effectively translated across settings. In this writing, we present a possible reason for this lack of uptake by focusing on differences and incompatibilities in meaning of words that are typically translated to and from the English “teaching” and “learning.” Specifically, we contrast the distinct webs of association in English, Czech, and Japanese as we explore the suggestion that subtle, language-rooted influences are at play in mathematics classrooms, enabling and constraining possibilities for (inter)actions of teachers and students. We situate these assertions amid preliminary findings from an ongoing study that brings together two distinct projects. The first revolves around examinations of distinct linguistic framings of “learning” and “teaching” in different languages, and the second is concerned with the development of a fine-grained analytic tool to track and interpret classroom dynamics in video-recorded mathematics lessons.

As we explain, the blending of these two projects was triggered by the coincidence of both having addressed topics in English, Czech, and Japanese. We introduce some of our preliminary observations and interpretations within and across lessons conducted in those languages, noting that our purpose is not to “demonstrate,” but to draw attention to some as-yet underexplored questions around translations of educational practices—not across cultures, but between linguistic frames. In our closing discussion, we offer that efforts to transpose educational practices from one linguistic frame to another are necessarily fraught, but not necessarily impracticable.

Owing to gaps in the available data, we acknowledge that aspects of this discussion are speculative. Our project is to provoke, not to prove. We end the writing with a discussion of this purpose.

1. Overview

Our orienting concerns for this discussion are unproblematized translations of the words “teach” and “learn” to and from English within the school mathematics research literature. We argue that such translations sometimes operate in ignorance of the grander webs of association that afford the particular richnesses of meaning in English and other languages. We worry that uncritical translations might obfuscate subtle and significant differences in educational practices across cultural and linguistic settings.

We situate this work in translation studies, a domain that draws on history, linguistics, cultural studies, philosophy, and computer science, among other disciplines (Munday, 2008). As might be expected, given this range of influence, there are diverse perspectives within translation studies. However, to our reading, all seem to be articulated in relation to the construct of “equivalence”—that is, taking an idea expressed in one language and representing it faithfully in another. In contemporary translation studies, expectations of equivalence are seen by most as problematic because, as Sfard (2008) explained, “discourses in one language may sometimes be not fully ‘isomorphic’ with discourses in another” (p. 100).

Phrased differently, the metaphor of “translation” is troublesome. Derived from the Latin trans + lātus “carried across,” the word is grounded in a presumption of equivalence. In some usages of the word, that presumption is justified. For example, within transformational geometry, equivalence is preserved across translation. However, with regard to what goes on when attempting to communicate across languages, the past half-century of translation studies has been developed around the realization that the metaphor of translation—that is, of “carrying across”—is a poor one.

Attentive to this issue, scholars of “cultural translation” (e.g., Bhabha, 1994; Lefevre, 1992) have offered a different metaphor, transformation, according to which the negotiation of meaning across languages is not a matter of carrying across (translating) that meaning by exchanging corresponding sets of words. Rather, the process involves attempts to activate correlated-but-never-corresponding webs of association. That is, cultural translation asserts that what is commonly perceived as an act of translation is actually an exercise in reformatting associations that are coherent in one language into spaces in another language that may be coherent, but that might also be rife with conflict and demanding of adaptation, negotiation, or other transformative process (Bhabha, 1994). The priority in this act of transformation is not establishing coherence between languages, but of maintaining coherences within respective languages.

Thus oriented, the research reported here had two emphases, which we separate into distinct discussions. In Part 1, we identify and analyze the webs of association around the English words “learn” and “teach” and with the Czech and Japanese words that are most commonly translated to English as “learn” and “teach.” Those analyses are then synthesized into “transcreations”—that is, as elaborated below, they are reformatted in ways that we hope make their original meanings more available to English speakers.

In Part 2, we report on the testing of those transcreations. Working from the hypothesis that they should enable us to anticipate distinct patterns of (inter)activity in mathematics lessons conducted in different linguistic settings, we compared the expectations conditioned by our transcreation to the results of fine-grained analyses of classroom-based activities and interactivities.

2. Part 1: Transcreations of the work of learners

2.1 Theoretical frame

Research on the role of language in school mathematics is complex and diverse, and it includes the learner's linguistic capabilities (e.g., Barwell, 2018), the use of language in the mathematics classroom (e.g., Rowland, 1999), and the language of mathematics itself (e.g., Pimm, 1987). In a recent review of such research, Planas et al. (2018) reflected the complexity and diversity of foci and approaches by noting that language is increasingly viewed as inseparable from culture and that mathematics classrooms themselves are “configurations of discursive activity” (p. 197). These points align well with the research reported here, although our unit of analysis is not an individual's use of language, for example, but the linguistic structures that give rise to the discourses in which individual actions occur.

That is why we turn to “transcreation,” which is a portmanteau of “translation” and “creation.” Transcreation was originally proposed as a more transformation-focused alternative to the notion of “translation,” aimed at highlighting that “trading words” and “negotiating meanings across languages” are dissimilar processes. Transcreation is thus about adapting articulations from one language to another in manners that maintain their semantic and affective qualities (Gaballo, 2012).

The notion of transcreation aligns with a structuralist model of language (de Saussure, 1916), whereby the truthfulness of a statement is not a matter of how precisely it reflects “the way things are” but the extent to which it summons and contributes to an established web of expectations. Kelly (2010) made the same point more poetically in the suggestion that “ideas never stand alone.” Rather:

They come woven in a web of auxiliary ideas, consequential notions, supporting concepts, foundational assumptions, side effects, and logical consequences and a cascade of subsequent possibilities. Ideas fly in flocks. To hold one idea in mind means to hold a cloud of them. (pp. 44–45)

Through the twentieth century, this manner of thinking about thinking was strongly represented within the structuralist movement—which, in the part of the education world that we inhabit, was especially influential in the form of constructivist and constructionist theories. Exemplars include Piaget's (1970) genetic epistemology and the subsequent radical constructivism (Glasersfeld, 1995). Structuralist sensibilities also infuse the sociocultural theories of learning and knowing that rose to prominence among mathematics education researchers toward the end of the twentieth century. Those perspectives displaced the principal site of meaning-making, away from the individual and toward social and cultural structures. However, the orienting principle was the same: Coherence within such dynamic cognitive, social, and cultural structures is what is most important in matters of knowledge and knowing. Our interests in this writing are the linguistic structures in different languages that might give form to distinct teaching practices.

Once again, such sensibilities are well represented in the notion of transcreation. It is grounded in a similar assumption, that each language is a unique structure—that is, each comprises webs of association that are combined in a distinctive manner. An illustration might be useful here. Each of us, the authors, lives in Canada, where French and English are designated as “official languages.” That means that political messaging is routinely presented in both, affording many occasions for translation-triggered tensions. For example, in a recent election campaign, a political party represented its position on a current international rivalry with a billboard reading “Supporting Our Ally; Appuyer notre allié.” To many with limited knowledge of French, that juxtaposition of phrases was jarring—and we turn to Google Translate to explain this disquiet: At the time the billboard was up, the top translations for “appuyer” were “to lean on” and “to push on.” To a naïve reader, these notions might seem diametrically opposed to the English “support.”

Of course, this tension resolves by delving a little deeper into appuyer's web of associations. Other translations include “to rest,” “to put up with,” “to rely on,” and “to lean against,” along with “to support.” Across this cluster, there is a clear sense of reflexiveness and reciprocity, somewhat in contrast to English's more unidirectional notion of “support.” Concisely then, and drawing on Sfard (2008), “discourses in one language may sometimes be not fully ‘isomorphic’ with discourses in another” (p. 100). Moreover, a lack of equivalence can have significant consequences for understanding and action. As Sfard further noted, “speakers of different languages are also likely to differ in the way they do things or interact with one another” (p. 100).

This suggestion has been a focus of academic inquiry for more than a century now, with some accounts tracing the idea back to Plato (e.g., Koerner, 2000). Currently, there is near-universal agreement that the imagery and associations that are structured within a language must have some non-trivial effect on what primary speakers of that language think and do (Wolff & Holmes, 2010). ¹ As for the nature of these effects, according to Sfard (2008), an empirical study oriented by this insight would entail “multidimensional comparative analysis of discourse” (p. 100).

Regarding efforts to make sense of cultural distinctions in school mathematics, we see transcreations as necessary elements. In particular, we believe a focus on webs of metaphoric associations to be a critical tool of investigation. Our transcreations are thus informed by conceptual metaphor theory (CMT; Lakoff & Johnson, 1999). CMT looks at implicit metaphor as an essential aspect of human thinking. It is concerned with the irrepressible and automatic human tendency to interpret one “conceptual domain” (e.g., idea, cluster of related experiences, set of interdependent interpretations) in terms of another. For example, interpreting “school mathematics” in terms of “progress along a pre-defined trajectory” is commonplace in English, even though these two categories of activity have few literal similarities.

Metaphorical thinking is a subset of “analogical reasoning,” which encompasses “any type of thinking that relies upon an analogy” (Bartha, 2019). Studies of analogical thinking tend to spread a much wider net than inquiries oriented by CMT, it also includes formal and explicit uses of non-literal argumentation (e.g., based on similes, or allegory), critical interrogations of implicit associations (e.g., in science, or politics), and reasoning based on any type of non-logico-deductive conceptual association (e.g., metonymic, or paradigmatic). Thus, while we identify metaphors as the focus of this writing, the work is perhaps more properly described as an examination of analogical reasoning in the teaching of mathematics. That is, we engage with metaphor as both a ubiquitous implicit process in human thinking and as a productive explicit tool for rethinking.

CMT asserts that the implicit use of metaphor is not only common, but it is also consequential. One's language orients one's attentions, and there is empirical evidence that this is the case. On a basic level, auditory similarities among words are associated with different patterns of noticing across different languages (Chabal & Marian, 2015; Tanehaus et al., 1995). On a more complex level, Thibodeau and Boroditsky (2013, 2015) have demonstrated that metaphors play distinct and observable, but often covert, roles in reasoning by channeling attentions to chains of inference and plans of action. Significantly, shifts as subtle as changing a single noun can trigger major reconsiderations—an observation that Thibodeau and Boroditsky (2015) used to underscore that a metaphor “does not act alone. … [It] instantiates a knowledge frame that then can coerce other information … into the frame” (p. 14).

Metaphors orient attention, and they do not act alone. Advocates of CMT press these points further, asserting that metaphors also prompt action and serve as uncritical justifications for subsequent actions and interpretations (Lakoff & Johnson, 1999). Such insights orient our transcreations of “learning” and “teaching” across multiple languages.

2.2 Our methods for transcreation, and some cautions

Our strategy for making sense of and representing diverse webs of association involves multiple processes. Davis and Francis began this work by focusing on English. The first stage of their process was to examine etymological meanings (i.e., the original metaphors) of the words “learn” and “teach.”

Importantly, the initial reliance on etymologies should not be interpreted as having succumbed to the “etymological fallacy”—that is, the belief that a word's original meaning must be similar to its current meaning (Zenker, 2002). On the contrary, this work was grounded in the conviction that meanings evolve, sometimes in ways that separate current usages from original referents—as, for example, in the cases of “taping” sounds and “filming” actions. For us, then, etymological meanings offered a starting place, not a focus. At the same time, we were attentive to the possibility that aspects of original metaphors may persist—not necessarily as explicit meanings, but (more likely, as illustrated below) as influences in aggregates of activity.

The next stage in our efforts was concerned with uncovering broader webs of meaning by considering images and other words associated with educational structures and practices, especially those that resonated with the metaphors identified in the first stage. Then, in our final stage of the English-focused analysis, we attempted to re-present what we had found as a concise description in which both the implicit network of metaphors and their pragmatic influences were made explicit. For ease of comparison across languages, we elected to focus on the “work of learners” in this description.

The same basic process was followed for the transcreations of the Czech and Japanese words that are most frequently translated as “teach” and “learn.” These efforts were undertaken in collaboration with Boháč-Clarke and Takeuchi who, respectively, are native speakers of Czech and Japanese. Importantly, given the increasing movements—even pressures—toward internationalized standards in school mathematics (cf. Kaiser et al., 2002), we anticipated the work might be frustrated by cross-cultural influences on vocabulary and practices. However, as we revisit our closing remarks, we were pleasantly surprised: Across the three languages we examined, each appears to have maintained a strong and distinctive internal coherence.

2.3 English: The work of learners as “following the leader”

The English “learn” is derived from the Proto-Indo-European (PIE) root -lois “furrow, track.” ² That is, a track-based metaphor of learning seems to draw in and draw on conceptions of learning as moving along, making progress toward goals, getting somewhere, and so on. It might be tempting to argue that “learning is moving along a furrow” is an instance of what Rorty (1989) called a “dead metaphor”—that is, one that has been rendered literal through centuries of usage, losing all its original figurative meaning. That certainly seems to be true at the level of individuals; very few English speakers are likely aware of this original image.

However, on the levels of social institutions and collective enactments, this foundational meaning of “learn” is in no way forgotten. It is vibrantly maintained in a net of associations that entangles concordant metaphors of learners, teaching, teachers, and education along with structures and practices that define modern schooling. For example, a track-based conception of learning summons and supports a conception of the learner as an entity-in-motion along a trajectory—fitted precisely to the root of the word “student,” from PIE root *(s)teu—“pressing forward, pushing toward.” That learner may be quick or slow, ahead or behind, accelerated or delayed, making progress or losing ground, on track or lost, passing (from Latin *passare “to step beyond”) or failing (from Vulgar Latin *fallere “to trip”), motivated (from Latin motivus “moving”) or lacking agency (from Latin agere “to set in motion, drive forward”). This cluster blends seamlessly with a metaphor for knowledge as a field, area, or domain that is traversed by courses and curricula (both from Latin cursus “a running, a journey”) that have been plotted with basics (from Greek basis “that on which one steps”), benchmarks (literally, surveyor's points of reference), and grades (from Latin gradi “to walk, step, go”). Correspondingly, teaching in this frame is about leading, guiding, directing, and orienting toward goals.

Especially cogent here is the grounding metaphor of the English “teach.” Derived from the PIE root *deik “to point, show,” the word's original sense meshes with knowing the field, looking ahead, reviewing, orienting attention, and directing motion—senses that, like the original meaning of “learn,” are explicitly available to few but vibrantly enacted by many. Provocatively, these notions are fitted to both traditionalist and reformist/progressivist sensibilities, evident in the emphasis on “teaching as directing” in the former and on “teaching as guiding” in the latter. Although these two attitudes are sometimes cast as opposites in the “math wars,” in terms of grounding metaphors and surrounding webs of association, they are much more alike than different.

To this detail, an exercise that we have found useful in efforts to better understand implicit-and-enacted assumptions on “teaching” is to conduct internet image searches of “teach” and “teacher.” For every search engine we have tested, an overwhelming majority of the generated images revolves around acts of pointing. We have collected a few images that we feel are indicative in Figure 1 (Google.com, July 24, 2023), and we encourage readers to conduct similar searches. That is, the grounding metaphor of teaching is not buried in time; it persists, even dominates, in representations of modern English-moderated classrooms.

Figure 1.

Stylizations ³ of typical images of “teaching” from an English image search.

2.4 Czech: The work of learners as “familiarizing oneself”

The sorts of images that are associated with “teaching” can be surprisingly different when similar searches are conducted in different languages. For instance, when we did an image search using Czech Google (Google.cz, July 24, 2023; see Figure 2) to search for “učit,” the Czech word most often translated to English as “teach,” only one of the first 40 images actually included a teacher, and none of those 40 images involved pointing.

Figure 2.

Stylizations of typical images of “Učit” from a Czech image search.

Učit is derived from the Proto-Balto-Slavic au kī tei and Proto-Slavic *vykno˛/ “to acclimate, to habituate.” ⁴ As well, the Czech word most often translated as “learn” is the reflexive verb, “učit se,” which is more appropriately translated as “familiarize oneself.” That notion is strongly reflected in the results of corresponding image searches. Overwhelmingly, learners are pictured alone, engaging with a text or similar artifact, apparently getting used to the material placed before them. The intensely personal nature of this construct is also evident in some of its synonyms, such as “osvojit si,” meaning “make it your own,” “dospívat,” meaning “to grow up, to ripen,” and “maturovat,” meaning “to mature” (commonly used in the sense of “becoming prepared” for high-school-completion examinations).

Učit se, osvojit si, dospívat, and maturovat are more than descriptive. For students in school, the notions of “familiarize oneself,” “make it your own,” “grow up,” and “mature” also operate as personal imperatives and social requirements. They are about obligatory self-transformation—that is, the mastery of the knowledge and skills deemed necessary for the functional adult. As might be inferred from the expressions and postures in Figure 2, the onus for learning here is on the individual, who is afforded no choice in what must be mastered but must nonetheless become proficient to transition from a youth—a beginner, novice, entrant, fledgling, newcomer, apprentice—to an adult. Significantly, once that self-transformation has been achieved, a different word for learning, “studovat,” comes into play. Studovat is associated with pursuing interests and elaborating abilities, but such self-selected and self-directed learnings fall into a different category than the formal, formative, and self-transformative foci of učit se. (Importantly, učit se continues to be used in adulthood to refer to undeveloped competencies that are contextually necessary, such as learning to operate an automobile.)

2.5 Japanese: The work of learners as “imitating the master”

Japanese presents a further contrast. One of the words that means “to learn,” 習う (narau) shares the same pronunciation with 倣う(narau). 倣うmeans “to imitate or follow” and the hieroglyph of the Chinese character 倣 represented an act of compelling a child learn or imitate hand gestures according to the norms of society. ⁵ While 習うand 倣うare distinctively used, 習うcarries the meaning of “to imitate” as seen in the case of 見習うthat literally means “watching to learn.” Another word that means “to learn” is 学ぶ. This word can be pronounced both as “manabu” and “manebu.” When it is pronounced as “manebu,” it emphasizes the meaning of imitating or echoing others. Across these etymologies, there is an implication of at least two individuals (i.e., one to be imitated, and one imitating—or one to be watched, and one watching). Here, the teacher's role is to model or to offer a model, a skill, wisdom, or knowledge, and learning is the responsibility of the learner. This expert-novice structure is common across Japanese traditional arts—a detail that will be familiar to many westerners in the example of the tea ceremony. The meaning of the word for teaching in Japanese, 教える (oshieru) is inseparable from these meanings associated with the word for learning. Learning is understood to be about observing and imitating the teacher's model. At the core of the development of expertise are acts of watching, listening, and imitating. In terms of contemporary meanings and enactments, imitation still sits at the core of this web of association.

When we conducted image searches for the Japanese word for teaching “教える” and associated terms using Japanese Google (google.co.jp, July 24, 2023; see Figure 3), we were initially taken aback. At first glance, the results seemed reminiscent of the English searches, with many depictions of teachers in front of students and most involving pointing. Closer examination, however, revealed something quite different across the English and Japanese images. For the former, students’ gazes generally follow the teacher's point while, for the latter, learner gazes are much more often on the teacher than on whatever the target of the point might be. Indeed, when a Japanese teacher is pointing, the point often appears aimed at nothing in particular—more a pedagogical gesture than a directive act. Consistent with the web of associations, attention is more often on the person who is to be modeled than on a targeted object.

Figure 3.

Stylizations of Typical Images of “教える” from a Japanese Image Search.

As well, many of the images show the teacher holding what appears to be a pointer in the air, although it is not clear what, if anything, is being pointed at. That lack of target prompts the thought that this act might not be about pointing at all. While we cannot be certain, an alternate meaning might be found in another phrase that means “to teach”: 教鞭を執る, literally meaning “to pick up a teaching whip,” and actually including a character (教) symbolizing the gesture of holding a whip (Todo, 1965). Thus, the gesture might be a proxy for raising a whip. As illustrated by Naka Kansuke (1913–1915) popular autobiographical novel, 銀の匙 (Gin no saji; Silver Spoon), whips were used to discipline students in the not-too-distant past. While they may not be used anymore, the expression is still heard, and the associated sensibility is still vibrant—evident in regular news stories of in-class violence by teachers in Japan and the persistence of a demanding and alienating school culture (Yoneyama, 2012).

These details link to another distinguishing feature of the images returned from the Japanese search—namely, orderliness, which is presented in such elements as the carefully organized desk, the evenly spaced chairs, the consistently directed gazes, and the uniformity of activity. There is a sense of control, but not just the top-down variety. As Shimizu (2009) describes, a “good mathematics lesson” perceived by Japanese students is the space coordinated and co-constructed between the teacher and students; the space that affords students to carefully listen to the teacher as they synthesize diverse views that emerged during in-class discussions. Stated differently, the phrase “to pick up a teaching whip” and the accompanying posture of a raised “whip” perhaps operate more as prompts than threats. Such notions are integral to the imitating-the-master paradigm. That is, that monitoring of students’ bodies, along with consequent self-discipline and collective order, permeate this web of association.

2.6 Comparing English, Czech, and Japanese webs

Three distinct images arise from the etymological roots of the English words “learning” and “teaching,” and the Czech and Japanese words typically translated as learning and teaching—having to do with, respectively, students following a path under the direction or guidance of a teacher/leader, supporting individuals as they familiarize themselves with something new, and apprentices imitating the actions of a master.

Before turning to our analyses of different classroom enactments, it bears mention that the last two of these meanings share a consequential entailment: They place the onus for learning (i.e., familiarizing oneself, or mimicking the master) on the learner (i.e., the self-teacher, or the apprentice). In contrast, in English, the burden of effective learning falls on the teacher, who is tasked with knowing and showing the path. This contrast is consistent with a common conclusion of cross-cultural comparisons of educational attitudes. As Nisbett (2003) developed, there are clear tendencies among west-European sensibilities to cast the teacher as principally responsible for effective learning and to fault the speaker if meaning is not readily grasped. In contrast, as Nisbett reported, among those cultures in which east-Asian sensibilities prevail, responsibility for mastery of content tends to reside more with the student and the obligation to ensure clarity when meaning is unclear lies principally with listener.

It is interesting to note that such contrasts are consistent with the source metaphors for the words used to name the roles and responsibilities for the actors in classrooms. We turn now to a more empirical analysis of those actors and their actions, exploring the suspicion that foundational metaphors and their grander webs of meaning are associated with distinct patterns of (inter)activity in mathematics classrooms.

3. Part 2: Analyses of classroom (inter)activity

The blended study reported here arose in the fortuitous meeting of two projects that, initially, were never intended to intersect. While Davis and Francis were engaging with colleagues (including Boháč-Clarke and Takeuchi) in examinations of metaphors for learning and teaching in different languages, Towers and Markle were devising a strategy, dubbed “Bodymarking,” for fine-grained analyses and nuanced representations of bodily (inter)actions within mathematics classrooms. When the research teams became aware of one another's work, preliminary analyses of webs of meaning in English, Czech, and Japanese had been completed. Coincidentally, Bodymarking had been used to analyze TIMSS recordings of lessons in the United States, the Czech Republic, and Japan. While we initially perceived that concurrence as a happy accident, it has since proven serendipitous in the richness of contrasts offered across the three languages—doubtless rooted in the facts that English and Czech are on divergent branches of the Indo-European language family, and Japanese is in an entirely different family.

Bodymarking is concerned with the co-activity of classroom agents. That is, the method is explicitly about addressing embodied aspects of the sociocultural domain of teaching and learning, aimed as it is at making visible the cultural and social differences in nonverbal communication across different actors and spaces. ⁶

As the analyses unfold below, we link them to issues of language and metaphor. Hence, the two distinct approaches in the blended study—a focus on language and a focus on the body—serve to enable a close examination of the underlying social and cultural norms at play in classrooms in a way that would not be possible through either analysis alone.

3.1 Bodymarking

Towers and Markle are part of a team of researchers developing a suite of strategies to represent and interpret the coordinated and consensual actions of agents in mathematics classrooms (McGarvey et al., 2022). The overarching intention of this work is to find ways to ascertain and confirm when and how classroom communities can properly be characterized as “complex systems”—that is, collective knowers rather than collections of knowers.

Towers and Markle's (2022) contribution to that project is concerned with how bodies are oriented toward (e.g., gazing, pointing, turning toward) and around (e.g., gesturing, writing) artifacts and other bodies in the room. They refer to their strategy as “Bodymarking” to signal the combined intentions of identifying and interpreting—that is “marking” and “remarking on”—bodily (inter)action. Their categories, with descriptions, are presented in Table 1.

Table 1.

Bodymarking categories and color codes.

In their video analyses, Towers and Markle record observations for each of these strands at 15-s intervals for the duration of a lesson. Because the focus of these analyses is nonverbal patterns of collective action, as opposed to individual sense-making, the coding is done entirely without sound or subtitles. The absence of sound also enables each researcher to focus on the cadences of actions at the classroom scale. For each of the strands, except Shared Gaze (see Table 1), they record only the occurrence of a gesture, not its frequency. For example, there is no distinction made between a 15-s interval in which only one instance of pointing occurs and a 15-s interval in which there are many. If a gesture is observed, it is assigned a color-code to the relevant strand for that 15-s interval. The result of the coding process is a visualization in which the prevalence of each category manifests as a distinct strand of color.

A few additional remarks on some of the Bodymarking categories are relevant here. First, for both the Shared Gaze strands, an interval is coded if it is determined that the shared gaze occupies at least half of the interval—which is necessary because, of course, “gaze” is a continuous phenomenon. Fleeting shared gazes appear in almost every 15-s time interval, and so recording each such event would render Gaze strands meaningless. Second, although Boardwork is presented as a single strand in Table 1, Towers and Markle distinguish between student and teacher activity at the board in their coding. This distinction is preserved in the analysis below. Finally, as Towers et al. (2023) have detailed, Bodymarking's fine-grained and often-interpretive data analysis required an extensive process of inter-rater reliability, part of which involved the repeated coding of a selection of video segments until consensus was reached among multiple coders.

The quality of this sort of analysis, of course, is necessarily coupled with the nature and quality of the video data. The lesson recordings that we used reflect the idiosyncracies of the TIMSS videographers—or, perhaps more likely, the distinct webs of meaning in the languages of instruction. For example, in the US lessons, consistent with the notion of “following the leader,” the camera tended to follow the teachers’ movements throughout the classrooms and to focus on directions given to students. In the recordings of Czech lessons, the camera was frequently zoomed in on students working at the chalkboard, fitted with the notion that students were “familiarizing themselves” with the mathematical content. In the recordings of the Japanese lessons, the camera tended to alternate focus from teacher demonstrations at the board to student work, consistent with an educational culture developed around “imitating the master.” In all cases, the camera appeared to be tracking the locus of work in the mathematics classroom, which almost always involved writing, whether by teacher or student, and whether at the board or a desk. To some varying extents, these sorts of variations presented issues for coding some of the strands in Bodymarking. Shared Gaze, for example, relies on sustained, wide-angle shots of classroom activity, which are frequently not provided by the TIMSS video data. Even so, those data proved adequate for studying language-rooted patterns of activity, as we endeavor to show below.

3.2 Choosing videos from the TIMSS video study

The TIMSS video study comprised recordings of 53 eighth-grade lessons across seven countries. The original purpose of this archive was to provide “educators and policymakers a better understanding of how national, regional, and local policies related to curriculum and instruction are being implemented in the classroom” (National Center for Education Statistics, n.d.). More pointedly, the TIMSS video study was aimed at identifying effective practices, as Stigler et al. (2000) exemplified in one report comparing 50 lessons from high-achieving Japan and 81 lessons from modest-achieving United States.

Such a volume of recorded material might seem to present a problem when seeking to identify common or defining features of mathematics pedagogy in different countries. How does one select which lesson to highlight? How might one go about arguing and demonstrating the representativeness of a chosen lesson?

One matter that we rejected immediately as a criterion for video selection was topic of mathematical study. To be valid, our thesis that our transcreations should enable us to anticipate distinct patterns of (inter)activity in mathematics lessons must operate independently of specific concepts. As for the issue of representativeness of specific recordings, Stigler et al. (2000) helped to answer that question with an important observation:

Many U.S. educators are likely to be surprised to learn how similar teachers teach in our culture. What they previously saw as great variation in our classrooms will pale in comparison to how U.S. practices differ from others around the world. (p. 99)

We interpret this observation—that intra-nation differences among recorded TIMSS lessons “pale in comparison to” inter-nation differences—to indicate that those videos are adequate for our purposes here. In fact, the observation is anticipated by our thesis. We note, for example, that the metaphors “teaching as directing” and “teaching as guiding” are frequently presented as opposites within debates about school mathematics in the United States, even though each is anchored to the ubiquitous metaphor in English, “learning as path-following.” However, neither fits at all with our Czech transcreation of “familiarizing oneself.” Further, while both might be argued to be compatible with our Japanese transcreation, the imagined contrast between “directing” and “guiding” vanishes when considered in relation to “imitating a master.”

A further consideration for us in selecting videos for analysis was that creating and verifying a Bodymarking trace for a 45-minute lesson requires 5–10 hours of analysis. Our limited resources permitted comprehensive analyses of only two lessons from each country. So constrained, for consistency, we randomly selected the lessons indexed “1” and “3” in the TIMSS database for each of the United States, the Czech Republic, and Japan.

We turn now to the Bodymarking traces of the six lessons. While we include brief descriptions of each lesson, we do not dwell on anomalies or differences. That is because, consistent with the observation offered by Stigler et al. (2000), even though participants, locations, topics, and camera work varied, each trace is remarkably similar to its pair-mate—and remarkably different from all the other traces.

As well, for each pair of lesson traces, we offer commentaries that are focused on two criteria: Firstly, we draw attention to elements of each pair of traces that appear consistent with the web of metaphors around “learning” and “teaching” in the language under consideration. Secondly, we highlight especially sharp points of contrast between these traces and those of lessons conducted in other languages.

Figure 4.

Bodymarking Trace of Lesson US1, on graphing linear equations.

Figure 5.

Bodymarking Trace of Lesson US3, on exponent rules.

3.3 “Following the leader”: Two lessons conducted in English

3.3.1 US1

This 44-minute lesson on graphing linear equations consisted entirely of private, small group work. There was no whole-class public instruction. Instead, the teacher opted to give minilessons oriented around a list of ten linear expressions to small groups of three to four students. The teacher utilized student work as his instructional material.

3.3.2 US3

This 50-minute lesson was on exponent rules. Students were sitting at desks in pods, and the lesson consisted mostly of private, small-group work with some public whole-class instruction. This teacher did not model any solutions. Rather, the teacher spent time with small groups, provoking with questions about the new material. The lesson was consolidated for the whole class through multiple brief moments of whole-class instruction.

3.3.3 Commentary on the lessons conducted in English

For the English traces, the prominence of pointing is particularly noteworthy. As noted earlier, the original meaning of “teach” is “point.” In literal terms, teaching was originally about directing movements and orienting attentions, and that notion integrates seamlessly with path-based conceptions of learning and territory-based interpretations of knowledge. The extent of pointing in these traces also stands in contrast to lessons conducted in Czech and Japanese, presented below.

Another contrast between English and the other languages appears in Boardwork strands. There was little or none in the lessons conducted in English. At first, we were surprised with this result, because the teachers of US1 and US3 did, in fact, spend a good deal of time at the board (see Figures 4 and 5). However, as the process of generating the Bodymarking traces revealed, that time was spent gesturing, rather than adding or removing work. That is, the teaching in these lessons was much more about pointing at an already drawn map of the territory being covered than, say, working through a process (as in the Czech lessons, below) or modeling a process (as in the Japanese lessons, below).

Two other contrasts between Czech and Japanese lessons merit attention, which are developed in greater detail below. One appears in the Writing strand. Participants in US1 and US3 spent considerable time writing, but not much of it in a sustained manner. The other is a more global observation: We can discern no clear patterns of coordinated collective activity in these lesson traces.

3.4 “Familiarizing oneself”: Two lessons conducted in Czech

3.4.1 CZ1

This 45-minute lesson predominantly involved a concurrent mix of private and public work. Selected students worked through problems on the chalkboard and were expected to narrate their thinking for the rest of the class while fielding questions from the teacher. During this time, seated students worked individually in their notebooks, and they could choose whether to work alone or to follow along with the public work. This structure is evident in three tranches along the Boardwork (Student) strand of Figure 6. Throughout the lesson, the teacher rarely worked at the board. Instead, the instruction consisted mostly of reading questions out to students or verbally discussing solutions. The teacher never explicitly modeled a solution; rather, she had students work on problems and questioned them to encourage correct calculations, notation, and so on.

Figure 6.

Bodymarking trace of lesson CZ1, on exponents and square roots.

3.4.2 CZ3

This 45-minute lesson included a mix of public and private instruction, with students both working at the chalkboard at the front of the classroom and on their individual exercises. The teacher did little work at the board and never formally modeled any solutions. Rather, teacher-led instruction mainly involved reading questions and discussing solutions, with a significant emphasis on inviting students to talk their way through problems. In this lesson, the talking-through took on a collective call-and-response style.

3.4.3 Commentary on the lessons conducted in Czech

Despite the structural differences between the two Czech lessons (as seen in Figures 6 and 7), the Bodymarking traces suggest they were remarkably similar. For us, a distinguishing feature of these lessons was the long and sustained blocks of individual work by students, which appear in a combination of the Boardwork (Student) and Writing strands of CZ1 and the Writing strand of CZ3. Such activity is consistent with root meanings of the Czech words for teaching (učit, “getting used to”) and learning (učit se, “familiarizing oneself”), which place the onus on the learner.

Figure 7.

Bodymarking trace of lesson CZ3, on exponents.

As for the teachers’ actions, there was very little pointing during the Czech lessons. Indeed, pointing was for the most part pointless in these lessons, as teachers usually positioned themselves on the side of the classroom or behind learners while individual learners, for the most part, attended to their own independent work. In fact, it was not always obvious to us who or where the teacher was in these lessons, very much in contrast to our experience with watching the classes in English and Japanese. We also see this feature as consistent with the Czech web of associations. If learning is about teaching oneself, one might expect that the designated teacher would routinely fade from center of attention.

None of this is to suggest that the lessons were not under the teachers’ control or that the teacher offered no instruction. On the contrary, classroom activity was clearly structured by the teachers who routinely inserted information. That information was often recited by the teachers, who read it from textbooks cradled in their arms. The teachers’ actions appeared to us to be about setting the boundaries for learners’ self-directed activity.

3.5 “Imitating the master”: Two lessons conducted in Japanese

3.5.1 JP1

This 50-minute lesson was focused on angles. The first half was a teacher-led review of previously learned material, during which students were seated at individual desks, working on problems posed by the teacher. The second half was dedicated to small group work on a set of six related problems, during which students applied the concepts from the review portion to novel contexts. Throughout both parts, the teacher modeled and compared different strategies and solutions, principally while working from the chalkboard.

3.5.2 JP3

This 52-minute lesson was on solving inequalities. Over the course of the session, students worked on two problems, one for approximately 45 minutes and the other for approximately 8 minutes. Students in this classroom were frequently invited to work on the chalkboard in front of their classmates. Attentions in the classroom were oriented to that public work, as individually seated students followed along closely.

3.5.3 Commentary on the lessons conducted in Japanese

For us, the most distinctive property of these two lessons is the coordination of participant activity, which is perhaps most evident in the combination of Boardwork and Writing strands of the traces (as seen in Figures 8 and 9). For the Japanese lessons, the contents of those strands rarely overlap, indicating collective synchrony—or, more accurately perhaps, of ordered, disciplined, controlled (inter)activity. Students were attending to the teacher when the teacher was writing, and the teacher was attending to students as they were writing.

Figure 8.

Bodymarking trace of lesson JP1, on finding the value of an angle.

Figure 9.

Bodymarking trace of lesson JP3, on changing shape without changing area.

Other categories of coordinated activity in the lessons conducted in Japanese can be readily discerned through row-by-row comparisons of the Bodymarking graphs, such as the predictably tight correlation between Hands/Body and Manipulating Tools in the JP3 trace (Figure 9). We find these remarkable, both because they are consistent with conceptions of teaching and learning that are about modeling and imitating and because of the dearth of collective synchrony in lessons conducted in English and Czech.

These traces also reveal that Japanese teachers did far more work at the board than their American and Czech counterparts and that students in Japanese-speaking classrooms were afforded extensive opportunities for uninterrupted engagement under the watchful eye of their teachers—consistent with teaching-as-modeling and learning-as-imitating metaphors. Looking at the same strands in the other analyzed lessons, it is apparent that students in Czech-speaking classrooms had comparable opportunities for extended engagement. However, that activity was not strongly coordinated with teacher activity, suggesting student work was not closely monitored by teachers—consistent with teaching-as-familiarizing and learning-as-self-teaching metaphors. In the English-speaking classrooms, there were also substantial opportunities for student engagement, but the Bodymarking traces reveal those opportunities were fragmented and routinely interrupted. The teachers were present and actively directing attentions throughout—consistent with teaching-as-pointing and learning-as-path-following metaphors.

Finally, the lessons in Japanese involved decidedly more pointing behaviors than the lessons in Czech and markedly fewer than the lessons in English, as we anticipated based on our analyses of webs of metaphor.

3.6 Checking our work

As we describe above, insights from CMT tell us something about the flocks of association at play around the etymological roots of the principal word for learning. In English, the original metaphor for learning is about following a well-defined path, and teaching is associated with keeping students on that path. In Czech, learning is about habituation, and teaching is associated with supporting students’ habituation. Finally, in Japanese, students and teachers take on the roles of apprentice and expert, respectively. Learning consists in imitating what is modeled by expert teachers. Although the Bodymarking analysis focuses on only a handful of lesson recordings and, as described above, those recordings presented some challenges in terms of analysis, we argue these linguistically grounded webs of association are manifested in the visualizations. The most prominent strands in the US (English) lessons are Pointing and Writing, consistent with the track/point metaphor. In the Czech lessons, the web of associations manifests in part through what is not present: although the Writing and Boardwork (Student) strands are prominent, Pointing and Boardwork (Teacher) are not. And in Japanese, while we see an abundance of pointing, similar to the US lessons, we also see the coordinated, non-overlapping bands of teacher activity and learner activity.

After completing the analyses presented above, we checked our interpretations in two ways: Firstly, when interpreting the results of the Bodymarking process, Towers and Markle often revisit intervals of interest in the mappings and their associated video segments (and vice-versa). Drawing on this approach, we identified intervals of interest in these six mappings in light of the country-specific strands discussed above—for example, Pointing in the US and Japanese lessons, and Writing and Boardwork (Student) in the Czech lessons—and returned to the relevant segments in the TIMSS video recordings. Secondly, we reviewed notes on lesson anomalies compiled by TIMSS researchers and attempted to assess the extent to which those anomalies affected (or did not affect) the Bodymarking traces.

The results of the first check are presented in Figures 10–12, which include two screenshots from each of the six lessons reviewed. At first glance, it may seem that none of the images appears unusual—that is, one could reasonably expect to encounter similar scenes in a nearby mathematics classroom. However, for us, layers of nuance are presented as we consider each cluster of images in relation to the webs of association in the languages of instruction. Consider, for instance, the presence or absence of pointing, the orientations of student gazes, the locations of the teachers, and the foci of teachers’ attentions across the four images in each cluster. Consider, as well, the similarities and dissimilarities between these clusters of images and the search engine image analyses presented in Figures 1–3.

Figure 10.

Screenshots of “typical instants” from lessons US1 and US3. ⁷

Figure 11.

Screenshots of “typical instants” from lessons CZ1 and CZ3.

Figure 12.

Screenshots of “Typical Instants” from lessons JP1 and JP3.

We are compelled by the ways that combinations of image search results, Bodymarking traces, and screenshots of typical instants seem to mesh so well with webs of metaphor embedded in the three languages examined. Not only do these results appear to lend credence to Stigler et al.'s (2000) assertion that “great variation in [U.S.] classrooms … pale[s] in comparison to how U.S. practices differ from others around the world” (p. 99), they embolden us to suggest that similar might be said of the “great variation” in classrooms in any country or lessons conducted in any language.

This conjecture is supported by the second of our checks, involving lesson anomalies identified by TIMSS researchers, which concerned such matters as lesson foci, ratios of private-to-public interactions, and amount of time spent by students on the chalkboard. ⁸ We examined the Bodymarking traces to assess how and how much they were influenced by noted anomalies. Our conclusion, as might be interpreted from the structural similarities of each pair of Bodymarking graphs, is that within-group differences are minor in comparison to between-group differences. That is, if any of the anomalies noted by TIMSS researchers affected our traces, it was only in inconsequential ways. While TIMSS commentators identified every lesson as in some way anomalous, our analyses suggest the two lessons in each pair were predictably similar—an observation that we believe bolsters our thesis that classroom (inter)activity is conditioned by the webs of association knitted into the language of meditation.

4. Discussion

4.1 Verifying and elaborating the research

We have already identified several constraints on the sort of research presented in this writing. For example, regarding the conceptual base, it can be difficult to mine original meanings of key words and to trace their many veins of influence. Fortunately, English, Czech, and Japanese have well-developed etymological dictionaries. For languages without such resources, this type of investigation may not be possible.

Regarding data analysis, to our knowledge, there are no methods or protocols designed to track and interpret language-specific influences on classroom activities. While the Bodymarking tool proved sufficient for our needs, it was designed for more general purposes. Moreover, it was designed and applied by English speakers, which presents two issues: Firstly, it may be implicitly biased toward English-based interpretations of life in classrooms—evident in the fact that Pointing stands alone as a principal category. Secondly, the tool may be better applied by speakers of the language of instruction, who might make different decisions than would an English speaker about, for example, what counts as “pointing.”

Regarding the data base, as already noted, there are issues with the TIMSS video archive, such as variations in camera angles and other factors that render it impossible to assess consistently where gazes are landing, what each actor is doing, and other aspects necessary for fully comparable traces across lessons and languages.

We are not the first to comment on such limitations. Indeed, for the designers of the LPS (Clarke et al., 2006), these concerns motivated the use of multiple cameras to glean information on teacher vantages, learner perspectives, and classroom overviews. They also prompted the decision that lessons would be analyzed only by native speakers of the language of instruction. Regarding the topics addressed in this writing, we interpret the former as a useful contribution, especially for mitigating the sorts of videographic inconsistencies noted early. That said, we see the latter point as risking an echo-chamber effect. We grant that nuanced information from experts on the specifics of classroom cultures is vital. However, as highlighted within CMT, expertise can be blinding. Its profound insights can come at the expense of unnoticed habits of interpretation and association—habits that are often more apparent to non-experts.

With these considerations in mind, we re-emphasize our opening acknowledgment that there is a speculative component to the reported work. At the same time, we feel that more robust analyses are not far out of reach. With more consistent video data, it would be straightforward to extend and verify analyses. Further, a level of validity might be generated by comparing traces from multiple countries that share a language. (For example, we have analyzed TIMSS lessons conducted in English from Australia, and the results were consistent with the U.S. lessons.) As well, it might be instructive to compare lessons conducted in related languages. For example, the web of associations in German is similar, but not identical, to the English web. The German “lernen,” for example, has the same roots as “learn,” but the word most often translated as teach, “unterrichten,” is derived from notions of straightness, rather than pointing. We do not yet know how the similarities and differences might register in lesson traces. Finally, we wonder whether the patterns of classroom activity that we have highlighted are present across all subject areas. Preliminary analyses of U.S.-based science and physical education classes suggest that the observed patterns are not specific to mathematics.

4.2 On the many metaphors for learning, and problem of ignoring the “dead” ones

We would be remiss were we not to address the fact that the path-following metaphor is just one among many employed by English speakers. As Davis and Francis (2021) have reported in their review of thousands of contemporary discourses on learning, other currently popular metaphors for learning include acquiring, encoding, connecting, constructing, construing, adapting, participating, and growing. The list goes on. As a team, we devoted much discussion to this matter as we assembled this report. Why, we wondered, does the path-making metaphor seem to exert such influence on structures and practices in English-speaking classrooms when there are so many other prominent metaphors for learning in play? And, assuming that there are similar diversities of interpretation in Czech and Japanese, why does classroom practice seem so tethered to the etymologies of the words that are most often translated to English as “learn” and “teach”?

Constraining our remarks to the case of English, we begin by noting the ubiquity of the path-following metaphor: It is prominent in everyday mentions of learning. It is assumed within virtually all formal discourses on learning in English (Francis et al., 2023). It is physically instantiated in many aspects of schooling. For the most part, these invocations and instantiations are implicit, and that implicitness takes us to a second observation. As mentioned earlier, the path-following metaphor is an example of what Rorty (1989) labeled a “dead metaphor.” It is no longer recognized—and, in some cases, no longer recognizable—as either figurative or descriptive. Rather, in most usages, learning is understood literally as making progress toward predetermined goals.

We feel compelled to highlight the irony here. For us, the path-following metaphor seems too present and too powerful to be characterized as “dead,” although we appreciate Rorty's (1989) intended sense of “not present to conscious awareness.” The irony deepens as we consider the other metaphors of learning mentioned a few paragraphs back. Each of those is less dead, in Rorty's terms. Compared to the path-following metaphor, they are much less pervasive, not as institutionalized, and, as Sfard (1998) demonstrated with the examples of “acquiring” and “participating,” much more readily interrogated and critiqued.

Thus, we offer the qualities of ubiquity and implicitness as reasons the notions of path-following and actions of pointing—that is, the original metaphors of “learn” and “teach”—are so much more obviously present in the Bodymarking traces of the U.S. (English) lessons than other popular contemporary metaphors. It is not that the original meanings have been lost; it is that the metaphors have been literalized in practice. We suspect that similar could be said of Czech and Japanese.

5. Conclusions

Much has been written over many decades about the difficulty of transforming teaching practices. For example, Cohen (1989) pointed to elements of educational systems—school organization, teaching conditions, and weak incentives—as major problems. More recent analyses (e.g., Ball, 2022) point to broader systems, linking the difficulties of change to social, cultural, political, and economic conditions. Stigler et al. (2000) also turned to differences in “culture” to account for why effective teaching practices tend not to carry over—that is, they do not translate well, in the mathematical sense, of preserving equivalence when shifting locations.

There is no denying the compelling influences of these factors. However, there is also no denying that, overwhelmingly, the reasons provided for difficulties in translation tend to fall outside educators’ control. Teachers have little agency across the considerations listed in the previous paragraph.

We might thus frame the foregoing discussion as an effort to restore some power to teachers. With regard to incorporating effective classroom practices from other cultures, we are convinced that transforming one's practice is not a matter of translating (copying) what someone else is doing. Rather, it is more a matter of transcreating. It relies on getting into others’ minds—specifically, learning about the tools of interpretation and the habits of association that render some activities sensible and others impractical.

To that end, consider this assertion: Educators in Czech mathematics classrooms do almost no teaching. Understood in the space of transcreation, the suggestion is that the predominant activities of Czech educators cannot be described as teaching/pointing; they do not fit the follow-the-leader model of teaching observed in our analyses of U.S. classrooms. That subtlety, it seems, is an easy one to miss for English-speakers. For example, Stigler et al. (2000) imposed the path-following metaphor onto lessons from around the world—evident in their mentions of making “progress,” “ways” of learning, “approaches” to teaching, “movements” through lessons, “following” plans, and “leading” students, to mention just a few of their repeated turns of phrase. Clarke et al. (2006) invoked similar phrases in framing their cross-national work, along with mentions of “entrenched” assumptions, varied “approaches,” student “progress,” and lesson “sequences” (from Latin sequi “to follow”). We acknowledge that we have done the same in this writing, despite deliberate effort not to do so. Importantly, these are not criticisms; they are observations. It would be debilitating to be continuously conscious of the metaphors invoked, even if it were possible. Put somewhat differently, while it is true that people have their languages, it is also the case that their languages have them.

We would thus argue that the familiar and seemingly banal observation that “math is taught differently in the United States, the Czech Republic, and Japan” is profoundly misleading. Such remarks focus attentions on common practices and deflects attentions from key meanings. We contend the critical detail is not that people are doing things differently in mathematics classes, it is that the phenomena associated with the English words “learning” and “teaching” are not the same phenomena as those associated with “učit se” and “učit” in Czech, or with “習う” and “教える” in Japanese. The notions summoned and the actions supported by these words are distinct.

To bring this detail into sharper relief, in another report that emerged from the TIMSS video study, Hiebert et al. (2002) concluded by pointing to “two formidable barriers” (p. 13) to significant and enduring change to classroom practices: “the natural cultural conservatism of both public schools and research universities … [and] … the proclivity of Americans to look for quick solutions” (p. 13). Doubtless, such inclinations matter. However, we suspect they are minor relative to ignorance of the root meanings that frame different enactments of school mathematics. To enact other practices, it is necessary to attend critically to webs of association that constitute the ground of current practice. That is difficult work, simply because conceptual metaphors are most powerful and resilient when they are farthest from conscious awareness (Lakoff & Johnson, 1999)—that is, after metaphors have become enactments (Varela et al., 1991).

Based on our analyses, we align ourselves with the conviction that efforts to translate educational practices from one cultural context to another are not just naïve, they are nigh on impossible in most contexts, as such transference would compel participants to engage in activity that may be linguistically incoherent for them. Embracing the notion of transcreation, we propose that efforts to adopt and adapt powerful classroom practices from other settings may be enabled by attending to broad webs of association that frame knowledge, learning, and teaching for the practitioners in both source and target settings. The issue, we argue, is not that entrenched practices are resistant to change, but that the structures that infuse and enable those practices are usually more subtle, pervasive, and robust than is commonly appreciated.