Guided Inquiry into a Physics Equation

Abstract

We present the theoretical argument that the use of mathematics in physics can be productively conceptualized as using a language and that learning to make sense of physics equations and appropriating them into novel scientific inquiry can be understood as a process of learning to read fluently with a high level of reading comprehension and to express one’s thoughts in writing. The data are drawn from a longitudinal ethnographic account of a high school student’s year-long research apprenticeship, in which a significant part of the work required the derivation and interpretation of an advanced physics equation (the phase velocity of waves in liquids). The analysis examines in different timescales the extended learning process as a complex activity system. The findings illustrate empirically how this type of learning can be productively analyzed by researchers and instructionally implemented by teachers or mentors as engagement in practices, a perspective that complements more standard cognitivist accounts of mathematical sensemaking. We highlight productive pedagogical and discursive instructional moves that facilitated this process and explicitly connect them to the learning they induced in terms of change in the student’s participation. While constrained by the scope of this case study, the concrete, microanalytic account of the learning process concretizes some of what physicists do with equations and suggests pathways to teaching students to meaningfully participate in this practice.

Physics equations often function as an object of inquiry in authentic scientific work. The uncovering of their “hidden” meanings leads to novel insights into the physical world (Dirac, 1984; Maxwell, 1990; Steiner, 1998; Wigner, 1960). The relations between components of a physics equation and the outcomes of mathematical manipulations on these components can be interpreted in ways that lead to the generation of explanations and predictions about the physical world (Gingras, 2001). Many physics students struggle to generate such interpretations (Meltzer, 2005; Pepper et al., 2012; Tuminaro & Redish, 2007).

This study examines a series of episodes drawn from a year-long videotaped participant observation of authentic laboratory sessions, in which a high school student worked on an open inquiry project that was mentored by his physics teacher, who acted in this context as a research mentor. A significant part of the work required the derivation and interpretation of an advanced physics equation (the phase velocity of waves in liquids). Students’ use of mathematics in physics has mainly been studied from a cognitive perspective (Kapon & Berland, 2023). In contrast, the current analysis examines an extended learning process as a complex activity system (Greeno, 2013). It focuses on the interaction between the student and the teacher research mentor in the meetings that laid the groundwork for the appropriation of the equation in the project. We present the theoretical argument that the use of mathematics in physics can be productively conceptualized as using a language and that learning to make sense of physics equations and appropriating them into novel scientific inquiry can be understood as a process of learning to read fluently with a high level of reading comprehension and to express one’s thoughts in writing. The analysis illustrates empirically how this type of learning can be productively analyzed by researchers and instructionally implemented by teachers or mentors as engagement in practices, a perspective that complements more standard cognitivist accounts of mathematical sense-making. We highlight productive pedagogical and discursive instructional moves that facilitated this process and explicitly connect them to the learning they induced in terms of change in the student’s participation.

Students’ use of mathematics in physics has been studied extensively in physics education research and the learning sciences. The section titled “Students use of Mathematics in Physics—A Literature Review” reviews the main insights from these studies. The section titled “The Study” explains how our work diverges from these previous studies and concludes with the research questions that guided the analysis. The section titled “Modeling the Learning Process” articulates the model that we developed to explain the complex learning process we witnessed. The empirical details of the study follow this theoretical introduction.

Students’ use of mathematics in physics—a literature review

Several studies have examined and suggested models for the underlying knowledge structure that affords the unique use of mathematics in physics. For example, Sherin (2001a) conducted a fine-grained analysis of physics undergraduates’ reasoning while engaging in problem solving. Sherin suggested that the knowledge system of successful physics students includes knowledge elements that he termed “symbolic forms.” Each symbolic form associates a simple conceptual schema with a pattern of symbols in an equation. Thus, their activation facilitates a deep understanding of the meaning of equations—a deep sense of what they have to say about the physical world. One example of a symbolic form is □ ± Δ (Base ± Change). Its associated conceptual schema is two terms that contribute to a whole but play different roles: one (□) is a base value and the other (Δ) is a change to that base. Sherin and other scholars have articulated lists of various symbolic forms (Jones, 2013; Meredith & Marrongelle, 2008; Sherin, 2001a). Sherin suggested that symbolic forms function like vocabulary elements for understanding the entailed physical meaning of algebraic notations. In a different study, Sherin (2001b) asked his informants to write a simulation for a falling object in the Boxer programming environment (diSessa et al., 1991). Based on a fine-grained analysis of their reasoning, Sherin (2001b) argued that the activated resources in this case were “interpretive devices”—action schemas that allow individuals to mentally run an equation for a particular purpose. He concluded that if programming language replaced algebra as the principal representational system for physics instruction, students’ knowledge system would significantly differ.

The focus of Sherin’s studies (2001a, 2001b) was the characterization of the activated knowledge elements rather than their genesis. Izsak (2000) conducted clinical interviews with pairs of 8th grade students to trace the cognitive processes that lead to the development of symbolic forms. He showed his interviewees a physical device (a winch) and engaged them in solving a set of problems related to this device, which required the use of algebra. The interviews involved minimal intervention from the interviewer. A fine-grained analysis of one of the pairs’ discourse captured the construction of a particular symbolic form and highlighted two cognitive mechanisms involved its generation: “notation variation” and “mapping variation.” Notation variation “consists of experimenting with and adjusting the number of terms and the placement of plus and minus signs in algebraic representations” (p. 36), and mapping variation “consists of experimenting with and adjusting correspondences between features of algebraic representations and attributes of the physical world” (p. 36). Note that Izsak was interested in the genesis of the knowledge structure at the cognitive level—the interaction with the interviewer was intentionally limited to the minimum possible. Note also that the students in Izsak’s study generated the algebraic representation for the winch from scratch. Hence, this assignment is less representative of physics learning in school, where students are often taught about existing algebraic models for a class of physical phenomena, must make sense of these models, and then must use and refine them to make sense of phenomena and solve problems in the physical world.

Several physics education researchers have drawn on cognitive linguistics to describe the interrelations students build between physics and mathematics. One of the models used to explain this process at the individual cognitive level is “conceptual blending” (Fauconnier & Turner, 1996). Fauconnier and Turner suggested that individual reasoning takes place in a network of mental spaces. Input counterparts are selectively projected into a new mental space—the blend—together with existing schemes. Inferences and novel conceptual structures emerge from the blend through composition, completion, and elaboration of its components. Physics education researchers who have examined undergraduate students’ use of math in problem solving sessions and interviews suggested that students ascribe physical meaning to mathematical representations by blending resources drawn from the mental spaces of mathematical machinery with resources from the mental space of the physical world (Bing & Redish, 2007; Hu & Rebello, 2013b). Another model that was used to explain this process at the individual cognitive level is known as “conceptual metaphors” (Lakoff & Johnson, 1999)—intuitive knowledge elements that capitalize on implicit mappings between a source domain (i.e., the physical world) and an abstract target domain, such as mathematics, often expressed in language. For example, Hu and Rebello (2013a) reported that their students viewed the mathematical term “dx” in a physics equation as a small segment of a bar, while the mathematical term “dq” was considered as a small amount of charge. Note that here too the unit of analysis is the students, and the interaction with the teacher is not examined at all.

Tuminaro and Redish (2007) focused their analysis on what physics students do when they engage in problem solving. Inspired by studies in the learning sciences, they described physics students’ engagement in problem solving as an engagement in epistemic games (Collins & Ferguson, 1993)—a set of goal-oriented activities that characterize disciplinary inquiry, which is shaped by a set of rules and strategies (i.e., game rules). The nature of these activities is epistemic since they result in a generation of an epistemic form—a legitimate representation of new knowledge in a particular field. Tuminaro and Redish (2007) examined undergraduate physics students’ authentic problem solving. They argued that students’ productive use of mathematics in these settings reflects engagement in two epistemic games. The first is mapping meaning to mathematics, an epistemic game that is characterized by the following sequence of moves: (1) develop a story about the physical situation, (2) translate quantities in the physical story to mathematical entities, (3) relate the mathematical entities in accordance with the physical story, (4) manipulate symbols, and (5) evaluate and interpret the story. The second is mapping mathematics to meaning, an epistemic game that is characterized by the following sequence of moves: (1) identify target concepts, (2) find an equation relating the target concepts to other concepts, (3) tell a story using this relationship between concepts, and (4) evaluate the story. Tuminaro and Redish argued that in both games, students’ particular mapping depends on the content and epistemic resources the students activate. Note however, that while their analysis explicitly focused on what students do as they solve problems, it did not attend to the nature of guidance the students received or their interactions with the teacher.

Several studies have shown that students’ mathematical sensemaking in physics is affected not only by their knowledge of physics and mathematics but also by their epistemological stance about physics and mathematics. For example, Hammer (1994) demonstrated how an epistemological belief that physics is a set of fragmented unrelated formulas rather than a conceptual and coherent knowledge structure, led an undergraduate physics student to ignore his discomfort with an algebra-based solution of a problem in mechanics that contradicted his common sense and stick with his initial incorrect answer. Gupta and Elby (2011) analyzed the mathematical sensemaking of an undergraduate physics student engaged in physics problem solving during a clinical interview, in which he reached and stuck with an incorrect solution that violated common sense. Their analysis demonstrated that their interviewee had the physics and mathematics knowledge resources to solve the problem correctly. They explained his behavior by his epistemological stance that physics equations do not express meaning that can tractably connect to common sense, and that in the context of physics problem solving, they are more trustworthy than common sense. The analytic approach in both studies was knowledge analysis that aimed to decipher the individual knowledge system that guides the student’s reasoning, whether content or epistemic knowledge. Gupta et al. (2015) reanalyzed key segments from the interview in Gupta and Elby (2011) original study, focusing this time on the interaction between the interviewer and the interviewee. The attention to the interaction dramatically changed their interpretation of their interviewee’s behavior. They noticed that the interaction between the interviewer and the interviewee generated and sustained an epistemological frame in which formal knowledge was foregrounded, and the interviewee’s role was to display this knowledge. Hence the epistemological stance that originally was attributed to the interviewee was revealed to have been socially co-constructed during the interaction with the interviewer. In our view, this revised analysis highlights the power of choosing a unit of analysis that includes the student and the teacher.

Several studies have highlighted the importance of guiding students’ mathematical sensemaking in physics. Karam (2014) contrasted two didactic approaches: “a technical approach, which involves an instrumental (tool-like) use of mathematics, and a structural one, focused on reasoning about the physical world mathematically” (p. 1). Building on an extensive literature review of the history and the philosophy of science, he suggested that physics instruction should reflect the structural role of mathematics in physics: “derive equations from physical principles using logical reasoning, focus on physical interpretations or consequences, justify the use of specific mathematical structures to model physical phenomena, connect apparently different physical assumptions through logic/…/, recognize profound analogies and common mathematical structures behind different physical phenomena/…/, mathematics seen as reasoning instrument, mathematics seen as essential to define physical concepts and structure physical thought” (p. 3). Karam illustrated these didactic principles through an analysis of lectures in an undergraduate course on electromagnetism taught by a physics professor considered to be an outstanding lecturer. Kuo et al. (2020) developed assessment items that assess undergraduate physics students’ mathematical sensemaking: “reasoning that leverages coherence between formal mathematics and conceptual understanding” (p. 1). They compared two groups of undergraduate students studying calculus-based physics: a control group that studied in a traditional way and two experimental groups that studied in a revised course that was explicitly designed to nurture the development of students’ mathematical sensemaking skills. Students in the experimental condition engaged more often in solving “crossover” problems which required the use of calculations on qualitative problems or the use of conceptual arguments on quantitative problems. While the performance of both groups did not differ on standard problems, the experimental groups did much better on the crossover problems. This study thus points to the importance of the guidance and learning environment in supporting the development of mathematical sensemaking in physics. Other studies support this contention as well. For example, Sawtelle and Turpen (2016) provided an ethnographic account of the experience of a biology student studying physics in a reformed algebra-based course for life-science majors. The course (Redish & Hammer, 2009) was designed to help the students learn to think scientifically: for example, by encouraging students to explain claims, equations, and principles clearly in their own words, or by explicitly encouraging them to search for coherence using common sense. The lectures and discussion sections involved extensive group problem-solving tasks that were designed to build connections between chemistry, biology, and physics. Sawtelle and Turpen (2016) presented an excerpt from an interview with the biology student, in which the student described how she actively sought to understand the meaning of the Gibbs free energy equation while stating that before the course she was not a “symbol person.” Levrini et al. (2015, 2019, 2020) conducted a series of studies that examined the discourse of high school physics students and their teacher while learning a reformed curriculum in thermodynamics. The curriculum and its classroom implementation were explicitly designed to support the development of students’ individual identities with respect to the content they were studying by foregrounding the “multi-perspectiveness,” “multi-dimensionality,” and “longitudinality” of physics. Students were engaged in explicit discussions that examined and challenged the ideas learned from the macroscopic and microscopic perspectives by underscoring the conceptual, experimental, formal implications and philosophical distinctiveness of each perspective. The students were invited to systematically compare the new models and theories they learned against the models and theories they previously studied. Levrini and her colleagues showed how the design principles of a learning environment coupled with a talented teacher who carefully orchestrated the classroom activity led to students’ appropriation of the formal ideas they learned, including a central equation in thermodynamics. Unlike the main body of literature we have reviewed so far, Levrini et al. (2015, 2019, 2020) and to some extent also Sawtelle and Turpen (2016), employed an analytic lens informed by sociocultural approaches to learning rather than cognitive approaches.

The study

We present findings from a study that examined a long-term learning process in which a high school student made sense of and came to appropriate a complex physics equation while working with a research mentor on an extended open inquiry project. The educational context of the study is particularly interesting since research apprenticeships provide a more disciplinary authentic context to study the learning of mathematical sensemaking in physics than problem solving or classroom instruction.

The research approach, and in particular our decision to examine the student, mentor, and the learning environment as a complex activity system, was deeply influenced by situated sociocultural approaches to learning. We were particularly influenced by Barbara Rogoff’s conceptualization of learning as taking place in three planes. The first plane, participatory appropriation (Rogoff, 1995) or transformation of participation (Rogoff, 1997), corresponds to the personal processes of learning though participation in a sociocultural activity. It concerns how individuals change through their involvement in the activity and in the process of becoming prepared for subsequent involvement in related activities. The second plane, guided participation (Rogoff, 1995, 1997), corresponds to interpersonal processes while participating in this sociocultural activity. It concerns the available directions for productive participation offered by social partners, as well as the cultural and social values that are communicated and coordinated in face-to-face and side-by-side interactions along with more distal organizations of people’s activities. The third plane corresponds to the community. It concerns culturally organized activities, where institutional practices and development extend from historical events into the present and are guided by cultural values and goals. Rogoff sometimes terms this plane apprenticeship (Rogoff, 1995). Since we use the word apprenticeship for the particular sociocultural activity we explored here (i.e., research apprenticeship in school), when referring to this plane, we explicitly refer to culturally organized activities.

Our study aimed to answer the following research questions: (1) What were the purposeful activities the student engaged in as part of the process of appropriating a complex physics equation into his inquiry? (2) What were the mentor’s instructional decisions and discursive moves that guided and shaped the student’s participation? (3) How did the student’s participation in the activities identified in RQ1 change over time?

Modeling the learning process

The modeling presented here was inferred and developed in an iterative bottom-up analysis that attempted to explain the learning process we observed. We describe it first since we refer to its tenets in the analysis.

Reading through a physics equation

We parallel the practice of attribution of a physical meaning to mathematical equations with the performance of highly sophisticated and generative reading comprehension (Gerrig, 1993). In the reading comprehension literature, some scholars have operationalized this performance by the kind of questions that the “performing” individual can answer based on his/her understanding of the text (Basaraba et al., 2013). Going back to the mathematics equation, this can be paralleled to the nature, scope, and depth of the explanations and predictions students can produce based on their understanding of an equation. Thinking in this way about students’ reasoning through physics equations affords a focus on what students can do with the equation, instead of the knowledge and mental models they activate as they reason with the equation. It also makes it possible to relate what they do to the kind of guidance they receive.

Early research on reading comprehension suggested that readers begin the analysis of texts by extracting basic units of meaning (Kintsch, 1974). In texts, these are words, sentences, and paragraphs. We suggest that in an equation these are (1) individual symbols, variables, and parameters (“words”), (2) components that combine symbols, variables, and parameters into a subunits of meaning (i.e., “sentences”), and (3) combinations of several components into one equation (i.e., “a paragraph”). For example, consider the equation that describes the maximum kinetic energy ($K_{\max })$ of an electron ejected from a metal due to the absorption of light. According to Einstein’s explanation of the photoelectric effect: $K_{\max }=h\nu -\phi ,$ where $h$ is Planck constant, $\nu$ is the frequency of the incident photon, and $\phi$ is the work function of the metal; namely, the minimum energy required to remove an electron from the surface of the metal (i.e., these are the “words”). The expression $h\nu$ in this equation can be considered a subunit of meaning in this equation, a “sentence” denoting the energy of the incident photon, whereas the whole expression $K_{\max }=h\nu -\phi$ can be considered a short paragraph.

Note, however, that carrying out complex reasoning with respect to a text often calls for integrating information in the text with broader real-world knowledge (Johnson-Laird, 1983; van Dijk, 1977). Jerome Bruner (Bruner, 1986) suggested that texts become stories when they force “meaning performance” upon the reader (p. 27). The literature on reading comprehension highlights three levels of performance, which are operationally defined based on the kind of questions about the text that the reader is able to answer. The parallelism between equations and texts that was described above can be extended to these levels as well. The basic level of extracting meaning from a text relates to the textually explicit information in the text and is defined as the ability to answer questions that require an understanding of the literal information explicitly stated in it (Basaraba et al., 2013; Pearson & Gallagher, 1983). For example, think of the sentence “the boy wore a blue shirt.” A reader who can extract the textually explicit meaning will be able to answer the question “what was the color of the boy’s shirt?” In the case of physics equations, this can be translated into understanding the meaning of all the symbols and parameters in the equation. For example, in the case of $K_{\max }=h\nu -\phi$ this means the ability to conclude that the maximum kinetic energy of the ejected electron depends on the frequency of the incident photon and the work function of the metal. We refer to this level of performance as extracting the textually explicit meaning of a physics equation.

A higher level of performance in reading comprehension refers to the textually implicit information in the text and is exemplified when the reader is able to answer questions that are not explicitly stated in the text but can be logically inferred from the information that was stated in it (Basaraba et al., 2013; Pearson & Gallagher, 1983). In the case of mathematical equations, this can be translated into inferences that can be implicitly extracted from the equation by mathematical reasoning, such as positive/negative dependences, examining special cases (e.g., x = 0), etc. For example, a reader can infer from the physics equation $K_{\max }=h\nu -\phi$ that the higher the frequency of the incident photon, the higher the kinetic energy of the ejected electron. This information is not explicitly stated in the equation but rather stems from the mental mathematical manipulation readers can perform on the equation, based on their understanding of its syntax and structure. We term this kind of performance extracting the textually implicit meaning of a physics equation.

The highest level of extracting meaning from a written text relates to the scriptally implicit information in the text and is defined as the ability to answer questions that require the reader to make inferences based on the integration of information presented in the text and their prior knowledge (Basaraba et al., 2013; Pearson & Gallagher, 1983). In the case of physics equations, this can be paralleled to inferences that require the integration of knowledge about the physical conditions to extract new meaning from the equation. For example, in the case of $K_{\max }=h\nu -\phi ,$ knowing that the kinetic energy of the ejected electron cannot take a negative value implies the existence of a threshold frequency of the incident photon. We term this kind of performance extracting the scriptally implicit meaning of a physics equation.

The adverb scriptally is borrowed from the literature on levels of reading comprehension (Basaraba et al., 2013; Pearson & Gallagher, 1983), and is based on the concept of “script” that is often employed in this literature to refer to the activation of schemas that embody knowledge of stereotyped event sequences (Abelson, 1981). Here we think of it as the activation of prior knowledge about the physical system in question or the entities that are being considered.

Note that the use of equations in physics often requires the “reader” to make highly sophisticated inferences based on deciphering their scriptally implicit meaning, particularly when approximations are involved. For example, physicists often calculate the electric field very far from a charged conductor by treating the conductor as a point charge, based on the premise that the size of the charged conductor is much smaller than the distance between the point in which the field is calculated and the conductor. In contrast, when physicists calculate the electric field near the surface of the same conductor, far from its edges, they consider the conductor as being of infinite size.¹ Such moves require the integration of prior knowledge of the system (i.e., scripts) with the textually explicit and implicit information extracted from the equation. Moreover, such processes often result in new equations that emerge from the original ones, which require additional deciphering of their textually and scriptally implicit meanings.

From a situative perspective, language scaffolds “the performance of action in the world” as well as “human affiliation in cultures and social groups and institutions through creating and enticing others to take certain perspectives on experience” (Gee, 2018, p. 136). If mathematics is the language of physics (Wigner, 1960), then the intentional and purposeful engagement in deciphering the textually explicit, and particularly deciphering the textually implicit and scriptally implicit meanings of physics equations are central practices of doing physics, reflecting the socially situated identity of physicists. Consider for example the words of the eminent physicist James Maxwell:

.… as we are engaged in the study of Natural Philosophy we shall endeavor to put our calculations into such a form that every step may be capable of some physical interpretation, and thus we shall exercise powers far more useful than those of mere calculation—the application of principles and the interpretation of results (Maxwell, 1990, p. 672).

This study investigates the process of learning to meaningfully participate in these practices. We examine how a student learned to read through a physics equation, to engage as a physicist in deciphering its textually and scriptally implicit meanings, so that the “words” and “grammar” of the equation “would line up to express a specific perspective on experience” (Gee, 2018). Note that meaning in language is anchored in experiences of situated action in the material and social world (Gee, 2002). Hence the analysis specifically examines the interactive, intersubjective dialogue between the student and his research mentor and the nature of the specific activities in which they engaged.

Anticipating mathematical formalisms through intuitive sensemaking

A critical review of a series of studies (Nathan, 2012) demonstrated the benefits of starting with concrete and applied knowledge and moving from it to the formalism. Studies in mathematics and physics education suggest that employing mathematics formalisms as a form of sensemaking is not a habit of mind that is developed spontaneously (Gupta & Elby, 2011; Leron, 2003; Leron & Hazzan, 2009; Nathan, 2012); rather scientists learn to employ them in their reasoning. We think of sensemaking as a process in which individuals construct and reconstruct self-explanations that evolve, change, replace one another, or merge into a new self-explanation (Kapon, 2017). Explanations for a physical phenomenon make sense to novice learners when they cohere with their intuition when they provide a sense of mechanism, and when they fit their social and epistemic framing of the situation in which the explanation was generated or given (Kapon, 2017). But scientists also learn to trust and rely on mathematical and computational rigor in their sensemaking (Fischbein, 1987). Moreover, physics equations often model phenomena or laws of nature in a parsimonious representation that sometimes encodes mechanistic explanations, in the sense that one can decode the entities involved, and the causal structure of the connections between them from the equation (Russ et al., 2008). Hence, we argue that physicists can represent their intuition quantitatively using formal mathematical representations (Sherin, 2001a), and when they make sense of an inference drawn from an equation, this coheres with their intuitive sense of mechanism.

Given the above, a proficient “reading comprehension” of a physics equation is built on an intuitive sense of the quantitative account of the behavior of the system or phenomenon that the equation models, and how it is represented in the equation. Appendix A describes these relationships and identifies the act of tentative predictions or explanations of quantitative trends as a component of mechanistic reasoning.

Educational context

In the last few years, the first author’s research group has been conducting ethnographic studies (see the Method section) in schools associated with the Acheret Center in Israel (ACHERET, 2005; Cohn & Cohn Snir, 2019; Cohn & Trumper, 2019; Gluck, 2015; Reich et al., 2009). Acheret aims to nurture the incorporation of authentic physics research experiences at the secondary school level. It associates several schools with diverse cultural and ethnic backgrounds. It trains and supports high school physics teachers in becoming research mentors in their schools. As part of their advanced physics curriculum in high school, all 11th and 12th grade students in the associated schools conduct an individual or paired long-term research project (12–18 months.) The students carry out their research in the school laboratory, guided by physics teachers who act as research mentors and perceive themselves in this context as such (Kapon, 2016; Kapon et al., 2018, 2021). During a mentorship session (3 h per week) a teacher mentors 1 to 3 projects simultaneously.

The teacher-mentor whose work is discussed in this paper, Mr. Shay Harazi, was chosen for this analysis due to the time and effort he invested in supporting his mentees’ use of mathematical modeling as a tool for sensemaking. We refer to him throughout the paper as “the mentor.” The students address him by his first name (Shay). The filmed mentoring sessions took place in a large school that serves mainly middle class secular Jewish families from the surrounding rural area. Ten working sessions (3 h each) were filmed throughout a full school year, as well as the external oral exam at the beginning of the following year. The analysis focused on the interaction of the mentor with one student who worked alone on a project entitled “Waves.” The student at the time was in the 11th grade. He was an avid surfer, who wanted to do a project on waves because he was interested in finding out “when and why there will be good waves.” The mentor was also the student’s teacher for his “regular” physics classes (10th–12th grades).

In 10th grade, the student was not considered a “good” student. The mentor, who knew him at the time only in his capacity as his physics teacher, described in retrospect the student’s behavior in class as “classic ADHD.” He said that the student was frequently distracted, disturbed the class, and was considered as unruly by all his teachers. His average grade in the 10th grade was 85, which is considered a middling grade in the first year of an advanced level physics class. In contrast, his participation in the project (11th grade) and the learning trajectory he exhibited were outstanding in terms of commitment, as was his performance in the experimental and theoretical aspects of the work. During that year and the year after, the mentor reported that he gradually became a “very good” student in the regular class in terms of behavior, and his academic achievement also improved significantly (10th grade: 85, 11th grade: 93, and 12th grade: 97). In retrospect, the mentor noted that thanks to the project he got to know the student, and gave him alternative assignments that challenged and interested him during the “regular” lessons when the class was working on something that he knew the student would find hard to concentrate on.

Methods

Research approach

The research method combines ethnography and fine-grained interaction analysis of selected episodes. Ethnography is the study of social interactions, behaviors, and perceptions in communities, organizations, and other groups (Reeves et al., 2008). Its goal is to document the culture, the perspectives, and practices of people in these settings, for the purpose of generating an understanding of the way members in the group, organization, or community in question see and interpret the world (Hammersley, 1992; Pole, 2003).

The study takes a situated socio-cultural approach to learning, based on the premise that studying cognition in its cultural and social context as a unit can better capture the properties of both culture and cognition (Vygotsky, 1986), particularly in the context of learning. The current study aimed to unpack the practice of learning to make sense and interpret an advanced physics equation and appropriate it into an inquiry, through participation in a specific research apprenticeship. This ethnographic account formed the basis for focused analyses of specific practices that can be linked, at least in principle, to cognitive development (Saxe, 1991; Saxe & de Kirby, 2014). At the micro-level we examine the learning process in selected episodes through a fine-grained analysis of the unfolding of the in-the-moment interaction between the student, the research mentor, the materials, and the informational resources in the learning environment. At the macro-level we used ethnography to situate the episodes within the year-long learning experience. We compare the findings from the micro-analyses of episodes across time and place to trace the development over a lengthy period of time of the culturally organized activities, the guidance provided to the student, and the student’s learning as expressed in the transformation of his participation in the inquiry.

The underlying assumption was that the guidance the student received unfolded during the moment-by-moment interactions between himself, the mentor, the available tools, the materials and informational resources, and the structure of the tasks, through talk moves and tasks that drew his attention to the relevant attributes of the equation, the phenomena and the nature of the activity. These attributes were not transparently available to the student, and thus required “disciplining” his perception (diSessa, 2002; diSessa & Sherin, 1998; diSessa & Wagner, 2005; Levin & DiSessa, 2015; Stevens & Hall, 1998) as well as semiotic mediation (Wertsch, 1984, 1985). Hence, an analysis that simultaneously examines the learning process at the micro-level (fine-grained, in-the-moment interaction) and macro-level (across time and place) can serve to articulate instructional moves and relate them to changes in the student’s participation in practices (i.e., learning).

Data collection

The analysis drew on various data sources.

Videotapes of three weekly working sessions

One central method of data collection in ethnography is participant observation (Jorgensen, 2015). The adjective “participant” is not incidental. To achieve familiarity with members of a group and their practices, the researcher needs to become intensively involved with them in their cultural environment, over an extended period of time. In practice, this means that we attended the working sessions as well as other activities at Acheret on a weekly basis, and became an integral part of this environment. We also formed close friendly relations with its management team and the mentors. The second author filmed the sessions that are used as data in this paper with a POV camera that was attached to her shoulder. This left her hands free to write notes or take part in the meeting.

We chose three videotaped sessions for the current analysis, that were spread over a period of 6 weeks. These sessions were chosen for transcription and discourse analysis because of their intensive focus on deciphering the meaning of the mathematical equation that became the basic model from which other models and explanations were derived. The analyzed episodes of discourse are drawn from these meetings.

Artifacts that were generated during the inquiry process

These include the student’s research proposal, his final research report, spreadsheets and documents that were produced in each session, the student’s and mentor’s written notes during these meetings, and pictures of the whiteboard. These artifacts were used to contextualize the transcripts (i.e., environmentally coupled gestures), and as a basis for the stimulated recall interviews.

Stimulated recall interview with the student

A stimulated recall interview (Shkedi, 2005) with the student was conducted after the student completed the project, submitted the final research report, and defended it in an oral exam. The questions were based on the final research report. The report was available to the student throughout the interview. The student was asked to relate and explain interpretations, graphs, and equations in the report, and describe in retrospect how these were developed.

Informal conversations with the mentor

While the student and the mentor worked on the project, as well as during the analysis and the writing of this paper, we had numerous discussions with the mentor. The goals of these discussions were to better understand a particular event or a decision he made in a session, to fill in missing information, or to verify a particular interpretation. These conversations took place before, after, or during the working sessions that were filmed or recorded, as well as over the phone and via email at different times. They were also documented as research notes. The mentor also read and commented on the manuscript.

Analysis

The analysis was conducted on the original Hebrew transcriptions. The episodes presented in this paper were translated into English during the writing of this paper. The transcription conventions are detailed in Appendix B. Speakers are denoted in the transcripts as follows: “S:” denotes student, “M:” denotes mentor, and “I:” denotes interviewer.

Inferring a learning process from an analysis of interaction

One of the key differences between cognitive-scientific and psychological informed studies of learning and situative informed studies of learning is how the level at which learning processes take place is perceived from each perspective. In the former approach (which has shaped most studies that have examined students’ use of mathematics in physics) learning is “hypothesized to occur at the level of individual information processing, thinking, and learning.” In the latter approach (which shaped the current study) it is “hypothesized to occur at the level of activity systems and joint participation in communities of practice” (Greeno, 2013, p. 43). As Greeno writes “constructive processes of understanding and reasoning occur in conversation through the joint actions of the participants.” (p. 44).

Unpacking culturally organized activities

The analysis aimed to articulate what the participants were doing and referring to. At the discourse analysis level, special attention was given to the interlocutors’ gestures (Goodwin, 2007b; Gregorcic et al., 2017; Kapon, 2015), particularly to what Charles Goodwin (2007a, p. 196) termed environmentally coupled gestures; i.e., “gestures that cannot be understood by participants without taking into account structures in the environment to which they are tied” (p. 196). Examples of “structures in the environment” in our context included the experimental apparatus, graphs or tables on the computer screen, written equations, etc. Each activity in the episode of interest was labeled with a descriptive title. The activities were compared and further contextualized using the information we gathered during our extended ethnographic studies at the Acheret Center (in the current study and others). During this process, their descriptive labels were refined.

The last phase of the analysis involved a theoretical reorganization of this list. The emergent list of activities seemed to reflect a process of reading comprehension. Thus we conducted a search on teaching and learning reading comprehension and came across a study that operationalized levels of reading comprehension (Basaraba et al., 2013). We found this study highly informative to our analysis since it seemed that we could hierarchically aggregate many of the activities that we identified into categories that analogically correspond with levels of reading comprehension. This sparked a reorganization of the activities and further refinement of their descriptive labels. A closer examination of the activities that could not be classified into this conceptualization of reading comprehension revealed that each could be related to engaging in mechanistic reasoning, and we used Russ et al. (2008) discourse markers of mechanistic reasoning to further refine the descriptive labels of the remaining activities.

Unpacking guided participation

The fine-grained discourse analysis of the selected episodes integrated several layers of fine-grained analysis of discourse, and attended to aspects of knowledge and interaction (Kapon, 2015). One layer searched for specific forms of the mentor’s recipient design; namely components of talk in a conversation, such as the selection of words and topics or the ordering of a sequence in a conversation constructed by the mentor with an orientation and sensitivity toward the student (Sacks et al., 1974). We particularly searched for recipient designs, or “talk moves,” such as probing, pressing, follow-ups, and re-voicing (Windschitl et al., 2018), through which the mentor might drew the student’s attention to particular aspects of the scientific work or try to prompt the activation of particular information or experience that he assumed the student may share. We examined the kind of questions that were asked, their timing and the way they were phrased, responses to questions and concerns raised by the student, etc. Other recipient designs we examined reflected how the mentor epistemologically framed the situation; namely, his moment-to-moment grasp of what was going on with respect to knowledge and learning (Hammer et al., 2005), and how he relatively positioned himself and the student.

Unpacking transformed participation (i.e., learning at the individual level)

We used various measures to infer transformed participation. The first was the chronological coding of the activities. However, engaging in a new activity is not an indication of learning in itself, since it may merely reflect the order of activities that the mentor facilitated. To infer learning, we searched for references in the student’s and mentor’s discourse to previous experiences that might suggest how previous experiences prepared the student to participate in the activity in question.

Another measure used to infer transformed participation was the changes in the ways in which the student positioned himself and his expressions of autonomy within a particular activity over time. This could be inferred from the discourse analysis by attending to the content and prosody of speech and the environmental gestures of the student. Examples of relevant discourse markers included who made the decision on the next move, the adverbs and pronouns used when the student discussed specific issues related to the inquiry with the mentor, expressions of affect, etc. It could also be inferred from explicit statements in the student’s retrospective account of his experience during the interview.

The third measure was the level of the student’s competent participation in a specific activity over time (e.g., follow orders vs. independent participation). While deeply influenced by Rogoff’s conceptualization of learning as taking place in the aforementioned three planes (Rogoff, 1995, 1997), our approach somewhat diverged from hers in the sense that we did not and (and could not) ignore individual cognition in the analysis of learning in interaction (diSessa et al., 2015). As shown in the findings, the mentor made assumptions about the student’s prior knowledge, and we highlight these assumptions in the analysis. Moreover, the student and particularly the mentor often used the verb “understand” in their speech, mainly in reference to their ability to explain.

Hence, the fourth measure we used to infer transformed participation was the level of scientific accuracy of the explanations and reasoning expressed by the student over time. Apart from analyzing the level of accuracy of the explanations, we also attended to the student’s explicit retrospective accounts of changes in his understanding during the interview.

Findings

A bird’s eye view of the project

Finding out “when and why there will be good waves,” is a complicated problem from a scientific perspective. After some negotiations, the student and the mentor decided together to focus on the influence of the depth of the water on the propagation velocity of the water wave. Since the student had not learned anything yet about waves, the mentor suggested that they should start by investigating a simpler problem first, and examine factors that influence the velocity of a propagating wave in a string. Equation (1)(1) $$v=\sqrt{\frac{T}{\mu }}$$(1) represents the mathematical model the student derived (with guidance). (1) $$v=\sqrt{\frac{T}{\mu }}$$(1) $v$—velocity, $T$—tension, $\mu$—linear mass density

The student designed and conducted an experiment with the mentor that put the model to test and found a reasonable fit (R² = 0.98). They hypothesized together that the small discrepancy stemmed from the fact that they had neglected the effect of gravity in the experiment. They did not continue to test this hypothesis since their target was for the student to learn the basic concepts related to the phenomenon of waves, and they wanted to get to the main part of the project: the study of waves in water.

The student and mentor decided together to focus the inquiry on the influence of the depth of the water on the propagation velocity of the water wave. Specifically, the project examined (1) the propagation velocity of waves at different depths, (2) changes in the propagation velocity, wavelength, and amplitude due to abrupt changes in depth (transmission from deep to shallow water); (3) changes in the propagation velocity, wavelength, and amplitude when the depth of the water changes linearly.

Some sessions involved experimental work, whereas others involved more theoretical work, but most involved both. The student and the mentor designed and built the experimental apparatus for each case. The initial experiment was run in a regular aquarium (∼0.5 m long), but all the other experiments were done in a larger aquarium (∼4 m long, ∼0.3 m deep) that was placed in the yard near the school laboratory. The oscillating disturbances were created manually and were timed using a metronome, and the generated waves were videotaped. The films were analyzed using Tracker software to measure wavelength, velocity, and amplitude.

Focused meetings

Here we examine the work related to mathematical modeling and focus on three meetings that involved deep sensemaking related to the central equation in the project: the phase velocity of waves in liquids (Equation 2(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) ). The student and the mentor referred to the phase velocity as “the propagation velocity,” or simply “the velocity,” and we use their terminology throughout this paper. The mentor and later the student referred to specific expressions in Equation (2)(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) as separate units of meaning (i.e., “gravity component,” “surface tension component,” and “depth component” –see Equation 2(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) ). (2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2)

• $\frac{g\lambda }{2\pi }$—“gravity component”

• $\frac{\sigma 2\pi }{\lambda \rho }$—“surface tension component”

• $\tanh \left(\frac{2\pi h}{\lambda }\right)$—“depth component”

• $v$—“propagation velocity” (phase velocity)

• $g$—gravitational acceleration

• $\lambda$—wavelength

• $\sigma$—surface tension coefficient

• $\rho$—density

• $h$—depth

In contrast to Equation (1)(1) $$v=\sqrt{\frac{T}{\mu }}$$(1) , Equation (2)(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) is not an equation that an 11th grade student can derive from scratch. The student technically derived Equation (2)(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) from the dispersion relation of waves in a liquid which the mentor and student found in Wikipedia (Equation 3(3) $${\omega }^2=\left(gk+\frac{\sigma }{\rho }{k}^3\right)\mathit{tanh}\left(kh\right)$$(3) ). The student had no idea what the term “dispersion relation” meant. The mentor introduced the basic relation of periodic waves (see the box of Equation 3(3) $${\omega }^2=\left(gk+\frac{\sigma }{\rho }{k}^3\right)\mathit{tanh}\left(kh\right)$$(3) ) into the equation and told the student to derive the expression for the velocity (i.e., Equation 2(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) ) from them. This technical derivation took the student <10 min. In contrast, deciphering the meaning of Equation (2)(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) , interpreting it, and starting to use it to formulate explanations and predictions took hours and days, building on extensive preparation in previous meetings. (3) $${\omega }^2=\left(gk+\frac{\sigma }{\rho }{k}^3\right)\mathit{tanh}\left(kh\right)$$(3) $\omega$—angular frequency $\omega =\frac{2\pi }{T}$

$T$—period $T=\frac{\lambda }{v}$

$k$—wave number $k=\frac{2\pi }{\lambda }$

• $g$—gravitational acceleration

• $\sigma$—surface tension coefficient

• $\rho$—density

• $h$—depth

Focus episodes

The student had to notice or infer three related physical properties that are explicitly and implicitly expressed in Equation (2)(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) : (1) the propagation velocity is independently influenced by gravity, surface tension, and the depth of the water; (2) in their experimental conditions, the influence of the surface tension was negligible relative to the influence of gravity; (3) the propagation velocity increases with depth, and reaches its maximum level in deep water.

In terms of reading through the equation, perceiving property # 1 in Equation (2)(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) mainly involves the extraction of the textually explicit meaning of Equation (2)(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) , but making sense of these relations is far from trivial. Inferring properties #2 and #3 also require extracting the textually implicit and scriptally implicit meanings of Equation (2)(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) . Moreover, developing an intuitive sense of these quantitative relations is very challenging. Due to space considerations, we only illustrate this process for property # 2.

The student started to work on the project in early October 2016. Here we analyze excerpts from three sessions in which an intuition as to the structure of Equation (2)(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) , and how to read through it was mediated. Session 1 (3 h) took place on December 5, 2016. In this session Equation (2)(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) had not yet been presented. However, the mentor was planning to employ it and spent a lot of time nurturing the student’s intuitive sense of the quantitative behavior of the phenomenon (i.e., nurturing the generation of tentative predictions or explanations of related quantitative trends.) Session 2 (3 h) took place on January 2, 2017. Two hours of that meeting were dedicated to a discussion of previous empirical measurements. Here we focus on the third hour, in which the equation of the dispersion relation (Equation 3(3) $${\omega }^2=\left(gk+\frac{\sigma }{\rho }{k}^3\right)\mathit{tanh}\left(kh\right)$$(3) ) was introduced to the student. Session 3 took place on January 16, 2017. Most of this session was devoted to the derivation and investigation of the different meanings of Equation (2)(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) . During this time the student was also working on his research proposal which was submitted at the end of January. The analysis draws on the transcripts of sessions 1–3, the submitted research proposal, parts of the final research report that related to this phase of the work, and a stimulated recall interview that the second author conducted with the student after the student had submitted the report and defended his findings (September 25, 2017).

A bird’s eye view of the analysis of learning in three planes

In the following section, we present a longitudinal microanalytic account of the learning process, which considers the student, the mentor, and the resources in the learning environment as interacting parts of a complex activity system. We do so through a fine-grained discourse analysis of selected episodes from the corpus, which are presented longitudinally. The analysis attends to the three planes suggested by Barbara Rogoff in conceptualizing learning as taking place in three planes (Rogoff, 1995, 1997): the culturally organized activities in which the student and the mentor engaged, the guidance the mentor provided for the students’ participation, and the transformation of the participation of the student. To support the readers in tracking these three planes in the longitudinal analysis of episodes, we start by a summary of the features of the learning process that were uncovered in each plane. Table 1 lists the practices in which the student learned to participate during the process of appropriating a complex physics equation into his inquiry. Table 1 also lists the coordinated purposeful activities that afforded the student’s learning to participate in each practice or sub-practice. Table 2 lists the mentor’s instructional decisions and discursive moves that guided and shaped the student’s participation. Table 3 summarizes the chronological changes in the student’s participation in the extended inquiry. The right column in each table lists the episodes in which every feature is illustrated.

Table 1. The socioculturally organized activities in which the student learned to participate.

Practices (1, 2), sub-practices (A, B, C), and related activities (#)	Episodes
1. Anticipating the mathematical formalism through intuitive sensemaking
1.1. Crafting mechanistic explanations (identifying entities/variables, relations between entities, using analogies, etc.(Russ et al., 2008)	1, 2a, 2b, 2c, 3
1.2. Generating qualitative descriptions of quantitative trends of the system (e.g., if the value of entity X increases then the value of entity Y decreases)	1, 3, 5
1.3. Hypothesizing a crude or metaphorical mathematical representation that qualitatively describes a quantitative trend (e.g., “$g$ should be in the numerator”)	1, 4, 5
1.4. Using dimension analysis to derive a physics equation up to the value of its constants	1
2. Reading through the equation
2.A. Deciphering the textually explicit meaning of the equation
2.A.1. Identifying the conceptual meaning represented by each variable and parameter in a mathematical representation	4, 6a
2.B. Deciphering the textually implicit meaning of the equation
2.B.1. Generating inferences based on a mental simulation of how a change in one variable affects the behavior of another variable/component.	7a, 7b, 9
2.B.2. Generating inferences based on analytical processing of mathematical representations.	9
2.B.3. Solving the equation/model for special cases (e.g., the calculation of the wavelength in which the gravity and the surface tension components are equal).	9
2.B.4. Using a numerical simulation of the equation to find quantitative trends in the behavior of the variables	11b
2.C. Deciphering the scriptally implicit meaning of the equation
2.C.1. Identifying compound components in an equation as representing conceptual meaning	6b
2.C.2. Substituting “real” known values into the equation	10, 11a
2.C.3. Explicating the meaning of a symbolic and/or numerical result in words, and attending to its units	10, 11b
2.C.4. Designing a numerical simulation that affords an investigation of the behavior of the equation with real values	11a

Table 2. The mentor’s instructional practices and discursive moves.

Instructional move	Episodes
1. Disciplining the student’s perception and practice
1.1. Draw the student’s attention to important features of the phenomenon or the problem	2b, 3, 11a
1.2. Draw the student’s attention to important features and aspects in a representation	1, 4, 7a, 10, 11a, 11b
1.3. Press for mechanistic explanations	2a, 2b
1.4. Problematize	7b
2. Telling
2.1. Re-explicate important ideas	4, 7a, 8, 10
2.2. Explain	4, 6b, 7a, 11b
2.3. Mediate meaning	7a
3. Explicating continuity and progress
3.1. Explicitly connect the student’s current work with work and ideas expressed previously	3, 4, 7a, 11a, 11b
3.2. Mark conceptual progress (e.g., achieving coherence)	2b, 11b
3.3. Press for detailed documentation of results and current thinking	5, 8, 10
4. Deliberate attentiveness and responsiveness to the student’s thinking
4.1. Carefully examine the student’s written work and respond to it.	10
4.2. Ask clarifying questions to either check whether a seemingly correct student’s answer indeed reflects a shared understanding, or the nature of a student’s misunderstanding reflected in a seemingly incorrect answer	3, 4
4.3. Formulate questions that require the student to explain or predict, to generate a shared “problem space” for co-construction of a shared understanding.	4
4.4. Attend and respond to the student’s difficulties	4
4.5. Attend and respond to the student’s temporal framing of the situation	10
4.6. Press for articulated explanations by asking clarifying questions or by engaging the student in tasks that facilitate the generation of these explanations	2b, 11b
5. Capitalizing on the student’s ideas
5.1. Elaborate and build on the student’s questions	2a
5.2. Re-voice the student’s ideas to mark important ideas	2a, 2b, 7a,7b, 11a
5.3. Develop and elaborate on the student’s insights	5, 7a, 11a
5.4. Rephrase student’s ideas more accurately	2b, 3, 6b, 7a, 7b, 11b
5.5. Elicit relevant prior knowledge and embodied experiences	2b, 2c, 3, 7a, 11a
6. Responsive sequencing and timing
6.1. Sensitive timing and sequencing of activities attuned to the student’s progress	2c, 3, 11b
6.2. Plan a coherent sequence of activities	7b, 9, 10, 11a, 11b
7. Encouraging the student’s autonomy
7.1. Encourage and praise inquisitive and independent thinking	2a, 2b, 9
7.2. Frame the situation as a joint authentic inquiry	1, 2a, 5, 11a
7.3. Position the student as an equal partner, and encourage the student to take the lead	1, 5, 11a
7.4. Invite the student to correct his own mistakes instead of correcting them for him	10
7.5. Give the student time to think	2b

Table 3. Changes in the student’s participation over time and activities.

Area of learning	Developmental trend	Episodes
1. Theoretical explorations	1.1. Increased sophistication of mechanistic reasoning (i.e., considering not just entities but their explicit properties, “chaining”—Russ et al., 2008)	3, 6b
	1.2. Moving from explorations that are based solely on intuitive mechanistic reasoning toward increasingly sophisticated mathematical modeling-based explorations	5
2. Approximations	2.1. Moving from concern and disbelief to conviction	12
	2.2. Starting to use adjectives of quantity in a relative sense (relative to other quantities) rather than an absolute sense; i.e., talking about a neglected quantity in the approximation as an influence that is always present but becomes very small relative to other influences in these conditions, rather than as an influence that ceases to exist.	3, 11b
3. Numerical investigations	3.1. Instead of blindly following instructions related to the preparation and analysis of the numerical simulation, explaining the meaning of each of the simulation components.	10
	3.2. Moving from blindly following the mentor’s direction in preparing the spreadsheet for the numerical calculation, to independently performing the task, and eventually to reasoned participation motivated by what the simulation can potentially demonstrate and that reflects some anticipation of what it may show before the actual programming.	11b
4. Agency and sense of autonomy	4.1. Gradual moving from merely following the mentor’s instruction, to motivated agentive participation in the decisions about the progress of the investigation.	6a, 6b, 9, 10, 11a, 11b
	4.2. Becoming emotionally invested in the inquiry process, expressing excitement, interest, and a sense of authentic discovery.	10, 11b

Longitudinal analysis of learning in three planes

The analysis of the episodes is presented in longitudinal order. When referring to specific features listed in Tables 1–3, the relevant terms are highlighted in italics with reference to the table and the specific item number.

Warmup—waves in a string

We were not present at the meeting in which the equation that models the velocity of the propagating wave in a string (Equation 1(1) $$v=\sqrt{\frac{T}{\mu }}$$(1) ) was derived and discussed. However, the student retrospectively discussed its derivation in the stimulated recall interview at the end of the project. We examine his account here since the mentor and the student made many references to waves in a string while they were discussing waves in water.

Episode 1. (September 25, 2017, interview with the student)

Table

229.	I:	You wrote here that at the beginning you had a hypothesis that the larger the tension, the higher the velocity, and that you hypothesized “based on common sense” (reading from the report) that the velocity would be the tension divided by the density, and then you wrote this down, but without the square root (in Equation 1(1) $$v=\sqrt{\frac{T}{\mu }}$$(1) ).
230.	S:	Right.
231.	I	But then you said that the power of the velocity was missing. Could you explain what exactly you did here?
232.	S:	So, we said, the logic was, that the only force that can transform the wave is the tension. So if the tension were larger, the force would be larger, and the wave would (propagate) faster. So this means that if there is a fraction (meaning: the relation will be represented by a fraction), /…/ the tension will be in the numerator. /…/ and then I thought, what else might be, and I said that the density of the string. Because if the density is greater, it will be really heavier, and then you need a lot of force to lift it. So I said that the larger the density, the heavier it will be, and the force, you need more force, so this means that it (the wave) will be slower. So I put the density /…/ below (i.e., in the denominator), and that’s it. And then I told Shay (the mentor’s name) here is the formula, I found it. And he said, great, but did you check the units? I said I hadn’t so he told me to check. And I did, and I saw that a square root was missing here (points to T in Equation 1(1) $$v=\sqrt{\frac{T}{\mu }}$$(1) ) and here (points to µ in Equation 1(1) $$v=\sqrt{\frac{T}{\mu }}$$(1) ) /…/ At first I showed this (to the mentor) with that (the velocity) squared, and he said “OK, would you agree to take the square root?.” And we took the square root from everything /…/

While episode 1 provides only a retrospective account, rather than demonstrating an in-vivo learning experience, the students’ account discusses a set of activities that form a practice that we term anticipating the mathematical formalism through intuitive sensemaking (Table 1, 1). This practice is composed of several activities which the student described. The first is the crafting of a mechanistic explanation (Table 1, 1.1) where the student’s account presents references to different facets of this activity including identifying the entities that form the phenomenon (e.g., tension, density), identifying and articulating the properties of these entities that are necessary for the particular mechanism to run (“the only force that can transform the wave is the tension.”). The second activity was the generation of a qualitative description of a quantitative trend (Table 1, 1.2) (“So I said that the larger the density, the heavier it will be, and the force, you need more force, so this means that it (the wave) will be slower”). The third activity was hypothesizing a crude metaphorical mathematical representation (Table 1, 1.3) which the student used to describe the previous trend (“So I put the density/…/below (i.e., in the denominator)”).

The student also referenced the guidance and instructional moves of the mentor. For example, he recalled that when he showed the equation he developed to the mentor, the mentor asked him “did you check the units?” The mentor here was making an important instructional move, he was drawing the student’s attention to important features of the representation (Table 2, 1.2): a physics equation must be balanced in terms of its units; that is, the units on one side of the equation should be the same as on the other side. The student interpreted this prompt as guidance to perform “dimension analysis” to the formula he developed; that is, to identify the quantities and describe their units in terms of base units,² and check whether both sides of the equation resulted in the same unit. The students described how he did this manipulation and found that he should correct his equation and square the velocity (i.e., $v^2=\frac{T}{\mu },$ instead of $v=\frac{T}{\mu }$). Hence, he describes an activity in which he was using dimension analysis to derive the physics equation up to its constants (Table 1, 1.4). At this point, he made another reference to the guidance he received from the mentor. He noted that the mentor examined the equation and said “would you agree to take the square root?” Here again, the mentor makes an instructional move of drawing the student’s attention to important features of the representation (Table 2, 1.2) by calling the student’s attention to the velocity as the dependent variable, which implies that it would be more convenient to isolate it. Note also how the student remembered the way the mentor made this move, by asking him “would you agree to….” This is an instructional discursive move that positions the student as an equal partner in the inquiry (Table 2, 7.3). The student reported that indeed “we took the square root from everything” (i.e., $v=\sqrt{\frac{T}{\mu }},$ reaching Equation 1(1) $$v=\sqrt{\frac{T}{\mu }}$$(1) ). The word “we” in the last sentence reflects the student’s sense of joint inquiry with the mentor (Table 2, 7.2), which came across throughout the interview.

The prevalent use of dimension analysis in physics is mainly to check the validity of a developed mathematical model. However, the mentor prompted the student to perform dimension analysis to develop a mathematical representation. This particular use of dimension analysis characterizes a practice implemented by many mentors in Acheret. They learn this strategy as part of their professional development and they often use it with their mentees, since in the 11th grade, students do not have the mathematics required to analytically derive many of the equations they use in their inquiry. Hence, we can consider using dimensional analysis in this way as reflecting a socio-culturally organized activity.

To sum up, Episode 1 illustrates several socio-culturally organized activities that form the practice of anticipating the mathematical formalism through intuitive sensemaking (Table 1, 1) by crafting mechanistic explanations (Table 1, 1.1), generating qualitative descriptions of quantitative trends of the system (Table 1, 1.2), hypothesizing a crude or metaphorical mathematical representation that qualitatively describes a quantitative trend (Table 1, 1.3), and using dimension analysis to derive a physics equation up to the value of its constants (Table 1, 1.4). The student’s retrospective account also highlights a few entailed instructional decisions and discursive moves of the mentor. These included drawing the student’s attention to important features and aspects in a representation (Table 2, 1.2), which we consider part of the instructional practice of disciplining the students’ perception and practice (Table 2, 1), framing the situation as a joint authentic inquiry (Table 2, 7.2), and positioning the student as an equal partner (Table 2, 7.3), which we consider part of the instructional practice of encouraging students’ autonomy (Table 2, 7).

Anticipating mathematical formalisms through intuitive sensemaking

Before introducing Equation (2)(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) , the mentor spent a great deal of time helping the student intuitively see that gravity, surface tension, and depth are factors that influence the propagation velocity of waves in liquids, and that the relative influences of gravity and the surface tension should be considered. This section draws on excerpts from the mentor-student discussions to illustrate how an intuitive sense for the later formal mathematical representation of the relative influence of gravity and surface tension (Equation 2(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) ) was cultivated.

Episode 2 depicts how the mentor helped the student perceive that the force of gravity affected the outcome of the phenomenon; namely, that it influenced the propagation velocity. We divide this episode into three sub-episodes, based on their different functions and the mediational moves employed by the mentor.

Episode 2a. (mentoring session, December 5, 2016, 0:10:36)

Table

1.	M:	/…/ So here (the case of waves in a string) we measured the velocity of the waves, right?
…
8.	S:	But here (water) there is no tension (i.e., the force of tension in a string) /…/
9.	M:	OK, OK! Now you are asking interesting questions. Where is the tension here?
…
11.	M:	Hmmm… Why was the tension important?
12.	S:	Here (pointing to the aquarium) or there (pointing to the results of the experiment with the string on the computer)?
13.	M:	There (the string)
…
18.	S:	The stronger (the tension), the faster it (the string) returns (to equilibrium)

In turn 1 the mentor is starting to make an analogy to the previous investigation of waves in a string. In turn 8 the student questions the validity of the analogy between waves in a string and water waves (turn 8, “But here (water) there is no tension”). The mentor’s reaction in turns 9 and 11 capitalizes on the student’ ideas (Table 2, 5): In turn 9 he revoices the student’s idea to mark it as important (Table 2, 5.2). In turn 11 he poses a question that elaborates and builds on the student’s question (Table 2, 5.1) (“Why was the tension important?”). The last question presses for a mechanistic explanation (Table 2, 1.3), as it pushes the student to explain how the tension in the string affected the velocity of the propagating wave in that case (turn 11). These moves prompted the student to craft a mechanistic explanation (Table 1, 1.1) in the case of the waves in a string (turn 18), which set the groundwork for searching for a mechanism in the case of water waves.

The mentor not only mediated an important feature involved in identifying a mechanism (i.e., articulating the causal effect of a particular behavior of an entity); i.e., press for mechanistic explanation (Table 2, 1.3), but also by framing the situation as a joint authentic scientific inquiry (Table 2, 7.2) (e.g., “we measured,” turn 1) and praising the student’s inquisitive and independent thinking (Table 2, 7.1) (e.g., turn 9, “OK, OK! Now you are asking interesting questions”). The mentor explicitly positions the student as a legitimate member of the research team that has important ideas to contribute.

To sum up, Episode 2a illustrates the student’s engagement in crafting mechanistic explanations (Table 1, 1.1). Several instructional and discursive moves of the mentor nurtured and supported this engagement: he pressed for mechanistic explanations (Table 2, 1.3), he capitalized on the student’s ideas by elaborating and building on the student’s questions (Table 2, 5.1), and by re-voicing the student’s ideas to mark important ideas (Table 2, 5.2), and he encouraged the student’s autonomy by praising inquisitive and independent thinking (Table 2, 7.1) and by framing the situation as a joint authentic inquiry (Table 2, 7.2).

In episodes 2b and 2c, the mentor started to push the student to consider gravity as an optional influence.

Episode 2b. (mentoring session, December 5, 2016, 0:12:45)

Table

25.	M:	So what is disturbed (in the case of water)?
26.	S:	What is disturbed (7)? I think the water’s uniformity (gesturing flatness with his hand)
27.	M:	What is uniformity? When you sit (on your surfboard) and wait for waves, and a wave goes past you, what do you feel?
28.	S.	WOW (gestures with his hand up and down)
29.	M:	(the mentor in reaction lifts himself from the chair and sits down again, imitating sitting on a surfboard) What did you just do with your hand?
30.	S.	The height of the water
31.	M:	OK. The height of the water. It’s a change in the height of the water.
32.	S:	You know what I was thinking? If there is a wave and there is water here, then something should also go down.
33.	M:	Now you are talking! /…/
…
41.	M:	Write a note to yourself (in the research journal) “does the water have a ‘down’?”

In this episode (2b) the mentor drew the student’s attention to the fact that downwards is a key direction in the context of the experiment—he disciplined the student’s perception (Table 2, 1), by drawing the student’s attention to important features of the phenomenon (Table 2, 1.1). In so doing he employed a variety of talk moves. He started by orienting the discussion toward a mechanistic explanation (Table 1, 1.1), asking the student to identify a relevant entity in the phenomenon; i.e., pressing for a mechanistic explanation (Table 2, 1.3) (turn 25, “What is disturbed?”). Notice the 7 seconds the mentor waited quietly for the student’s answer in turn 26; i.e., he gave the student time to think (Table 2, 7.5). When the student used a vague term (turn 26, “uniformity”), the mentor asked for clarification (“what is uniformity”?, turn 27) and continued with another question about the student’s embodied experience as a surfer (2nd sentence, turn 27). The clarifying question “what is uniformity?” pressed for an articulated explanation (Table 2, 4.6), and the appeal to the student’s embodied experience as a surfer supported the student in generating an articulated explanation by eliciting relevant prior knowledge and experiences (Table 2, 5.5) (i.e., surfing). When the student answered with an embodied gesture (turn 28), the mentor mimicked it and asked the student to verbalize his gesture (turn 29); hence he re-voiced the student’s idea (Table 2, 5.2), and his request for a verbalization of the gesture constituted additional pressing for articulated explanation (Table 2, 4.6). When the student described the disturbance as height (turn 29), the mentor rephrased it more accurately (Table 2, 5.4) (turn 31, “change in the height of the water”). When the student suddenly realized that the water must at some point move downwards (turn 32), the mentor marked the “downwards direction” as an important feature of the phenomenon (i.e., draw the student’s attention to important features in the phenomenon—Table 2, 1.1). He also praised the student’s inquisitive and independent thinking (Table 2, 7.1) (turn 33, “now you are talking”) and further marked the “downwards direction” as an important feature of the phenomenon by asking the student to write it down in his research journal (turn 41, “Write a note to yourself ‘does the water have a ‘down’?’”). The last move explicitly marked conceptual progress (Table 2, 3.2), and in so doing explicated the continuity and progress of the inquiry (Table 2, 3).

In summary, episode 2b presents an instance in which the student was engaged in guided crafting of a mechanistic explanation (Table 1, 1.1). This process was guided by several instructional and discursive moves by the mentor: He disciplined the student’s perception by explicitly pressing for a mechanistic explanation (Table 2, 1.3), and by drawing his attention to important features in the phenomenon (Table 2, 1.1). He attended and responded to the student’s thinking by asking clarifying questions that pressed for articulated explanations (Table 2, 4.6). He capitalized on the student’s ideas by eliciting a relevant prior experience (Table 2, 5.5), by re-voicing a student’s idea to mark it as important (Table 2, 5.2), and by rephrasing it more accurately (Table 2, 5.4). He encouraged the student’s autonomy by giving him time to think (Table 2, 7.5), and by praising his independent and inquisitive thinking (Table 2, 7.1). He also supported the student’s awareness of his progress and his sense of coherence by marking his conceptual progress (Table 2, 3.2).

Episode 2c. (mentoring session, December 5, 2016, 0:16:38)

Table

49.	M:	/…/ Now I want us to think for a moment together (2) So let’s say that I have a straight pool (gestures to indicate a horizontal surface), and I try to pile up water with my hands (gestures gathering something into a pile). Why doesn’t it stay like this (‘piled’)?
50.	S:	Well (smiles) it’s a liquid.
…
67.	M:	Press play (on a YouTube clip of a floating drop of water in a space station) let’s watch for a moment
70.	S:	WOW! This is water?
71.	M:	Yes, this is water. Think about of what I asked earlier. What causes the wave (‘piled’ water in a pool on Earth) to move downwards?
…
82.	S:	But why doesn’t the air go up (referring to an air bubble in the water drops in the YouTube clip)
83.	M.	There is no gravity /…/

In episode 2c the mentor explicitly mediated gravitational force as an entity that influences the behavior of waves in water (i.e., crafting mechanistic explanation, Table 1, 1.1). The mediation was mainly achieved by the choice of the task he gave the student (Table 2, 6.1): examining the behavior of a drop of water in a space station (weightless conditions) and comparing it to the student’s experiences on Earth. The mentor’s talk moves and gestures in turns 49 and 71 (e.g., “I try to pile up water with my hands (gestures gathering something into a pile). Why doesn’t it stay like this (‘piled’)?”) were explicitly aimed at activating and eliciting the student’s knowledge of relevant experiences (Table 2, 5.5).

The intuitive sense of the influence of surface tension was mediated through a similar process as the one described in episodes 2a, 2b, and 2c for the influence of gravity. This mediation highlighted the surface tension’s increased influence as the smaller the size of the drops.

Episode 3 took place after this phase. It illustrates an initial attempt to nurture the development of an intuitive sense that the surface tension and gravitational force contribute independently to the propagation velocity of waves in liquid and that these influences should be considered relative to each other.

Episode 3. (mentoring session, December 5, 2016, 0:39:07)

Table

1.	M:	The waves we are messing with are big
2.	S:	Relative to drops
3.	M:	Yes. This force (the surface tension) still exists
4.	S:	Aha, but it is much weaker (in large waves)
5.	M:	Weaker than what?
6.	S:	Than a small drop
7.	M:	No, no. What are the other important forces? Let me return to (my) original question. When I pile water (gesturing grabbing water) why would this (the shape of the “piled” water) change at all?
8.	S:	The force of gravity works hard (in pulling the water downwards)
9.	M:	So, relative to the force of gravity
10.	S:	Oh, it (the influence of the surface tension) is getting smaller
…
11.	S:	But why if there is more water, should this force (the surface tension) become smaller?
…
12.	S:	It’s really weird, because in the case of a drop in a faucet, this force gets larger the more…

In turn 1 the mentor refocused the student’s attention back to the actual problem they were studying (large waves) (draw the students’ attention to important features of the phenomenon—Table 2, 1.1). The word “relative” in the student’s response in turn 2 mark a subtle move toward a more sophisticated participation in a physics discourse, where adjectives, such as big and small should be stated relative to a known quantity (Table 3, 2.2). We believe that this change in the student’s participation emerged from the activities described in episode 2, reflecting the mentor’s plan of a coherent sequence of activities (Table 2, 6.1), as well as his in-the-moment instructional moves. In turn 3 the mentor drew the student’s attention to an important feature of the phenomenon—the force (Table 2, 1.1). The student’s answer in turn 4 was “but it is much weaker.” Many teachers would have interpreted this answer as evidence that the student understood that in the water waves under the experimental conditions, unlike the case of drops, the force of the surface tension is much smaller than the force of gravity. Instead, in turn 5 the mentor asked, “Weaker than what?.” This clarifying question verified whether the seemingly correct student’s answer indeed reflected a shared understanding (Table 2, 4.2). The student’s response (turn 6, “than small drops”) shows the power of this move, as it revealed that the student was not comparing the force of gravity to the force of the surface tension but the size of the waves to the size of the small drops. In turn 7 the mentor again tried to draw the student’s attention to an important feature of the phenomenon (Table 2, 1.1). This was done first by explicitly noting that there are competing forces in question (“What are the other important forces?”), and then by trying to elicit relevant prior knowledge, by explicating the connection to a previous question on which they were thinking (Table 2, 3.1 and 5.5): the “pile of water” question from turn 71 in episode 2c. The goal of this move was to prompt the student to think of gravity. The student indeed activated the notion of gravity (turn 8), saying that “the force of gravity works hard.” Nevertheless, in turn 9 the mentor rephrased the student’s answer more accurately (“So, relative to the force of gravity”) (Table 2, 5.4), to emphasize the idea that gravity and the surface tension contribute independently to the propagation velocity, and thus their influences should be considered relative to each other. In turn 10 the student seemed to correctly respond to this line of reasoning (“the influence of the surface tension is getting smaller”). However, as the student’s reaction in turns 11 and 12 clearly show, despite his “positive” reaction in turn 10, that the notion of the relative influences of gravity and surface tension in the context of the water waves he was studying did not make much sense to him (“But why if there is….,” “it’s really weird…”). It also shows that he was genuinely trying to make sense of the phenomenon rather than give the right answer (i.e., agentive motivated participation—Table 3, 4.1). As shown later, while at this point the student’s speech only partially reflected a correct explanation of the relative influences of gravity and the surface tension in the investigated conditions (Table 3, 2.2—initial stage), the insight gained from this discussion was important seeds for later discussions of the scriptally implicit meaning of the equation.

In episode 3 the student was engaged in crafting a mechanistic explanation (Table 1, 1.1), and generating a qualitative description of a quantitative trend (Table 1, 1.2) regarding the relative influences of gravity and surface tension. We argue that this marks a subtle change in the student’s participation in physics inquiry, relative to episode 2. First, the level of mechanistic reasoning the student presented was higher (increased sophistication of mechanistic reasoning—Table 3, 1.1): he was starting to consider the properties of the underlying entities and trying to “chain them” i.e., explaining how the properties and their organization created the particular behavior of the system (Russ et al., 2008) (e.g., “But why if there is more water, should this force (the surface tension) become smaller?”). As the discourse analysis above shows the mentor facilitated this process by the planning of the sequence of activities involved (Table 2, 6.1), by drawing the student’s attention to important features of the phenomenon (Table 2, 1.1), by asking clarifying questions to check whether a seemingly correct student’s answer indeed reflects a shared understanding (Table 2, 4.2), by eliciting relevant prior knowledge (Table 2, 5.5), and by rephrasing the student’s answer more accurately (Table 2, 5.4).

Episodes 4 and 5 that followed present the first attempt to connect the intuitive discussion in Episode 2 (the influence of gravity on the propagation velocity) to a mathematical representation. Episode 4 illustrates (one of many examples) how the mentor prepared the groundwork for the extraction of the textually explicit meaning of Equation (2)(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) . We remind the reader that at this stage the student had not seen yet Equation (2)(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) or Equation (3)(3) $${\omega }^2=\left(gk+\frac{\sigma }{\rho }{k}^3\right)\mathit{tanh}\left(kh\right)$$(3) .

Episode 4. (mentoring session, December 5, 2016, 0:42:30)

Table

1.	M:	So on the moon the velocity (in the case of water waves—see Equation 1(1) $$v=\sqrt{\frac{T}{\mu }}$$(1) ) is different
2.	S:	In the moon it is…
3.	M:	For the string it will be the same. If I took the string to the moon, I would measure the same (wave) velocity. Does $g$ (gravitational acceleration) appear in the formula (wave velocity in a string—see Equation 1(1) $$v=\sqrt{\frac{T}{\mu }}$$(1) )?
4.	S:	Not according to the formula
5.	M:	OK. So do you agree that $g$ should appear somewhere in the formula of the velocity of water waves?
6.	S:	Oh, we have it (points to the derivation of the propagation velocity in a string), it was just reduced
7.	M:	Are you talking about the string?
8.	S:	The string
9.	M:	If it was, and it was reduced, then it isn’t (meaning, $g$ does not appear in the formula of the wave velocity in a string) /…/
10.	S:	Wait, but it should have been in the string, it should have been in the string even if it was reduced
11.	M:	If it is reduced then it has no effect
12.	S:	But it should have been there, and since it was reduced
13.	M:	It’s like when mass appears in the calculations of free fall, but is then reduced. It has no effect on the acceleration. I’m pushing you hard (smiles) to say that gravity has an effect (in the case of waves in water), that $g$ would definitely be there (in the formula for the propagation velocity of water waves).

The mentor (turns 1 and 3) initially aimed to engage the student in hypothesizing a crude mathematical representation of a phenomenon (Table 1, 1.3): whether the variable $g$ that symbols the gravitation acceleration should be included in the equation that describes the velocity of waves in water. In turns 1, 3, and 5, the mentor formulated a question that required the student to explain or predict, in order to generate a “shared problem space” for the co-construction of a shared understanding (Table 2, 4.3). In doing so he explicitly related the current inquiry to the work done in the previous meetings (Table 2, 3.1) by building on the previous examination of wave in a string and asking the student to compare the influence of gravity in both cases. In turns 3 and 5 he was explicitly drawing the student’s attention to important features of the representation (Table 2, 1.2) (whether the formula in each case should include the variable $g$). The student’s response in turn 6 reveals that he incorrectly assumed that the formula he derived for the propagation velocity in a string was applicable to water waves as well. He also incorrectly assumed that the gravitational acceleration $g$ affects the velocity of waves in a string, even though it was reduced in the dimension analysis derivation, and thus did not appear in Equation (1)(1) $$v=\sqrt{\frac{T}{\mu }}$$(1) . This latter point does not cohere with an important structural aspect of the textually explicit meaning of equations (Table 1, 2.A.1); namely, that reduced variables are no longer part of a physics equation which models a real phenomenon, and thus have no influence. The student reiterated this mistake in turns 10 and 12 as well, which suggests that this confusion was not only discursively based but rather something that the student had difficulty grasping. The mentor noticed this confusion (the clarifying question in turn 7—Table 2, 4.2), and after the student’s answer confirmed the suspected confusion (turn 8) the mentor changed the activity to deciphering the textually explicit meaning of an equation (Table 1, 2.A) (the velocity of waves in a string—Equation 1(1) $$v=\sqrt{\frac{T}{\mu }}$$(1) ). In turns 9, 11, and 13 the mentor responded to this difficulty (Table, 2, 4.4) by telling—he restated the idea (Table 2, 2.1) that if a variable is reduced from an equation it means that the quantity it represents has no effect (turns 11) and explained (Table 2, 2.2) with an analogy (turn 13). We cannot tell at this stage how much the student understood from this exchange.

To sum up, in episode 4 the mentor, in response to the student’s confusion about the meaning of reduced variables in an equation, changed the activity from hypothesizing a crude mathematical representation of a phenomenon (waves in water) (Table 1, 1.3) to deciphering the textually explicit meaning of an equation, or more specifically, identifying and naming variables and parameters in the equation (waves in a string) (Table 1, 2.A.1). The episode presented several instructional moves by the mentor: drawing the student’s attention to important features in the representation (Table 2, 1.2), telling (i.e., restating important ideas and explaining them) (Table 2, 2.1 and 2.2), relating the current work to work done in previous meetings (Table 2, 3.1), clarifying questions (Table 2, 4.2) and questions that required the student to explain or predict (Table 2, 4.3) which helped the mentor to co-construct and assess the extent of the shared understanding, and respond to the student’s difficulties (Table 2, 4.4).

Episode 5 illustrates (one of many examples) how the mentor planted the seeds for supporting the extraction of the textually implicit meaning of Equation (2)(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) before it was formally introduced. The mentor asked the student to summarize the conclusions they had drawn in his journal, and they talked about the precise articulation of these conclusions.

Episode 5. (mentoring session, December 5, 2016, 0:52:59)

Table

1.	M:	/…/ I’m nagging but I want to insist that $g$ must be in the formula, it is important
2.	S:	But it is reduced
3.	M:	No. I think that water waves on the moon should be different
4.	S:	Oh yes.
5.	M:	I am not sure whether they will be slower or faster
6.	S:	Slower
7.	M:	As I understand water waves, and I do not understand them well enough, $g$ has to be there
8.	S:	Yes, yes, yes, I say that too. It will be something constant but
9.	M:	It will be the same $g$ we know (gravitational acceleration). Not the $\mathrm{g}$ from $\mathrm{kg}.$
10.	S:	I also agree. Yes.
11.	M:	So let us write it down. OK?
12.	S:	Wait, if the wave velocity equals something with $g$
13.	M:	With $g$
14.	S:	I think $g$ should be in the numerator (gesturing balance of the two sides of the equation). Do you also think that?
15.	M:	Yes.
16.	S:	Yes. It means that if it (the aquarium) were on the moon, it (the wave velocity) would be smaller, the velocity
17.	M:	I agree, we think alike (8). Maybe if we understand more, we will be able to write a formula with units (dimension analysis). I think this will be harder since water waves are really complicated.
18.	S:	This is going to be really hard.
19.	M:	Gravitational force, would you like to write it in the numerator? But maybe you can be more precise (5), maybe (3) maybe the gravitational acceleration, although the gravitational force is also correct. Another thing that will be there is the size of the wave, the wavelength, and …

Since the student’s reaction in turn 2 seemed to wrongly return to applying Equation (1)(1) $$v=\sqrt{\frac{T}{\mu }}$$(1) (waves in a string) to the case of waves in water, the mentor’s reaction in turn 3 re-directed him toward a sensemaking process. Note how the mentor positioned himself as someone who is also trying to make sense of the situation with the student (turn 3, “I think that,” turn 5, “I’m not sure whether,” turn 7 “…and I do not understand them well enough”). These moves positioned the student as an equal partner and encouraged him to take the lead (Table 2, 7.3). They also framed the situation as a joint authentic inquiry (Table 2, 7.2). The student responded to this framing and participated in the discussion as an equal partner (e.g., turn 8, “Yes, yes, yes, I say that too.”) Note that the mentor knew much more than the student. He was familiar with the dispersion relation of waves in liquids from his undergraduate coursework in physics and knew Equation (2)(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) . However, this was an inquiry for him as well. In one of the informal discussions we had with the mentor, he said that he learned a great deal from this project, since “there’s a difference between knowing an equation and really understanding its meaning and implications for real-world messy phenomena.” Note that the mentor’s use of “understanding” here corresponds to the ability to explain.

It was in this atmosphere that student generated an important mathematical insight: He translated his intuitive sense that the larger the gravitational acceleration, the larger the wave velocity should be into a metaphorical mathematical representation (Table 1, 1.3)—“I think $g$ should be in the numerator” (turn 14), and generated a qualitative prediction that expressed a quantitative trend (Table 1, 1.2)—“on the moon, it (the wave velocity) would be smaller” (turn 16). Compare this for example to the student’s discourse in Episode 2b (how the water should “go down”) and in Episode 3 (“the force of gravity works hard”). We consider this change in the students’ participation as an indication of learning; i.e., moving from an intuitive mechanistic reasoning to initial engagement in mathematical modeling (Table, 3, 1.2), although at this stage the mathematical representation (i.e., g in the numerator) is still metaphorical. The mentor’s agreement was expressed in a manner that further supported the frame of joint authentic inquiry and the positioning of the student as an equal partner (turn 17, “I agree, we think alike”) (Table 2, 7.2 and 7.3), and in turn 19 he further elaborated on the student’s insights (Table 2, 5.3) by sharing his thoughts on how to proceed.

Note how the mentor pressed the student to document the insights in his research journal (Table 2, 3.3): turns 11—“let us write it down,” turn 19—“would you like to write it in the numerator? But maybe you can be more precise…” Apart from developing and elaborating on the student’s insights (Table 2, 5.3), the mentor’s speech also again reflected egalitarian positioning (Table 2, 7.3) is also apparent here (“let us,” “would you like…”). Note that the egalitarian positioning of the student and the mentor in a joint inquiry is not an idiosyncratic feature of this teacher-mentor but a culturally valued goal of the Acheret community of teacher-research mentors. Kapon (2016) documented it in the practice of a different mentor, and it explicitly came up in a series of interviews conducted with 14 core members of the community (Faraj et al., 2021).

The student’s participation in this episode clearly progressed relative to the previous episodes. He successfully participated in the activity of hypothesizing crude metaphorical mathematical representation of the phenomenon (Table 1, 1.3), which was too challenging for him in episode 4, reflecting a process of leaning, namely, moving from reasoning that is limited to intuitive mechanistic reasoning toward early attempts at engaging in mathematical modeling (Table 3, 1.2). Moreover, the student’s role in this activity became more central in that he contributed the ideas and insights to the inquiry. The change in the student’s participation here builds on his participation in the previous episodes, and in response to the mentor’s instructional moves: his insistence on explicating continuity and progress, always documenting their current thinking (Table 2, 3.3), and most notably, from framing of the situation as a joint inquiry (Table 2, 7.2) with deliberate positioning of the student as an equal partner (Table 2, 7.3).

Reading through a physics equation—discussing the equation

The following episodes illustrate the discussions that followed the introduction of Equations (2)(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) and (3)(3) $${\omega }^2=\left(gk+\frac{\sigma }{\rho }{k}^3\right)\mathit{tanh}\left(kh\right)$$(3) . Episode 6 is drawn from the third hour of Session 2 (January 2, 2017). The mentor asked the student to read the Wikipedia page that discusses the dispersion of water waves. This is where the student first saw Equations (2)(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) and (3)(3) $${\omega }^2=\left(gk+\frac{\sigma }{\rho }{k}^3\right)\mathit{tanh}\left(kh\right)$$(3) . He first examined the page in the Hebrew version of Wikipedia, and immediately, taking the mentor’s advice, switched to the more detailed English version of that page (Wikipedia, n.d.). Equation (2)(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) is presented in Wikipedia without the surface tension expression, and the mentor drew the student’s attention to this fact and expressed his criticism since the influence of the surface tension is explicitly mentioned in the introduction (“capillary”), and in the dispersion relation that appears later on the page (Equation 3(3) $${\omega }^2=\left(gk+\frac{\sigma }{\rho }{k}^3\right)\mathit{tanh}\left(kh\right)$$(3) , also presented in Wikipedia). Episodes 6a and 6b illustrate the student’s first attempts to make sense of the equation.

Episode 6a. (mentoring session, January 2, 2017, 2:01:20)

Table

1.	S:	OK. Now let us say what this means
2.	M:	OK /…/ so lambda is the wavelength, we know that. This $\mathrm{c}$ is actually $\mathrm{v}$ (the velocity in Wikipedia is expressed by the letter $\mathrm{c}$ instead of $\mathrm{v}$)
3.	S:	Fi (refers to the Greek letter $\mathrm{\rho }$) is the density
4.	M:	This ($\mathrm{\rho }$) is Ro, the density /…/
5.	S:	Ro. Yes
6.	M:	It’s the mass for the surface unit
7.	S:	Volume unit
8.	M:	Right, volume unit. Kilograms per cubic meter. For water this is 1010–1013 kg per cubic meter. Meter, over meter, over meter weigh about a ton.

The student is engaged here in deciphering the textually explicit meaning of Equation (2)(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) ; namely, he identifies and names each variable and parameter in the equation (Table 1, 2.A.1). The mentor’s reaction in turn 8, further elaborates on this practice by explicating the units of the variable $\rho .$ He participated in the discourse as an equal partner in the inquiry: he initiated the activity (turn 1) and he corrected the mentor (turn 7) when the mentor made a mistake (turn 6); i.e., presenting motivated agentive participation (Table 3, 4.1) In contrast, his participation in Episode 6b that started a few turns later, while representing agency and self-regulation as well, required much more guidance.   

Episode 6b. (mentoring session, January 2, 2017, 2:02:59)

Table

21.	S:	$2\mathrm{\pi }$ (in Equation 2(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) ) means one period?
22.	M:	So yes, $2\mathrm{\pi }$ really helps us when we write /.../ say I want to draw something vs. x, I will write sine of $\mathrm{x}$ divided by the wavelength and multiplied by $2\mathrm{\pi }$ (refers to the equation^3 $y\left(x,t\right)=A\sin \left(\frac{2\pi }{\lambda }x\pm \frac{2\pi }{T}t\right),$ which was presented to the student in a different meeting)
23.	S:	Aha, and this will be one rise and one fall
24.	M:	No, look, every time $\mathrm{x}$ equals a whole number of wavelengths I will get a multiplication of $2\mathrm{\pi }.$
25.	S:	Right. And then how many waves.
26.	M:	This is how I write it. And I call $2\mathrm{\pi }$ divided by $\lambda ,\mathrm{}k.$
…
34.	M:	So $\mathrm{k}$ is like the $\mathrm{\omega }$ of the wavelength (refers again to the equation of a one-dimensional traveling wave $y\left(x,t\right)=A\sin \hspace{0.17em}\left(\frac{2\pi }{\lambda }x\pm \frac{2\pi }{T}t\right)=A\sin \hspace{0.17em}\left(kx\pm \omega t\right),$ where k is the angular wavenumber and $\omega$ is the angular frequency.^4)
35.	S:	Ah… not sure. Say that again?

Turn 21 mark a change in the activity. The student’s question “$2\mathrm{\pi }$ (in Equation 2(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) ) means one period?” indicates a move from engaging in deciphering the textually explicit meaning (Table 1, 2.B) of the equation to engaging in deciphering the scriptally implicit meaning of the equation (Table 1, 2.C)—the student identifies $2\pi$ as a complex component with conceptual meaning (Table 1, 2.C.1). The identification of $2\mathrm{\pi }$ as a unit of meaning was based on the student’s prior knowledge that the sine function is a periodic function, which repeats its values every 2π radians. The mentor’s explanation (“sine of $\mathrm{x}$ divided by the wavelength and multiplied by $2\mathrm{\pi }$” in turn 22), rephrased the student’s idea more accurately (Table 2, 5.4), by noting that the relevant unit of meaning is actually the wave number $\frac{2\pi }{\lambda }$ (see footnote 4 for an explanation of the physics). The student’s response (turn 23) did not reflect a shared understanding with the mentor since he was merely reiterating how the period of a sine function which he referred to in his previous turn can be seen in a graph. The mentor noticed this lack of intersubjectivity and tried to explain (Table 2, 2.2) the meaning of wave number to the student in turns 24, 26, and 34. However, the student’s reaction in turn 35 (“Ah… not sure. Say that again?”) was an explicit expression of difficulty accepting the expression $2\pi /\lambda$ (wave number) as a unit of meaning. This is not surprising given the fact that the student had learned the basic tenets of wave theory on the fly as he was working on the project.

Episodes 6a and 6b illustrate a process of change in the student’s engagement. In 6a he engaged in deciphering the textually explicit meaning (Table 1, 2.A) of the equation, by identifying the conceptual meaning of the variables and parameters in Equation (2)(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) (Table 1, 2.A.1), whereas in 6b he engaged in deciphering aspects of its scriptally implicit meaning (Table 2, 2.C), by identifying compound components in an equation as representing conceptual meaning (Table 2, 2.C.1). He could not have engaged in deciphering the latter before he had successfully deciphered the former. Thus, we consider this as evidence of (slight) progress in his engagement in mathematical modeling-based exploration (Table 3, 1.2). Moreover, his participation was self-directed and agentive (Table 3, 4.1).

Episodes 7a, 7b, and 8 are drawn from the same session a little later.

Episode 7a. (mentoring session, January 2, 2017, 2:07:03)

Table

1.	M:	/…/ So we were talking about the gravity component (in Equation 2(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) ) that told us that larger waves (larger wavelengths) propagate faster.
2.	S:	Right.
3.	M:	Try to think. We have two more components (in Equation 2(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) ) that we have to consider.
…
11.	M:	So consider something interesting for a moment. Where did we say that the surface tension has an influence? Do you remember what surface tension is?
12.	S:	Yes. When the water wants to become closer
13.	M:	It’s the force of the water. /…/
…
19.	M:	When did we say that this thing (surface tension) matters?
20.	S:	In tiny-tiny waves.
21.	M:	In tiny waves. When lambda (the letter $\mathrm{\lambda }\mathrm{}$ that denotes wavelength in Equation 2(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) ) is small. Lambda is in the denominator here (points to the surface tension component in Equation 2(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) ).
22.	S:	Aha, so it has an effect.
23.	M:	So this (the surface tension component, in the case of small wavelengths) is very big, and this is very small (the gravity component, where $\mathrm{\lambda }$ is in the numerator)

In turn 1 the mentor re-explicated important ideas (Table 2, 2.1) by reiterating that the larger the wavelength $\lambda ,$ the smaller the gravity component, and thus the smaller the propagation velocity; i.e., drawing a textually implicit meaning from Equation (2)(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) by calling for mental simulation of how a change in $\lambda$ affects the relative size of the gravity and surface tension components (Table 1, 2.B.1). In turn 3 the mentor drew the student’s attention to important features and aspects in the representation (Table 2, 1.2) by pointing out the additional components of Equation (2)(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) . In turn 11 the mentor explicitly tried to activate and elicit the student’s relevant prior knowledge (Table 2, 5.5) required for inferring the scriptal meaning of the surface tension component in Equation (2)(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) . He did so by referencing their intuitive discussions in previous meetings (Table 2, 3.1) on the influence of the surface tension and the conditions in which it can become relatively influential. In turn 12 the student activated this knowledge but expressed in a teleological manner (“Yes. When the water wants to become closer”). The mentor’s reaction in turn 13 was to re-voice this answer but phrase it more accurately (Table 2, 5.4) by pointing to the force as the agentive entity (“it’s the force of the water”). In turn 19 the mentor specifically tried to remind the student about the mechanistic explanation they created (Table 2, 3.1) in the previous discussions. He asked the student to recall the setup conditions they identified (Russ et al., 2008) for the large influence of the surface tension. This prompt led to the student’s recall of this information (“tiny-tiny waves,” turn 20). In turn 21 the mentor re-voiced the student’s idea (Table 2, 5.2) and further elaborated it (Table 2, 5.3) by explaining how this information comes to cohere with the textually implicit meaning of the equation (“Lambda is in the denominator here”). The student’s acknowledgment “Aha, so it has an effect” was answered by the mentor’s further explication (Table 2, 2.1) of the textually implicit meaning in turn 23.

Episode 7b. (mentoring session, January 2, 2017, 2:09:50)

Table

40.	M:	So look, we have here a surface tension component that is mainly important for small waves. OK? And saying “small” is not enough. We have to be able to say how much “small” is. And it depends on this sigma (referring to $\mathrm{\sigma },$ the surface tension coefficient in Equation 2(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) ).
41.	S:	It’s a constant this sigma.
42.	M:	It’s a constant for a specific liquid.
43.	S:	Yes. For a particular liquid.
44.	M:	Exactly. But we have to understand how much this is (the value of the surface tension component). From which wavelength this (the influence of the surface tension) becomes insignificant. At some point, it becomes really small compared to this (the value of the gravity component) or vice versa. OK. This is something we are going to do soon.

In turn 40 the mentor first highlighted the need to quantitatively estimate the relative influence (“We have to be able to say how much ‘small’ is”) and then hinted that to be able to do so they needed to know features of the system that are not explicated in the equation (“it depends on this sigma.”); i.e., the mentor was problematizing (Table 2, 1.4) the situation for the student. This problematization explicitly invited the student to decipher the scriptally implicit meaning of the equation (Table 1, 2.C), by highlighting the need to consider the real known values of the system-specific parameters in Equation (2)(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) (Table 1, 2.C.2). The student did not see this at first glance and noted that sigma denotes a constant (turn 41). The mentor’s response both revoiced the student’ answer to mark it as important and rephrased it more accurately (Table 2, 5.2 and 5.4), by emphasizing the system’s specificity (“it’s a constant for a specific liquid,” turn 42). The student expressed his agreement by re-voicing the last sentence (turn 43). The mentor’s response (turn 44) further problematized the activity (Table 2, 1.4) by articulating a new problem that they should address (“From which wavelength this (the influence of the surface tension) becomes insignificant”). The last problematization explicitly invited the student to decipher the textually implicit meaning of the equation (Table 1, 1.B), by highlighting the need to generate an inference based on the result of the mental manipulation of the mathematical representation (Table 1, 2.B.1). The mentor’s suggestions also hint that he had a plan for a coherent sequence of activities for the student (Table 2, 6.2).

Episodes 7a and 7b illustrate engagement in deciphering the textually and the scriptally implicit meanings of Equation (2)(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) . However, while the student was clearly attentive, the mentor was the one who led the conversation and the student followed. The mentor’s instructional and discursive moves reflects the practice of an experienced teacher: he explained and re-explicated important ideas (Table 2, 2.1), he re-voiced the student’s ideas (Table 2, 5.2) and rephrased them more accurately (Table 2, 5.4), he reminded the student about relevant insights they arrived at in previous meetings (Table 2, 3.1), he disciplined the student’s perception and practice by problematizing the situation and highlighting the issues that should be resolved (Table 2, 1.4). During these episodes, the student engaged in the activities following his lead but did not take a leading role at this point.

The conversation drifted at that point to related issues. At some point, the mentor sent the student to summarize their discussion in writing in his research journal. The student worked alone on his writing. When the student signaled to the mentor that he had finished, the mentor was surprised.

Episode 8. (mentoring session, January 2, 2017, 2:15:45)

Table

1.	M:	Already finished writing (surprised tone that reflects disbelief)?
2.	S:	The main points. That $\mathrm{\sigma }$ is the surface tension
3.	M:	And what happens in big and small waves?

The mentor suspected that the student’s summary was incomplete given how quickly he wrote it. The student’s answer in turn 2 confirmed the mentor’s suspicion. It also suggested that the student did not fully appropriate the conclusions from the discussion. The mentor’s comment on turn 3 re-explicated the important features (Table 2, 2.1) of the conclusion that should be addressed in the summary; namely, the different effects of gravity and surface tension in large and small waves. The mentor’s pressing for a detailed documentation of the results and current thinking (Table 2, 3.3) in episode 8 was an instructional move that not only supported a sense of continuity and progress but also required more active participation from the student than his participation in episodes 7a and 7b in generating an inference based on a mental simulation of how a change in one variable (the wave length) affects the behavior of the gravity and surface tension components (Table 1, 2.B.1).

Reading through a physics equation—investigating the equation

About half of the third session, which took place two weeks later, was dedicated to an explicit inquiry of Equation (2)(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) . The participation of the student in this process was guided by mediating tasks and several talk moves. Episode 9 presents the first mediating task.

Episode 9. (mentoring session, January 16, 2017, 1:49:17)

Table

1.	M:	Wait, wait, let me ask you this question—for which wavelength will this be equal to this? (points to the gravity and surface tension components in the notebook)
2.	S:	So we have to do the function of this
3.	M:	Go for it (the student starts to write in the notebook)
…
17	M:	Let’s open an Excel file and do the calculation there since we will have to do it again. Let’s define the constants (as parameters in the spreadsheet).

In turn 1, the mentor invited the student to engage in deciphering the textually implicit meaning of Equation (2)(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) (Table 1, 2.B). Specifically, he suggested finding the wavelength in the special case in which the surface tension and the gravity components in Equation (2)(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) are equal (Table 1, 2.B.3), by deriving a new equation based on Equation (2)(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) and analytically solving it (i.e., generating an inference based on the result of an analytical manipulation of the equation—Table 1, 2.B.2). Although the mentor introduced the goal of the task through the questions in turn 1, turns 2 and 3 show that the transformation from a question into a concrete mathematical task was achieved in collaboration with the student. For example, the student articulated one possible pathway to meet the challenge (“we have to do the function,” turn 2). We interpret this move as an expression of an increased agency (Table 3, 4.1), relative to the students’ participation in the mathematical exploration in episodes 7a, 7b, and 8. The mentor encouraged him to proceed (“Go for it,” turn 3), hence he was encouraging the student’s inquisitive thinking (Table 2, 7.1) and thus nurtured his sense of autonomy (Table 2, 7). The mentor’s request from the student in turn 17 to do the calculation on a spreadsheet instead of his calculator, and define the constants of water on the spreadsheet as parameters that can take different values rather than numbers (turn 17), was targeted toward the next mediating tasks, in which the spreadsheet afforded further investigation of the meaning of Equation (2)(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) by substituting different values into the parameters. Thus, we interpret turn 17 as evidence for the mentor’s planning of a coherent sequence of activities (Table 2, 6.2).

Episode 10 starts when the student had completed the first part of the previous task, which resulted in an analytic derivation of the wavelength in which the gravity and surface tension components are equal. The student started to prepare the spreadsheet according to this derivation, and the mentor asked to examine his derivation.

Episode 10. (mentoring session, January 16, 2017, 1:58:19)

Table

1.	M:	Let’s see your calculation (examines the student’s written work while the student works on the spreadsheet for about 10 seconds. The word “calculation” refers to the parametric derivation the student did in Episode 9—numerical solutions were calculated in the spreadsheet)
2.	S:	(the student finishes typing the formula into the spreadsheet, turns to the mentor who is still looking at the notebook) Are you ready Shay (the mentor)? This would give me (1) the wavelength that (1), that will influence (1)
3.	M:	(looks up at the student and explains) that they will be equal to one another, right? That [the velocity
4.	S:	[The surface [tension
5.	M:	[/…/, that the gravity and the surface tension will influence the velocity to the same extent
6.	S:	Right
7.	M:	(smiles and speak quietly in a tentative tone:) Estimate how much is it going to be?
8.	S:	Like this! (expressively presses the keyboard, turns his gaze from the computer to the mentor—see Figure 1—while speaking in a pleased tone) Come on, it should be something very-very small.
9.	M:	So wait a minute, what do we get now?
10.	S:	A velocity of 0.02
11.	M:	No. You just calculated…
12.	S:	Wavelength
13.	M:	Wavelength. Write next to (points to the appropriate cell in the spreadsheet)
14.	S:	In centimeters?
15.	M:	No. What value did we substitute?
16.	S:	In meters
17.	M:	So let’s call this (cell D2 in the spreadsheet—see Figure 2) wavelength
18.	S:	(uncertain tone, looking at the mentor) That the velocity (continues with a more assured tone) that the gravity and the surface tension will equally influence the velocity (the mentor nods as the student speaks, and the student writes in the appropriate cell (E2) in the spreadsheet—see Figure 2)

Turn 1 in which the mentor carefully examined the student’s work reflects a recurring path: the mentor’s deliberate attentiveness to the student’s thinking (Table 2, 4), through carefully examining the student’s written work (Table 2, 4.1). The student’s participation reflects his sense of joint inquiry. As turn 2 revealed, the student was excited and curious about the result of the calculation and wanted to share this excitement with his mentor (“are you ready Shay (the mentor’s name)?”). The pauses in his speech in turn 2, when he explained what the result aimed to demonstrate, show that though the student had a good idea of what he was doing, he was still struggling to clearly express it in words. The mentor did this for him in turns 3 and 5 by re-explicating the main idea (Table 2, 2.1).

In turn 7 the mentor asked the student to estimate the value of the wavelength in which the gravity and the surface tension components of Equation (2)(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) are equal. This was an invitation to engage in deciphering the scriptally implicit meaning of the equation the student derived (Table 1, 2.C), since it required integrating knowledge of the physical world (experience), by substituting real known values of the parameters into the theoretical equation (Table 1, 2.C.2). However, while the mentor epistemologically framed the in-the-moment interaction as an “estimation based on the student’s experience,” the student framed it as a “calculation.” Specifically, as the prosody and gestures of the student in turn 8 reveal, he became genuinely excited and curious about the result of the calculation he meticulously programed in the spreadsheet (Table 3, 4.2). The mentor attended and responded to the student’s temporal (mis)framing of the situation (Table 2, 4.5): he noticed the difference in their epistemological framing, and in turn 9 he explicitly invited the student to make sense of the numerical value of his result (“So wait a minute, what do we get now?”). The student in turn 10 was momentarily confused and identified the calculated quantity as velocity, but when the mentor pointed out the mistake (turn 11), the student immediately corrected himself and identified it as wavelength (turn 12). Hence the student interpreted the mentor’s bid as invitation to correct his own mistakes (Table 2, 7.4).

Note that while the value in cell D2 in the spreadsheet was 0.0172 (see Figure 2), the student read it out loud as 0.02 (turn 10). This performance reflects a rather sophisticated physics disciplinary practice of attending solely to significant digits. The student’s question in turn 14 (“in centimeters?”) reflects another disciplinary physics practice of attending to units (Table 1, 2.C.3) which is part of the process of deciphering the scriptally implicit meaning of equation. Although the units of length the student used in turn 14 were incorrect (centimeters instead of meters), after a quick guiding note from the mentor (turn 15) the student was able to easily correct himself (turn 16). Again, the mentor did not tell the student the right answer but rather invited him to correct his mistake (Table 2, 7.4).

Figure 1. The student gestures in turn 8, Episode 10.

Figure 2. The original spreadsheet calculation in Episode 10. The order of the columns goes from right to left, as does Hebrew writing. The text in green is a translation into English of the labels, titles and notes—each corresponds to the cell beneath it. The formula above is the formula the student entered into cell D2 to get a numerical value according to the parametric derivation in the notebook.

The mentor’s explicit requests in turns 13 and 17 to write a label next to the calculated value in the spreadsheet while pointing to the specific cell both drew the student’s attention to important features and aspects in the representation (Table 2, 1.2) (the representation in this case was the numerical model in the spreadsheet) and pressed for a detailed documentation of the results (Table 2, 3.3). By doing so the mentor disciplined the student’s perception and practice (Table 2, 1). This move resulted in the student’s explication of the content of the label (turn 18, and cell E2 in the Figure 2, translation in E1). The student’s explication (“the wavelength in meters, in which the gravitational/…/will influence the wave velocity”), reflects progress in the student’s participation in deciphering the scriptally implicit meaning of the equation (Table 1, 2.C), as it went beyond the mentor’s request for label (“wavelength”)—the student was explicating the meaning of the symbolic and numerical result in words (Table 1, 2.C.3). The students’ prosody in turn 18 moved from an uncertain to tone to a certain statement. This change suggests that the student’s confidence in what he was saying increased as he was writing and speaking. The student’s participation described in this paragraph is indicative of progress in that he no longer merely followed the instructions related to the preparation and analysis of the numerical simulation, but was able to explain the meaning of each of the simulation components (Table 3, 3.1).

In summary, episodes 9 and 10 capture a change in the student’s nature of participation both at the level of the activities he was engaging in and the nature of his engagement. At the activity level, the student moved from engaging in deciphering the textually implicit meaning of the equation (Table 1, 2.B.2 and 2.B.3), to deciphering an important aspect of its scriptally implicit meaning (Table 1, 2.C.2 and 2.C.3). The analysis also highlights a change in the nature of the student’s engagement. The student was deeply engaged, and his conduct reflected enhanced agency relative to episodes 7a, 7b, and 8. His conduct also reflected a continuous sense of partnership with the mentor. The content of the student’s speech, as well as the product of the interaction (the spreadsheet) when compared across episodes 9 and 10 reflect a trend toward a more sophisticated participation in numerical investigations (Table 3, 3). For example, in episode 9 the mentor had to explicate the main ideas for the student, while at the end of episode 10 the student was the one who explicated and went beyond the mentor’s directions (Table 3, 3.1). The student’s prosody reflects increased confidence in what he was saying as well; i.e., reflecting increased agency) Table 3, 4.1). Moreover, as turn 8 shows, he became emotionally invested in the inquiry, and expressed an authentic sense of discovery (Table 3, 4.2) The mentor supported this change through the nature and sequencing of the tasks he designed for the student (Table 2, 6.2), by encouraging the student’s autonomy through explicit encouragement and praise of independent and inquisitive thinking and inviting the student to correct his own mistakes (Table 2, 7.1 and 7.4), by deliberate attentiveness to the student’s thinking, careful examination of the student’s written work, by attending and responding to the student’ thinking through careful examination of his written work, and by attending and responding to his temporal (mis)framing of the situation (Table 2, 4.5), by disciplining the student’s perception and practice by drawing his attention to important features and aspects of the representation (the spreadsheet) (Table 2, 1.2), and when required by re-explicating important ideas as well (Table 2, 1.2).

Episode 11 starts in the next turn and presents the second mediating task: the numerical simulation of the behavior of the gravity and surface tension components of Equation (2)(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) and their relative influence on the velocity. Episode 11a illustrates how the task was presented and construed. Episode 11b shows the related learning outcomes.

Episode 11a. (mentoring session, January 16, 2017, 1:59:46)

Table

1.	M:	/…/ so, from which wavelength can we say that this (surface tension) has no effect compared to this (gravity)?
…
3.	M:	Wait, I want to suggest something
4.	S:	Go for it.
5.	M:	Maybe we can make a column of wavelengths
…
9.	M:	So we can make a column of wavelengths and for each we can calculate this component and this component (points to the gravitational and surface tension components of Equation 2(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) in the notebook). We can see the size of each of them and decide together when we can neglect one relative to the other. What do you say?
10.	S:	OK. It’s going to be with a lot of dragging (dragging = copying cells in the spreadsheet)
11.	M:	Dragging is not a problem. You calculate once (in the spreadsheet) and then drag
12.	S:	Let’s do it!
…
15:	M:	Let’s do a column of wavelengths./…/which wavelengths do we want to check?
…
19.	M:	What is the smallest wave that you can make (in our experiment)?
20.	S:	Like that (∼30 cm, see the gesture in Figure 3)
21.	M:	We can do smaller than this
…
41.	M:	Let’s start (the simulation) at one centimeter
…		(the student works on the spreadsheet)
49.	M:	I think we can stop at 10 m
50.	S:	It’s OK.
…
55.	M:	/…/ Now let’s write the surface tension component and the gravity component here. We can also write here the depth component, right? This and that (points to the relevant components of Equation (2)(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) in the student’s notebook) we can (calculate) separately, and then we can calculate the velocity for each wavelength.
56.	S:	Yeah, but let’s start from this (points to the gravity components in Equation 2(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) )
…		(the student starts to enter the formulas for each component, while occasionally consulting with the mentor—see the first three columns—going from right to left as in Hebrew writing—in the spreadsheet in Figure 4)

We start by noting a pattern that was already present in Episode 9: the mediating task was not imposed on the student but rather constructed through discussion (turns 3–9). Note how the mentor framed the situation as a joint inquiry (Table 2, 7.2), and how he positioned the student as an equal partner, encouraging him to take the lead (Table 2, 7.3): He asked the student’s “permission” to suggest something (turn 3). He presented the potential outcomes of the task as valuable to him as well (e.g., “we can make,” turn 5) and suggested he numerically substitute real known values in the equation (Table 1, 2.C.2). The student responded accordingly, by taking the discursive role which was held by the mentor in episode 9; the student, rather than the mentor, was the one who made the decision on how to proceed (e.g., turn 4—“go ahead,” turn 12—“let’s do it”). This behavior marks another subtle enhancement in the student’s enactment of agency (Table 3, 4.1) while engaged in deciphering the scriptally implicit meaning of the equation (Table 1, 2.C) through the design of the numerical simulation (Table 1, 2.C.4)

The mentor’s plan of a coherent sequence of activities (Table 2, 6.2) was highly apparent in the design of the task, which used the spreadsheet the student created in the previous task to elaborate the inquiry and turn it into a numerical inquiry in which the numerical value of the gravity and surface tension components were calculated for different wavelengths. In turn 10 the student mentioned the affordance the spreadsheet offers for this calculation (“dragging”) and in turn 11 the mentor re-voiced the student’s idea to mark it as correct (Table 2, 5.2). The student’s enthusiastic decision to start working on the task (turn 12) emerged from the discursive and pedagogic moves of the mentor described above.

The mentor’s question “which wavelengths do we want to check” in turn 15, disciplined the student’s perception (Table 2, 1) by drawing the student’s attention to important features of the representation (Table 2, 1.2): in a numerical simulation, the range of the numerical values the simulation gets as an input is not infinite. The mentor’s question “What is the smallest wave that you can make?” (turn 19) aimed to draw the student’s attention to important features of the phenomenon (Table 2, 1.1) and explicitly elicited relevant prior information and embodied experiences (Table 2, 5.5) that were discovered in previous meetings (Table 2, 3.1), and were needed to determine the range of wavelengths. The student’s embodied reaction in turn 20 (see Figure 3 for the student’s expressive gesturing of the length) suggests that this prompt indeed worked. The mentor used the student’s reply as a basis for a discussion about the range of wavelengths that the numerical calculation should encompass (turns 21, 41, 49); hence, he was developing and elaborating on the student’s insight (Table 2, 5.3), and the student indeed went along (turn 50). In turn 55, the mentor guided the student on how to construct the spreadsheet for this calculation, but as turn 56 shows, the student further expressed his autonomy and agency (Table 3, 4.1)—e.g., the student was the one who decided which component he was going to calculate first (turn 56). The second and third columns of the spreadsheet in Figure 4 present the result of this task and show the calculated values of the gravity component and the surface tension component in Equation (2)(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) , for a series of wavelength that reflect the range of wavelength that could be produced in their laboratory conditions.

Figure 3. Gesturing the size of a characteristic wavelength (Episode 11a, turn 20).

Figure 4. The student’s spreadsheet calculation in Episodes 11a and 11b. The columns are read from right to left, as in Hebrew. The text in green is a translation into English of the column titles; each corresponds to the cell beneath it.

Episode 11b started about 4 min after episode 11a ended. During this time the student continued to work on the task and discussed it with the mentor.

Episode 11b. (mentoring session, January 16, 2017, 2:08:55)

Table

100.	S:	(examines the values in the gravity component and surface tension component columns on the spreadsheet—see Figure 4). I already see that this (gravity) has more influence (than the surface tension)
101.	M:	Wait, let’s see
102.	S:	Wow, this is cool!
103.	M:	Let’s also do the ratio between them. It will give us an estimation
…
110.	S:	Now it (the surface tension) influences three times more (than gravity—refers to the ratio of 2.89 between the surface tension and gravity components that was calculated for a wavelength of 1 cm—see Figure 4)
111.	M:	So the surface tension component is almost three times larger than the gravity component (for wavelengths of 1 cm)
	…
114.	S:	Wow, look! (approaches the computer screen enthusiastically and points to the ratios calculated for larger wavelengths on the screen—see Figure 5) it hardly has any effect! (the surface tension in larger wavelengths—see Figure 5)
115.	M:	In (a wavelength of) 2 cm (points to the corresponding ratio on the spreadsheet—see Figure 4) this is what we found earlier, right? They (the gravity and surface tension components) almost balance each other out. /…/ in 4 cm (points to the ratio calculated on the spreadsheet for a wavelength of 4 cm on the computer screen—see Figure 4) the gravity component is already 4 times larger /…/ but still you cannot neglect it.
116.	S:	But we are doing larger waves
117.	M:	OK, so let’s see. Here is 15 cm
118.	S:	It is really negligible.
119.	M:	0.01 (refers to the corresponding ratio for a wavelength of 15 cm—see Figure 4)
120.	S:	It is quite negligible. 100 times smaller, it (the value of surface tension component) is rather small (compared to value of the gravity component for a wavelength of 15 cm).
…
123.	M:	You know how we will be able to see this really well?
124.	S:	In 1 meter it does not have any effect
125.	M:	We can calculate the velocity for each wavelength
126.	S:	But then you will have to multiply by the hyperbolic tangent
127.	M:	Yes. Once with and once without it and see how it changes
128.	S:	With and without what?
129.	M:	The surface tension component
130.	S:	OK. OOOOOOKKKKKKK, WOW!

The student’s participation in turn 100 as an independent researcher demonstrates not only a deep sense of agency (Table 3, 4.1); rather, his participation in the next turns also demonstrates an increased sense of intrinsic interest in the particular process of inquiry (Table 3, 4.2): He was deeply engrossed in the visualization of the “hidden” meanings of Equation (2)(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) that the numerical simulation afforded, and was excited by the immediate comparison between the values of the gravity and surface tension components it provided. As turn 102 shows he was genuinely excited about its affordance for constructing insights (“Wow, this is cool!”).

In turn 103 the mentor started to press for articulated explanations by engaging the student in a task that could facilitate the generation of such explanation (Table 2, 4.6). He did so by elaborating on the task the student was engaged in to create an additional visualization that would illustrate the relative values of the gravity and the surface tension components for different wavelengths more explicitly. Specifically, in turn 103 the mentor suggested using the spreadsheet to calculate the ratio between the two components. This move again illustrates the coherent sequence of activities and the sensitive sequencing of their timing (Table 2, 6.1 and 6.2).

The student complied and in turn 110 he read out loud a ratio of 2.89 from the first line in the spreadsheet (see Figure 4, left column and first line of values). In turn 111 the mentor re-voiced this observation but rephrased it differently (Table 2, 5.4) (“almost three times larger”). In doing so, he disciplined the student’s perception (Table 2, 1), by drawing his attention to the essence of the term ratio when reading a particular representation of ratio, i.e., its numerical value (Table 2, 1.2). This move successfully disciplined the student’s perception: As turn 114 and the enthusiastic gesture in Figure 5 show, it led to the student’s insight (“Wow, look!”): the gesture and speech in this turn reflect a feeling of authentic discovery (Table 3, 4.2); that in large wavelengths the surface tension has a negligible effect relative to gravity.

Figure 5. Enthusiastic gesture, turn 114, Episode 11b.

In turn 115 the mentor drew the student’s attention to the coherence of his numerical calculation with his previous calculation of the wavelengths in which both components were equal (around 2 cm). In doing so he explicated the connection between the student’s current work and his previous work (Table 2, 3.1), and marked conceptual progress (e.g., achieving coherence) (Table 2, 3.2). The mentor continued to point to wavelengths in which the surface tension component was small relative to the gravity component, but not negligible (“2 cm,” “4 cm”), engaging the student in using the numerical simulation of the equation to find the quantitative relative behavior of components of Equation (2)(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) (Table 1, 2.B.3), while explicating the physical meaning of the numerical results (Table 1, 2.C.4); i.e., generating inferences about the relative influence of gravity and surface tensions for different wavelengths. Hence the mentor was leveraging an investigation of both the textually implicit and scriptally implicit meaning of Equation (2)(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) (Table 1, 2.B and 2.C). The student’s spontaneous remark (turn 116) that in his experimental setup, the waves were significantly larger than those the mentor was referring to, signal a further development in his participation in theoretical exploration (Table 3, 1). Specifically, he achieved a higher level of engagement in mathematical modeling (Table 3, 1.2). The mentor responded (turn 117) by asking the student to use the spreadsheet to estimate what the ratio between the surface tension and gravity components for a larger wavelength (“15 cm”) would be, which is more in line with their experimental settings. The student examined the spreadsheet (Figure 4, line 11 in the spreadsheet, where the wavelength is 0.16 m or 16 cm) and said that “It is really negligible” (turn 118). Here too, the mentor (turn 119) pressed for an articulated explanation (Table 2, 4.6): he read out loud the value of the ratio for the wavelength of 15 cm (“0.01,” turn 119). Doing this in response to the student’s seemingly correct response (“really negligible,” turn 18) explicated that negligible does not mean zero, and thus invited the student to explicitly discuss this issue. This move indeed generated a further articulation and explication from the student in turn 120 (“It is quite negligible. 100 times smaller, it is rather small.”) The student’s last interpretation reflects yet another growth in his participation: a gradual move from talking about the neglected quantity in the approximation as an influence that ceases to exist in the conditions of the approximation (Episode 3) to talk about it as an influence that is always present but becomes very small relative to other influences in these conditions; namely, starting to use adjectives of quantity in a relative, rather than an absolute sense (Table 3, 2.2). This change in his discourse manifests a more sophisticated participation in deciphering the scriptally implicit meaning of the equation (Table 1, 2.C). The mentor’s pedagogical decisions and discursive moves where highly influential in facilitating this process.

The impressive part is that the mentor did not stop here, but suggested a further elaboration of the task, to pursue a more articulated conclusion (“You know how we will be able to see this really well?,” turn 123)—closely sequencing the activities to the student’s progress (Table 2, 6.1). The student did not hear this suggestion. In turn 124 he was still engrossed in his novel insight, saying “In 1 meter it does not have any effect.” This behavior further reflects the student’s deep emotional involvement in the inquiry and his authentic sense of discovery (Table 3, 4.2). The mentor made another offer. He suggested (turn 125) that the student would also numerically calculate the value of the velocity with and without the surface tension component. The student’s response in turn 126 (“But then you will have to multiply by the hyperbolic tangent”) is a correct articulation of the central part of the design of the simulation the mentor suggested. Hence, it demonstrates an additional development of the student’s engagement with the simulation since at this stage, he was able to independently design and program the numerical simulation and no longer needed the mentor’s assistance in this process (Table 3, 3.2). The students’ fluent use of the term “hyperbolic tangent” in turn 126 regarding the depth component in Equation (2)(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) is a result of a separate extended mediation process that took part in these three meetings, which we do not discuss here. In turn 127 the mentor reiterated the need to calculate the velocity with and without “it,” referring to the surface tension component of Equation (2)(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) . But as the student’s response reveals (turn 128—“With and without what?”) that this reference was not transparent to him. The mentor (turn 129) explained (Table 2, 2.2) that the calculation of the velocity should be done with and without the surface tension. The student enthusiastic reaction in turn 130 (“OK. OOOOOOKKKKKKK, WOW!”), apart from reflecting his genuine sense of discovery (Table 3, 4.2), demonstrates another growth in his ability to participate in the investigation of the scriptally implicit meaning of the equation (Table 1, 2.C). At the beginning of episode 9 he followed the mentor’s directions when preparing the spreadsheet for the numerical calculation. In episodes 10, 11a, and the beginning of episode 11b, his participation became more independent in that he was becoming more capable of handling the technical aspects of the design of the numerical simulation, and his speech also included references to the meanings of various components and numerical results in the simulation. However, his enthusiastic response in turn 130 presents a higher level of engagement: his excited speech reveals that he foresaw what the simulation could potentially demonstrate. He was excited because he anticipated what the results would be of running the simulation before the actual programming and execution took place (Table 3, 3.2), thus achieving an impressive level of engagement in a numerical simulation (Table 3, 3).

From a purely mathematical point of view, the three elaborations of the task (1—comparing the components, 2—comparting their ratio, and 3—comparing the value of the velocity with and without the component that could be neglected) do not provide additional significant information. But from a physics education point of view, they reflect an impressive sensitive timing and sequencing of activities attuned to the student’s progress (Table 2, 6.1), as well as planning of a coherent sequence of activities (Table 2, 6.2). Each elaboration further guided the extraction of the scriptally implicit meaning related to the approximation of neglecting the surface tension component relative to the gravity component. While the second elaboration added to the insights from the initial task by a further explication of the relative values of the two components separately, the third elaboration explicated the relative influence on the velocity, thus pushing the student to consider the entire equation as a whole.

The name the student gave the Excel spreadsheet file (Figures 2 and 4) when he saved it at the end of the meeting was (translated from Hebrew) “very important special and complicated calculations.” In the theoretical background section of his final research report, the student decided to include two graphs that he later produced from the spreadsheet in Figure 4. The first presented the ratio of the surface tension and gravity components vs. the wavelength. The second presented the velocity vs. the wavelength, with and without the surface tension component. These actions reflect the student’s sense that what he was doing in these activities was highly significant and meaningful to his later participation in the project (Table 3, 4.2).

Episode 12 is drawn from the last interview we did with the student. The interview took place after the student had already submitted his final research report and successfully defended his project in an oral exam. The student could access the research report throughout the interview. The interviewer (second author) asked the student why although the surface tension component appears in the version of Equation (2)(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) in the theoretical background, it does not appear in the analysis of the empirical results in the report.

Episode 12. (stimulated recall interview with the student, September 25, 2017)

Table

212.	I:	How do you feel about all these approximations? You did this approximation, and deleted the surface tension expression here, you simplified the formula.
213.	S:	I’m totally happy about the surface tension (neglecting the expression) /…/ It is a really, really good approximation. All our wavelengths were larger than 0.5 m, and this approximation worked from 10 cm /…/ But yes, at the beginning I was really worried about this approximation.
214.	I:	That you kind of missed something?
215.	S:	Yes. It’s like you ignore something that is really important, something that might have a strong influence.

The student’s answer reflects a deep appropriation of the scriptally implicit meaning that led to the approximation (Table 3, 2). Compare how the student described his initial grasp of the approximation in turn 213 “at the beginning I was really worried about this approximation” and his current one “It is a really, really good approximation. All our wavelengths were larger than 0.5 m, and this approximation worked from 10 cm.” This difference reflects a process of a gradual move from concern and disbelief about the validity of the approximation (neglecting the influence of the surface tension in the conditions of the experiment), to conviction (Table 3, 2.1). As episodes 9, 10, and 11 show, this process was explicitly supported by the numerical investigation of the equation that provided a concrete sense of its textually implicit and scriptally implicit meanings, and was facilitated by the mentor’s impressive discursive moves.

Discussion

Analyses of cognition in activity differ in their levels of aggregation in two ways. One difference is in whether the analysis focuses on individual information processing, treating interaction with other people and resources in the environment as a context, or focuses on processes of an activity system composed of the several individuals present (if there are more than one) along with their material and informational resources. The other difference in aggregation involves the complexity and time scale of the activity that is analyzed. (Greeno, 2013, p. 41).

As the literature review at the beginning of the paper demonstrated, many studies examined students’ use of mathematics in physics. However, most of these studies focused on students’ cognition and knowledge in problem solving and not on how reasoning with physics equations can emerge from interaction with a mentor/teacher around the practices involved in authentic inquiry. The naturalistic nature of the settings and the extended time scale of the learning process, along with our high familiarity with the Acheret community of practice, convinced us that if we truly wanted to make sense of the beautiful learning process we witnessed, a focus on the process of the activity system would better serve our purposes than a focus on individual information processing. We found Barbara Rogoff’s (1995, 1997) conceptualization of learning as taking place in the community (i.e., culturally organized activities), the interpersonal process (i.e., guided participation), and the individual process of learning (i.e., transformation of participation) highly instrumental in the examination of our large corpus of data. The longitudinal fine-grained discourse analyses of the interaction afforded an articulation of the unfolding of the interrelations between these three planes in a learning process in which an 11th grade student came to appropriate a physics equation into his own inquiry. We believe that in so doing, this study contributes to previous work on the use mathematics in physics by further articulating the nature of an important practice of doing physics (e.g., Einstein, 1934; Maxwell, 1990) and the process of learning how to meaningfully engage in it. In the following subsections, we explain why.

Articulating the culturally organized activities

While the practices and activities listed in Table 1 also reflect the well-motivated instructional choices of the mentor, we wish to stress that they are historically situated cultural activities (Rogoff, 1995). First, many of these activities perpetuated and evolved from the authentic scientific practice of physicists, such as the use of numerical simulations in physics inquiry. Rahm et al. (2003) characterized this process in research apprenticeships of high school students who are collaboratively mentored by teachers and professional scientists as the emergence of authenticity: “authentic science is an emergent property of a dynamic system of learning precipitated by the interactions among students, teachers, and scientists that occur within the contexts defined by the internal and external constraints of the cultures of the schools and communities within which they operate” (p. 737). Hence, the purposeful activities in Table 1 reflect the historically situated culture of doing research with students in the Acheret community.

Acheret functions as a professional community (Wenger et al., 2002) of teacher research mentors (for a detailed account of the values, goals, and characteristics of this community see Faraj et al., 2021; Zohar et al., in preparation). The overarching goal of this community is to integrate extended authentic physics inquiry experiences into the formal instruction of physics in high school. The co-construction of theory and its dialogue with empirical work, which was manifested in the study, is considered an essential component of authentic students’ inquiry in this community. Prospective mentors are enculturated into the community through its extended training program (full day, once a week, for two years), and its professional workshop for active mentors (3 h, every 3–4 weeks). Training is organized in an apprenticeship model guided by core members of the community. The mentor in this study is a graduate of the first cohort of trainees and a core member of the community. The specific activities described under “anticipating mathematical formalisms through intuitive sensemaking” and “reading through the equation” are also employed in the training program. In fact, some of them, such as using dimension analysis to derive a physics equation up to the value of its constants, and designing numerical simulations to investigate complex equations, are explicitly taught in the training program. The specific mentor is considered one of the experts of the community in applying these methods in inquiry. He teaches them in the training program and often guides novice mentors and advises experienced mentors on ways to productively use them with students.

Some of the activities listed in Table 1 cohere with findings from previous studies of students’ use of mathematics in sessions of physics problem solving. Activities, such as “Identifying the conceptual meaning represented by each variable and parameter” (2.A.1.), “Identifying compound components in an equation as representing conceptual meaning” (2.C.1.), and “Explicating the meaning of symbolic and/or numerical result in words, and attending to its units” (2.C.2.) can be considered part of previously described epistemic games of “mapping mathematics to meaning” and “mapping meaning to mathematics” (Tuminaro & Redish, 2007). In the same manner, “Solving the equation/model for special cases” (2.B.2.) corresponds to the hypothesized cognitive mechanism of activating a particular interpretive device (action schema) from the cluster of interpretive devices termed “special cases” (Sherin, 2001b). It could be argued from a cognitive perspective that an activity, such as “Identifying compound components in an equation as representing conceptual meaning” (2.C.1.) may require an activation of a specific symbolic form, or can support its generation (Sherin, 2001a).

However, we would like to stress that the articulation of the culturally organized activities listed in Table 1 is not a mere “renaming” of known epistemic games and/or symbolic forms or the identification of a few more. Indeed, some of the components in this list have been discussed in cognitive terms in previous studies. However, the situative analysis afforded a reorganization of these activities into a coherent hierarchical structure that provides empirically based evidence supporting the claim that mathematics is used in physics as a language. Specifically, the analysis shows that learning to make sense of physics equations and appropriating them into one own’s inquiry can be productively perceived as a process of learning to read fluently with a high level of reading comprehension. We consider this to be the major contribution of this study and argue that it has important implications for the teaching and learning of physics and mathematics.

First, it suggests that physics and mathematics educators may have something to learn from studies that examine how children become good readers and how to support children who have difficulties in reading comprehension. Second, consistent with studies in the learning sciences that call for designing learning processes that develop intuition before formalism (e.g., Nathan, 2012), the analysis shows how the practice of anticipating mathematical formalisms through intuitive sensemaking supported and nurtured learning to participate in the practice of reading through a physics equation. This emerges from the discourse analysis in the mentor’s and the student’s specific references to intuitive experiences when discussing the formalism. It was also apparent in the classification of the episodes into the specific activities summarized in Table 1, which clearly shows that the formalism was only treated in later episodes.

The analysis also captures the complexity of learning to participate in the practice of reading through a physics equation, since it required the student to decipher the textually explicit meaning of the equation, its textually implicit meaning, and its scriptally implicit meaning, where each of these sub-practices entailed a coordinated set of activities which were thoughtfully orchestrated and facilitated by the mentor through discourse. To decipher the textually implicit meaning of the equation, the student had to understand its textually explicit meaning. Since he was diving into a very complicated mathematics equation in a relatively short time, and was doing so with very limited background, he often needed help deciphering the textually explicit meaning even during the more advanced stages of his reasoning. In the same manner, he would not have been able to seriously consider the scriptally implicit meanings of the equation without understanding its textually implicit meaning.

Guided participation

We found that the cognitive apprenticeship in authentic scientific inquiry that we studied employed many teaching moves and included many of the sense-making practices that have been documented (in more cognitivist terms) in studies of high-quality teaching in classrooms: the first two clusters of instructional practices in Table 2 (clusters 1 and 2) resonate with the didactic principles suggested by Karam (2014) for teaching the structural role of mathematics in physics, although Karam did not examine how the implementation of these practice affects students’ learning as we did. The instructional practices that compose clusters 1–5 in Table 2 resonate with the principles of ambitious science teaching (Windschitl et al., 2018), and the instructional practices that compose clusters 4–7 in Table 2 parallel the principles of responsive teaching (Richards & Robertson, 2016). This is not that surprising given that ambitious science teaching (Windschitl et al., 2018) and responsive teaching (Richards & Robertson, 2016) are instructional approaches in science education that foreground scientific practice rather than content and that the mentor was an experienced and talented physics teacher. We note, however, that to the best of our knowledge, engagement in mathematical sensemaking within a scientific inquiry is not a focused practice that has been studied in research on ambitious science teaching and responsive teaching.

The analysis contributes to this literature by a microanalytic articulation of the ways in which instructional practices unfold in discourse between mentor and student. The analysis provides detailed examples that illustrate how various orchestrated sets of instructional practices shaped and influenced the student’s participation, where the transformation of the student’s participation is described as changes in relative positioning, expressions of autonomy and involvement, competency of practice, and quality generated explanations. For example, the analysis provides multiple examples of how the mentor’s use of clarifying questions, even when at first sight it seemed that the student provided the right answer, and the mentor’s constant pressing for articulated and detailed explanations, revealed areas of difficulty that needed further attention on the one hand and supported the student in constructing more robust explanations (and understanding) on the other. Other examples illustrate how engaging the student in tasks that facilitated different articulations of an explanation (i.e., the three different ways of showing that the surface tension component is negligible relative to the gravity component when calculating the wave velocity in the conditions of the experiment), generated confidence about his understanding of the phenomenon. The analysis also presents multiple examples of how the student’s autonomy was facilitated in discourse: by consulting with him and allowing him to explore his suggestions, by praising and revoicing his insights, by inviting him to find his mistakes and correct them, and others. The analysis points to the student’s specific reactions to these bids, as manifested in the student’s expressions of agency and autonomy that was reflected in his self-positioning, actions, and speech. The analysis also underscores how challenging the appropriation of the advanced equation was for the student, and how the mentor’s various moves to discipline his perception led to improved understanding of the nuanced meaning of the equation, which came to bear for example on the explanations the student provided. We consider this part of the analysis to make an important contribution, particularly because most studies that have examined students’ use of mathematics in physics (see the literature review) have focused on students’ cognition and knowledge in problem solving rather than on how a sophisticated use of mathematics in reasoning and sensemaking processes in physics can emerge from the interaction between the students and a mentor/teacher, and how the available materials in the learning environment shape this process.

We believe that this study also contributes to scholarly work on mentorship in research apprenticeships. Interviews with mentors and mentees in research apprenticeships have highlighted mentees’ gains in research skills, attitudes, and motivation toward learning science, and learning to think and work like scientists (Barab & Hay, 2001; Hunter et al., 2007; Sadler et al., 2010). In their review, Schunk and Mullen (2013) described students’ experiences in these settings as cognitive apprenticeship (as opposed to a traditional apprenticeship). They argued that while traditional apprenticeships focus on the performance of specific tasks, cognitive apprenticeships (Collins et al., 1989) facilitate a deeper understanding of concepts, meaningful engagement in problem-solving processes, and ways to learn how to apply skills independently in diverse settings (Mullen, 2020). Other studies have used surveys to examine the relationships between students engaged in research apprenticeships and their research advisors (Robnett et al., 2018; Tenenbaum et al., 2001). These studies have underscored the different facets of mentoring students’ research. The term "instrumental mentorship" was used in these studies to describe guidance aimed at helping students to develop skills as well as facilitate access to resources that can help students succeed in scientific investigations and develop the professional identities of scientists. The terms "psychological" or "emotional" mentorship have been used to describe role modeling, empathizing, and counseling (Robnett et al., 2018; Tenenbaum et al., 2001). The mentor’s instructional practices listed in Table 2, and even more so their illustrative manifestation in the in-vivo discursive moves as documented by the microanalytic analysis of the discourse, contribute to this literature by articulating how cognitive apprenticeship and instrumental and psychological mentorship are facilitated in discourse. They also provide examples of how mentorship-discourse can nurture, facilitate, and sustain a student’s epistemological stance toward physics as a conceptual and coherent knowledge structure (Hammer, 1994), physics equations as reflecting meaning that can be tractably connected to common sense, and physics learning as a process of sensemaking (Gupta & Elby, 2011). Moreover, the parallels with accounts of high quality science teaching, such as ambitious science teaching (Windschitl et al., 2018) and responsive teaching (Richards & Robertson, 2016) suggests that teaching and mentoring have a great deal in common.

Individual transformation of participation

Rogoff (1995) conceptualized learning through participation at the individual level, as preparation for future participation; namely, becoming prepared for subsequent involvement in related activities. The summary in Table 3 highlights trends of change in the student’s participation, which are articulated in detail in the discourse analysis of the episodes. These changes were manifested in the nature of the theoretical explorations the student took upon himself over time, his treatment of approximations in physics inquiry, his competency in designing and employing numerical investigations, and his implicit and explicit expressions of agency and sense of autonomy. We believe that the articulated changes in the student’s in-vivo participation and the explicated in-the-moment connections between these changes and the mentor’s instructional moves revealed in the discourse analysis contribute to the understanding of the challenges involved in learning to use mathematics in physics inquiry and highlight pathways to support this process.

Note that the situative approach to the analysis has limitations. We cannot estimate with certainty, based on our data, the extent to which the changes in the student’s participation described above were stable changes that transcended the specific experience. However, there were several indications that suggest that at least some of these changes were indeed stable and profound. Take for example the excerpt in episode 12, from the retrospective interview with the student after the project. The student explicitly described his initial disbelief and suspicion about the approximation (neglecting the influence of the surface tension relative to the influence of gravity). The way he described it reflects a sense of “black magic” that many physics students often feel with regard to the approximations and decisions to neglect components of equations that are so casually done in physics textbooks and lessons during the mathematical modeling of physical systems (e.g., “$\sin \theta =\theta$ in small angles”). In contrast, the student explicitly expressed his current deep conviction in the validity of the approximation and provided a scientifically sophisticated and well-justified argument in support of this conviction. Further evidence comes from a note we received from the mentor after he read the first draft of this manuscript. He told us that the student continued to work with him on the project long after he was successfully tested on it. He worked on it voluntarily alone and with the mentor, without credit, after school hours, during most of the next school year (12th grade). The mentor wrote that as the project progressed, increasingly sophisticated analyses and derivations based on Equation (2)(2) $$v=\sqrt{\left(\frac{g\lambda }{2\pi }+\frac{\sigma 2\pi }{\lambda \rho }\right)\mathit{tanh}\left(\frac{2\pi h}{\lambda }\right)}$$(2) emerged. He provided impressive concrete examples. During this period the student had to work much more independently since the mentor was far less available.

Conclusion

The analysis constructs a theoretically and empirically based argument that the use of mathematics in physics can be conceptualized as the use of language. The analysis demonstrates that learning to make sense of physics equations and appropriating them into one own’s inquiry can be understood as a process of learning to read fluently with a high level of reading comprehension, a claim that has various implications for the teaching and learning of physics and mathematics (e.g., building a quantitative intuition before introducing the formalism). The analysis also illustrates how sensitive mentorship of research apprenticeships involves an orchestration of instructional practices and pedagogical decisions that also characterize high-quality practice-based and student-centered classroom instruction, and link these to changes in the student’s participation. In addition, the analysis illustrates how an individual learning process that involves making sense of and learning to use physics equations in a scientific inquiry can be productively conceptualized as a process of transformed participation in a community of practice.

Acknowledgments

We would like to note that the inspiring mentorship and instruction analyzed in this manuscript are the work of Mr. Shay Harazi, the talented and inspiring teacher and research mentor in this case study. We would like to express our sincerest gratitude to Shay and the student he mentored in this project for warmly and openly sharing their thoughts and experiences with us, and for their interest in the study. We would also like to thank Mr. Moshe Reich and Dr. Amos Cohn for inviting us to study the unique learning experience in physics that they nurture at the Acheret center. We would like to express our deep gratitude to Andrew Elby, the Executive Editor, and the two anonymous reviewers, for their thoughtful feedback and suggestions. Shulamit Kapon is grateful to Andrea diSessa, Bruce Sherin, Edward Redish, Andrew Elby, Chandra Turpen, Jennifer Radoff, and Erin Sohr for helpful discussions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This study was supported by the Israel Science Foundation under grant #325/17.

Notes

1 Readers with a background in physics may also recall how the existence of electromagnetic waves is inferred from Maxwell’s equations, or how mean field approximations are employed to simplify equations in cases of many particles.

2 Base units are the standard units of measurement defined by the International System of Units (SI) for the seven base quantities, from which all other units can be derived. For example, length is a base quantity, and its base unit is the meter. However, tension (T in Equation 2(1) $$v=\sqrt{\frac{T}{\mu }}$$(1) ) is force, which is not a base quantity, and can be expressed in terms of the base quantities mass, length and time. Thus, the Newton, the measurement unit of force, can be expressed by the base units of kilogram, meter, and seconds, as $1_N=1\frac{kg·m}{{\mathit{sec}}^2}.$

3 The equation $y\left(x,t\right)=A\sin \left(\frac{2\pi }{\lambda }x\pm \frac{2\pi }{T}t\right)$ describes how a one-dimensional traveling wave (y) changes spatially (over the x axis) and over time (t). The sine function is a periodic function with values bounded between −1 and 1. A is the amplitude—the absolute maximum value of the wave. T is the wave period—the time it takes the wave to complete one cycle and repeat its values. $\lambda$ is the wavelength—the minimum distance over which the wave's shape repeats itself.

4 The frequency is the number of periods in a unit of time. When multiplied by an angle of $2\pi$ radians ($2\pi$_radians = 360^o) it provides the angular frequency $\omega =\frac{2\pi }{T}.$ The wavenumber is the number of wavelengths in a unit of distance. When multiplied by the angle of $2\pi$ radians it provides the angular wavenumber $k=\frac{2\pi }{\lambda },$ which is commonly referred to as just the wavenumber. In wave theory the terms angular frequency and angular wavenumber are used since angles are the independent variable of the sine function sin(x), which is the simplest function that solves the wave equations.

Appendix A:

Tentative predictions or explanations of quantitative trends as a component of mechanistic reasoning

A proficient “reading comprehension” of a physics equation is built on an intuitive sense of the quantitative account of the behavior of the system or phenomenon that the equation models, and how it is represented in the equation. We model this process by elaborating on the Russ et al. (2008) model of students’ discourse that reflects mechanistic reasoning. Building on studies in the philosophy of science, and observations of first grade students’ reasoning about falling objects, Russ et al. suggested a framework for discourse analysis that aids in identifying and analyzing students’ mechanistic reasoning. They identified seven structural components (“codes”) for mechanisms that can be identified in students’ explanations, and listed them in a numerical order that reflects their logical connections and increased scientific sophistication (pp. 512–513): (1) describing the target phenomenon, (2) identification of setup conditions, (3) identifying the entities involved, (4) identifying the activities the entities engage in, (5) identifying the relevant properties of the entities, (6) identifying the spatial and local organization of entities, (7) chaining backward or forward; namely, using knowledge about the causal structure of the world to make claims about what must have happened previously to bring about the current state of things (backward) or what will happen next given that certain entities or activities are present now (forward). Russ et al. also pointed to the use of analogies and animated models in mechanistic reasoning.

Let us examine the Russ et al. (2008) notion of “chaining backward/forward,” which reflects the highest level of mechanistic reasoning in students’ talk according to their framework (p. 513):

When students chain backward, they answer the questions, “What activities could have given rise to entities with these properties?” or “What entities were necessary in order for this activity to have occurred?” When students chain forward, they answer the questions, “What activities could these entities with these properties be expected to engage in?” or “If this activity occurred, what changes would I expect in the surrounding entities and their properties?” For example, a student might say, “I know that objects fall straight to the ground in air but not in liquids, so there must be some force pushing up on objects in liquids that keeps them from falling.”

However, we suggest that there is a higher level of “chaining” that is not captured in the Russ et al. framework, and which Russ et al. (2008) probably did not include because they were observing first graders rather than high school or undergraduate students of science. We term this level tentative predictions or explanations of quantitative trends. We define this as hypothesizing a qualitative account of a quantitative description of the system’s behavior, based on common sense rather than a mathematical derivation. An illustrative example of this type of reasoning, using the Russ et al. example for chaining cited above, is the hypothesis that if the force (or any other relevant “entity”) increases, then the acceleration, velocity, or any other “entity” in question increases, decreases, remains unchanged, decreases or increases up to a minimal or maximum value, etc.; while grounding this hypothesis on mechanistic reasoning rather than on a mathematical derivation (e.g., more force leads to more result).

Thus, suggesting tentative predictions or explanations of quantitative trends requires, at the basic level, identifying the tenets of the Russ et al. (2008) framework, such as identifying entities, their properties, and their spatial organization. However, it also involves “chaining” that goes beyond what Russ et al. describe as “what must have happened previously to bring about the current state of things (backward) or what will happen next given that certain entities or activities are present now (forward)” (p. 513). Moreover, students can represent these trends metaphorically by mathematical representations (e.g., “like a sine function”). We use the adverb “metaphorically” here since at this stage there is no commitment to a particular mathematical representation; it is merely used to qualitatively express a quantitative trend.

Appendix B:

Transcription conventions

The analysis was conducted on the original Hebrew transcriptions. The episodes presented in this paper were translated into English during the writing of this paper. The transcripts use the following conventions, which are often employed in studies of conversation analysis (Sacks et al., 1974):

1. “(#)”—Indicates a length of a pause in seconds.

2. “…”—Indicates an untimed pause.

3. “/…/”—Indicates that part of the original transcript was omitted.

4. “[”– Indicates a point at which an overlap by another speaker starts.

5. “(text)”—Indicates text that provides additional information, such as references to related bodily movements and interpretations.

6. Speakers: “S:”—student, “M:”—mentor, “I:”—Interviewer