Using and connecting multiple representations is a widely promoted teaching practice for strengthening students’ understanding of complex information and mathematical concepts (occurring in many practical catalogs of good teaching practices, e.g., Lesh, 1979; Moschkovich, 2013; NCTM, 2014). Empirical studies on students’ learning with multiple representations, however, have revealed that the presence of multiple representations alone is not sufficient, thus many of students’ challenges with representations have been identified, and design features in learning environments have been developed and evaluated that can help to overcome these challenges (Ainsworth, 2006; Renkl et al., 2013).

Compared to the rich state of research on student learning and on design features for learning environments, it is surprising that teaching with multiple representations has received much less research attention (Stylianou, 2010). This research gap has been also emphasized by Bossé et al. (2011) and Dreher and Kuntze (2015), who studied teachers’ knowledge and views (in interview studies and vignette-based simulations) about different representations. They showed that many teachers were not aware or did not notice key instructional issues in dealing with representations; but these are an important part of teachers’ pedagogy for promoting mathematical reasoning (Herbert & Bragg, 2021). This raises the following research question:

What exactly characterizes teaching practices in managing classroom discussions that support students in dealing with multiple representations?

To examine this research question in the current paper, we develop a conceptual framework for characterizing teaching practices based on empirical insights on student learning and design features and a topic-specific navigation space. By qualitatively investigating cases of video-recorded whole-class discussions, we decompose the teaching practices into those teacher actions that can indeed strengthen students’ understanding in the negotiations of meanings of different representations (Steinbring, 2005). In doing so, we contribute to the program of decomposition of practice, which according to Grossman et al. (2009) involves “breaking down practice into its constituent parts for the purposes of teaching and learning” (p. 2058) in teacher education.

To conduct this qualitative study, we chose the mathematical topic of conditional probabilities, as typical challenges and most accessible representations have been carefully identified for this topic (e.g., Binder et al., 2015; Gigerenzer & Hoffrage, 1995; McDowell & Jacobs, 2017). However, these representations have so far been identified mainly in written tests, without investigating the enacted teaching practices supporting students’ processes of unfolding information and connecting multiple representations.

In the first section, the theoretical background of the current study is provided, starting with brief sketches of the state of research on conditional probabilities and on students’ challenges and design features for connecting multiple representations. In the second section, we present our conceptual framework for characterizing teaching practices as navigation between concept elements and different representations and refine the research question. In the Methods section, the methodological framework of the current case study is presented. Empirical insights are given in the Analysis section by comparing two cases with many others. In the last section, the identified navigation practices are discussed.

Theoretical background

Because investigations of teaching practices can dive deeper when the particular mathematical structure of the topic in view is taken into account (Jacobs & Empson, 2016; Prediger et al., 2022), the theoretical background in this study starts with the state of research on students’ processing of conditional probability information (first subsection). Subsequently, we present the general findings on students’ challenges with representations and design features to overcome these challenges (second subsection).

State of research on conditional probability information: relevant mathematical structures and accessible representations

Since Kahneman and Tversky (1972) pointed to problems with probability judgments, many empirical studies have identified multiple challenges that students and adults encounter, in particular with understanding conditional probability information in texts and solving Bayesian inference problems (Eichler et al., 2020; Gigerenzer & Hoffrage, 1995; Kahneman & Tversky, 1972; McDowell & Jacobs, 2017). Recent research reviews list, for example, the base rate fallacy or alternative subtraction strategies and often mention the confusion of conditional probability P(A|B) with joint probability P(A ∩ B) (Binder et al., 2020; Gigerenzer & Hoffrage, 1995). We focus on the latter for the case of statistical information on different parts and part-of-parts.

One example for this kind of statistical information that can cause confusion is depicted in Fig. 1: specificity and sensitivity of medical tests are regularly given as conditional probabilities under the condition of being infected/not infected. Epidemiologists might be more interested in the joint probabilities in question 1, while patients might be more interested in other conditional probabilities under the condition that receiving test results must not confuse question 1 with question 2. In this example, thinking about the involved parts, parts-of-parts, and wholes helps to process the given highly condensed information, and the reasoning about the involved parts and wholes is supported by the visual area model.

Fig. 1
figure 1

Conditional or joint probabilities? Example for the mathematical topic of complex probability information

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A strong research tradition in cognitive psychology and statistics education has developed on identifying accessible representations as an important design feature to help overcome the typical challenge of confusion between joint and conditional information: many studies have compared in which language and/or visual model written information should be given so that students and adults interpret the texts most successfully. For example, tasks in which the numbers are presented in natural frequencies tend to be solved more successfully than those with probabilities (Binder et al., 2015; Gigerenzer & Hoffrage, 1995; McDowell & Jacobs, 2017), which suggests avoiding the use of formal probabilities in favor of statistical frequencies. Leuders and Loibl (submitted) provide evidence that the focus on part-of-whole and part-of-part structures can also substantially support students in interpreting texts when percentages instead of natural frequencies are given.

Besides the language chosen for the given texts, many studies have shown that tasks presented with visual models tend to be solved more successfully than those in text-only format (McDowell & Jacobs, 2017). The various visual models that have been examined with respect to their accessibility include 2 × 2 contingency tables (e.g., Binder et al., 2015), tree diagrams (e.g., Binder et al., 2015; Sedlmeier & Gigerenzer, 2001; Yamagishi, 2003), frequency nets (Binder et al., 2020), and area models (as in the example in Fig. 1), which are also called unit squares (Böcherer-Linder & Eichler, 2017) or eikosograms (Pfannkuch & Budgett, 2017).

Some empirical studies have also determined the relevant mathematical structures that require the most attention for understanding conditional probability information, showing that visual models that point out the underlying nested sets (Böcherer-Linder & Eichler, 2017; Yamagishi, 2003) or part-of-part structures (Leuders & Loibl, submitted) best support the understanding of conditional probability information.

These empirical findings on differential accessibility of given texts and visual models with conditional probability information provide the rationale for our choices of the language of parts-of-parts (instead of probabilities, as suggested by Leuders & Loibl, submitted) and area models as the visual models with a high potential to visualize ratios in relationship to each other (Böcherer-Linder & Eichler, 2017; Pfannkuch & Budgett, 2017). They also hint at the need to make the mathematical structures of nested sets and the part-of-part structure explicit.

Based on this rich state of research on potential representations for conditional probability information, we have chosen the representations and summarized them in Fig. 2. Besides the given text, the chosen visual model, and the symbolic representations, Fig. 2 also lists three different language representations (in Halliday’s, 1978 sociolinguistic sense of register) needed to talk about the representations. The contextual language is mostly bound to the context situation in the given text and the technical language to the symbolic representation. The meaning-related language refers to the language needed to articulate the part-whole or part-of-part structures. As earlier research showed the need to also explicitly connect these different language registers, we include them as separate representations.

Fig. 2
figure 2

Multiple representations in our approach to conditional probability information (example from Fig. 1)

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Having summarized the rich state of research on students’ challenges and possible design features of conditional probability information, McDowell and Jacobs (2017) conclude their research review with the call for “future work [that] will focus not only on different performance criteria but on the processes leading up to the correct solution as well, with the aim to understand why many participants continue to have difficulties solving … [the] problems” (p. 1301). In our view, the focus on these processes should include the teaching practices enacted for supporting students’ pathways towards understanding.

State of research on students’ challenges and design features for using multiple representations

Far beyond conditional probabilities, the use of multiple modes of representation has been promoted in mathematics education since the 1960s for enhancing students’ understanding of abstract mathematical concepts and complex information with these multimodal resources (Bruner, 1966; Calor et al., 2020; Duval, 2006; Lesh, 1979; Mildenhall & Sherriff, 2018), including with dynamic representations through digital technologies (Falcade et al., 2007, and many others). In particular, the construct of semiotic mediation was used to explain the relation of external and internalized representations to promote the development of conceptual understanding (Falcade et al., 2007). So we can draw upon a rich state of research on student learning with multiple representations and design features that can enhance it, in particular in two important aspects.

Connecting rather than translating as a relevant student process for developing understanding

Various studies in mathematics education have identified students’ challenges in interpreting and using visual and symbolic representations (see Goldin & Shteingold, 2001), and increasing emphasis has been laid on their explicit introduction with the negotiation of meanings rather than simply assuming immediate access for all students (Duval, 2006; Falcade et al., 2007; Goldin & Shteingold, 2001; Steinbring, 2005).

These approaches have been substantiated by empirical findings in instructional psychology on students’ challenges in information processing from multiple representations (Ainsworth, 2006) and on the design features supporting overcoming these challenges. In her highly cited paper on learning with multiple representations, Ainsworth (2006) emphasized that solely juxtaposing multiple representations is not automatically effective, and calls for studying the “circumstances that influence the effectiveness” (p. 183) of multiple external representations. The most important circumstance is that learners “know how a representation encodes and presents information” (p. 186). In particular, when multiple external representations are used to support learners in complex information processing or constructing deep understanding of new concepts, the translation between representations must explicitly be scaffolded in the learning process and cannot be taken for granted. These insights are in line with mathematics education research emphasizing the need to initiate processes of deliberately translating between representations (Calor et al., 2020; Duval, 2006; Lesh, 1979).

Beyond translating, some mathematics educators have already mentioned the need to make the connections between representations more explicit (Duval, 2006; Kaput, 1989; Marshall et al., 2010; Prediger & Wessel, 2013; Sacristán & Noss, 2008). The teaching practice is therefore now often referred to as connecting multiple representations: for example, the practical paper by Marshall et al. (2010) suggested the instructional strategy that teachers engage students in dialog to make explicit the connections among representations (similarly NCTM, 2014).

Whereas the distinction between translating and connecting representations has not yet been elaborated in mathematics education research, instructional psychology has systematically investigated the design features that support students’ active creation and articulation of connections between representations: Renkl et al. (2013) outlined that connecting representations means making explicit how each relevant element of the represented concept in view is to be found in another representation. Without making these connections explicit for relevant elements, there is a risk that students’ information processing will remain on a superficial level without reaching a deep understanding. They also provided empirical evidence that students’ processes of connecting can be supported by design features such as color coding the relevant elements in the same colors, and by explanations that can be initiated by self-explanation prompts (ibid.).

For the mathematical topic in view, conditional and joint probabilities, Pfannkuch and Budgett (2017) explored six undergraduate students’ processes of connecting numerical and visual models and concluded that seeking and verbalizing connections “also assists in deepening conceptual understanding” (p. 308), having the same finding also for the non-trivial relation between the fractions and the absolute counts. The authors ended with a call for more research on learning processes of working with these connections.

In these ways, the existing findings empirically substantiate the distinction of four semiotic processes in dealing with multiple representations according to their degree of integrating the representation (see Fig. 3, adapted from Uribe & Prediger, 2021): when only one representation is used or multiple representations are juxtaposed (level 1), the students are not engaged in processes of connections. Students’ switches between representations achieve a first degree of flexible use as unconscious implicit transitions (level 2), but conscious translations (level 3) bear a higher degree of integration of representations (Lesh, 1979). The highest degree of integration of representations is achieved when students (or teachers) explain explicitly (level 4) how two representations of a concept are connected and eventually even justify the correspondence (Duval, 2006; Renkl et al., 2013; Uribe & Prediger, 2021).

Fig. 3
figure 3

Distinction of four semiotic processes of increasing degrees of integration of representations

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Need to unpack the involved concepts into more refined concept elements

In their perspective of focused information processing, Renkl et al. (2013) emphasized that it matters not only how representations are related, but also what is related, because the most relevant concept elements have to be addressed when connecting representations.

For example, many students speak about the correspondence of symbolic and visual representations for fractions only by addressing the numbers for denominator and numerator (as in the left example in Fig. 4), yet without explicating the meaning of the fraction in the part-whole relationship. Some students articulate this incomplete explanation despite holding a correct conception of fractions. For other students, it coincides with misconceptions such as “three quarters are smaller than three fifths as three fifths have more pieces,” which indicates a missing awareness of the part-whole relationship as the core meaning of fractions (Prediger, 2013). Hence, the explicit articulation of the part-whole relationship, as in the second example in Fig. 4, increases the chance of assuring a deep understanding for all students (ibid.).

Fig. 4
figure 4

Example of incomplete and detailed explanation: all elements must be identified in both representations to reach a deep understanding (red color marks the part, blue the whole, green the part-whole relationship)

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This example gives a first indication that the selection of relevant concept elements that need to be explicitly connected might depend on the individual state of understanding: many students who have completely understood fractions would only articulate a translation “three fifths can be seen in this bar model.” For those who have not yet acquired the compacted fraction concept, an unfolding into the correspondences of its concept elements (part, whole, and part-whole relationship) might be crucial.

According to this a concept “is understood if it is part of an internal network” (Hiebert & Carpenter, 1992, p. 67), concerning learning as building a network of concept elements. Whereas experts can work with compacted concepts, novices first need to relate refined concept elements to each other within this network and compact them into a new concept before being able to work with them (Aebli, 1981). Whereas this process of compacting has been described in many theoretical approaches (e.g., encapsulation, reification, and others, e.g., Sfard, 1991), it is a particular strength of Aebli’s (1981) account to also emphasize the inverse process: a compacted concept can be unfolded back into the refined concept elements by addressing the underlying concept elements and expressing the underlying relationships (Aebli, 1981; Drollinger-Vetter, 2011).

Figure 5 sketches how these compacting and unfolding processes play out for our topic of processing complex conditional probability information as shown in Fig. 1: whereas experts can directly translate the compacted information from the text into the visual area model and the statements with symbolic fractions, novices require further unfolding with an explicit articulation of how the part, the whole, and the part-whole relationship can be found in each of the representations (Leuders & Loibl, submitted). When we study the processes for using and connecting multiple representations, it is thereby important to also capture which concept elements are addressed in which way.

Fig. 5
figure 5

Topic-specific concept elements with unfolding and compacting processes

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Summing up, the state of research on processes of relating multiple representations in general and on accessible representations of conditional probability information (Fig. 2) allows us to choose representations that are potentially helpful for learning to process complex conditional probability information (Fig. 1), but also problematizes that the explicit connection of representation requires further support. It will serve in the next section as the theoretical and empirical foundation for a conceptual framework on semiotic and epistemic processes when we now turn the research attention from students’ processes and the design features that support them to teaching practices in initiating and supporting these semiotic processes.

Conceptual framework for capturing teaching practices with representations: navigating in the topic-specific space of concept elements and representations

Existing research on teaching with representations

Although using and even connecting multiple representations has occurred on many prescriptive lists of good teaching (e.g., Lesh, 1979; Moschkovich, 2013; NCTM, 2014), as have unpacking compacted concepts into more unfolded concept elements (Drollinger-Vetter, 2011), enacted teaching with representations has rarely been disentangled in more detail. Stylianou (2010), Bossé et al. (2011), and Dreher and Kuntze (2015) pointed to this research gap and investigated teachers’ underlying views and noticing. Their studies provided evidence that many teachers are not aware of typical challenges in dealing with multiple representations.

Additionally, Bossé et al. (2011) studied teachers’ self-reported practices of which modes of representations they usually translate in which direction (e.g., verbal representations were reported to be translated more often into symbolic representations than vice versa), but not how these semiotic processes were initiated and really enacted in classrooms. We will add to this body of research by investigating videotaped classroom practices.

Twenty years ago, interactionist research in sociocultural frameworks already hinted at the need to negotiate meanings of visual representations, having found that students assigned idiosyncratic meanings unless the conventional meanings were explicitly discussed (Meira, 1998; Steinbring, 2005). Even if the practices that teachers enact in these interactions have not yet been investigated in detail, we can draw from these works for an interactionist characterization of teaching practices as follows.

Interactionist perspective on teaching practices

In general, teaching practices are conceptualized as recurrent teaching behavior that teachers enact to reach certain curricular goals such as increasing students’ representational fluency (e.g., NCTM, 2014). For investigating these practices in depth, we adopt an interactionist perspective to locate the micro-sociological phenomenon in the fine-grained interplay of student–teacher interaction (Mehan, 1979; Steinbring, 2005).

Within this interactionist theoretical perspective, teaching practices are empirically characterized as teachers’ steering trajectories on a content-specific navigation space (Prediger et al., 2022), as will be explained in this and the next subsection.

In other areas, teaching practices have often been investigated with a focus on scaffolding (Makar et al., 2015), on the moves that teachers apply for initiating students’ cognitive or semiotic processes, and on the triplet of turns in IRE sequences, for instance, of teacher initiation, student responses, and teacher evaluation (Mehan, 1979). In a similar way, Velez et al. (2022) investigated teachers’ moves (actions and questions) while fostering the students’ handling of representations. Moreover, they investigated how these moves changed depending on students’ contributions.

However, Schwarz et al. (2021) argued that isolated teacher moves or IRE sequences form units of analysis that are too small to really capture the relevant practices. Instead, they called for studying teaching practices in units of analysis in medium grain sizes, which they call sense-making moments, to capture how teachers can expand, maintain, or shut down students’ opportunities for sense making along several utterances.

Following this call for medium grain sizes, Prediger et al. (2022) suggested characterizing teaching practices empirically as navigation practices that include several moves or IRE sequences to emphasize the central aspect of teachers’ steering in it. In order to explain this further, the navigation space first has to be specified.

Navigation practices in the topic-specific navigation space

According to the sketched state of research on students’ processes with multiple representations and the design features to support them, a conceptual framework for capturing teaching practices for dealing with multiple representations should entail at least four aspects in the navigation space:

  • the representations that are addressed (as studied by Bossé et al., 2011; see Fig. 2)

  • the semiotic processes by which the concept elements are related in the different representations (see Fig. 3, adapted from Uribe & Prediger, 2021)

  • the concept elements for which the representations are addressed (as studied by Renkl et al., 2013; see Figs. 4 and 5)

  • the processes of compacting and unfolding the involved concept elements (see Fig. 5)

Figure 6 depicts the topic-specific navigation space for the mathematical topic in view: conditional probability information. In its rows, it lists the ratio in which the pieces of conditional probability information are mathematized and the more refined concept elements into which the highly compacted ratios can be unfolded, namely, the parts, the wholes, and the part-whole relationships (see Fig. 5). For unfolding the ratio, grasping the underlying nested-set structure (Sloman et al., 2003) or part-of-part structure (Leuders & Loibl, submitted) is crucial, as Evans et al. (2000) argued: “to express a probability as ___ out of ___ it is necessary to specify a denominator—the set of people out of whom those with the disease should be measured” (p. 211).

Fig. 6
figure 6

Navigation space for capturing teaching practices as navigation practices between concept elements (in the rows) and representations (in the columns)

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These concept elements can be addressed and connected in different representations, so the columns of the navigation space distinguish the given texts (with conditional probability information), symbolic representations (of fractions or percentages), symbolic-related technical language, the visual model (area model), the contextual language (verbalizing parts, wholes, and part-whole relationships), and, finally, the meaning-related language (in which the part, whole, and part-whole relation can be addressed in a meaning-related but decontextualized way). In the empirical part of this paper, each of these representations will be shown to be addressed separately by teachers and students, so it is worth distinguishing them. Whereas processes of connecting representations are often only discussed for the compacted concepts (in our case, the ratios, in the first row only), a crucial characteristic of our navigation space is the consideration of how the compacted concept is unfolded into its concept elements and then expressed and/or connected in different representations (following the design necessities identified by Renkl et al., 2013).

Thus, the navigation space for our conceptual framework is thereby conceived as two-dimensional space in which all utterances of a whole-class discussion can be located. Within the adopted interactionist perspective, the navigation space is used to map students’ processes and teachers’ steering processes horizontally (for semiotic processes of switching, translating, and explaining connections of representations, including the language registers, see Fig. 3) and/or vertically (for compacting or unfolding processes, see Fig. 5). With focus on the steering prompts and activities of the teacher, we can capture certain movements or steering trajectories in this navigation space in medium-grain units of analysis. On this base, we compare the resulting movements on the navigation space from different classes and derive navigation practices, as will be further explained in the Method section.

Summing up, within our conceptual framework, a navigation practice is characterized as teachers’ steering in a typical pattern of processes co-constructed in successive utterances of the teacher-student interaction. Whereas the horizontal and vertical visualization of processes might suggest them to be independent, it is a main empirical task of this paper to explore how they are intertwined.

Refined research question

Within this conceptual framework, the general research question posed in the introduction can be refined for the particular topic in view, complex conditional probability information:

What exactly characterizes the navigation practices by which teachers steer the co-constructed processes of switching, translating, and explaining connections of representations to more compacted or more unfolded concept elements to support students’ processing of complex probability information?

Methodological framework for the qualitative study of teaching practices

Research context of the overarching design research project

Our qualitative, interactionist investigation of teaching practices is embedded in a larger project of topic-specific design research (Anderson & Shattuck, 2012; Cobb et al., 2003) in which two main research aims are combined: (a) designing curriculum material for a teaching unit that enables students to develop conceptual understanding of conditional probabilities and learn to process complex conditional probability information given in statistical frequencies and (b) qualitatively investigating the initiated teaching/learning processes for developing local topic-specific and overarching theories. For this, the overarching project conducted four design experiment cycles with successively improved designs for the teaching unit, enacted in seven classes (including 150 students in total) in grades 10–12 (mostly 15–18 years old). The teaching units were taught by the regular mathematics teachers, who were active volunteer partners in a research-practice partnership. The teachers could deviate from usual teaching as they were experimenting with the developed curriculum materials with perhaps unusual visual models and particular scaffolds. All lessons were video recorded (142 h of video, including several cameras to focus on students’ tables), names were anonymized, and videos were partly transcribed.

The successively developed curriculum material started with the context of a statistical survey on leisure activities of teenagers. The first learning goal was students’ processing of complex information on parts of different wholes, upon which the second part of the teaching unit builds for the second learning goal of formal conditional probabilities.

Data corpus in the current qualitative study of teaching practices in whole-class discussions

For the current paper, the data corpus for our analysis was restricted to whole-class discussions in the first two lessons of the teaching unit. Before the classroom discussions investigated in this paper, the students had already gained some experience with complex statistical information on parts of different wholes and the area model as their main visual model to visualize part-whole relationships. To engage students in rich discussions about the connection of representations between the given text and the visual model, the task in view (see Fig. 7, left) asked the students to discuss the statements of two fictitious students, Simon and Lara, who falsely connect the representations in highly compacted statements that need to be unfolded for processing the given information. Simon’s symbolic fraction in Fig. 7 refers to the conditional probability where the part, 45 teenagers who are female and sportive, is referred to the referent whole of all 54 teenagers who are female. In his statements with the wording “of the teenagers,” however, he refers the part of 45 teenagers to the complete whole of all 144 teenagers, so he articulates a joint probability. Lara’s statement refers to the whole of all 105 who are sportive and the part of 60 sportive people who are male, whereas her fraction would be nearer to 60/90, matching the description “60/90 of the boys are sportive.” Two central aspects are relevant for understanding: while unfolding, the students have to decode the right part-whole relationship from the given text and grasp the underlying part-whole relationship, where the recognition of the relationship words (“thereof”) is especially crucial. In the compaction step back to the ratio, the unfolded concept elements have to be carefully related to each other to achieve conceptual understanding of the ratio as a part-whole relationship.

Fig. 7
figure 7

Conditional or joint probabilities? Core task from the designed curriculum material and scaffolds for unfolding and connecting as design features (left from cycle 2, right from cycle 4)

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As a design feature providing support, scaffolds are given that unfold the highly compacted statements about ratios into the whole, the part, and the part-whole relationship (see Fig. 7, left). The scaffolds for unfolding the complex information into whole, part, and part-whole relationship were successively refined after first analyzing transcripts from cycle 1, in which students were shown to switch between the representations without explicitly connecting the different concept elements (Post & Prediger, 2020).

In the empirical part of this paper, we analyze two excerpts of transcripts from two whole-class discussions in which these kinds of tasks are discussed collectively after students worked on them independently. These two cases were selected from cycles 2 and 4 because they allow documentation of contrasting navigating practices: case 1 in a mathematically strong class and case 2 in a class in need of stronger teacher support. Both classes were taught by experienced mathematics teachers with more than 4 years of teaching experience who were volunteering in a research-practice partnership. The specific task formulation and scaffolds differ between case 1 (Fig. 7, left) and case 2 (Fig. 7, right), because case 2 involves a continuing exercise task with reduced scaffolds.

Methods for qualitative data analysis

To decompose the teaching practices for dealing with multiple representations, the transcripts were qualitatively analyzed with respect to the navigation practices by locating all utterances in the navigation space sketched in Fig. 6 and by characterizing the interactively emerging semiotic processes as steered by the teacher in the interaction. The following six analytic deductive coding steps were conducted:

  1. Step 1.

    The transcripts were segmented into sense-making moments (Schwarz et al., 2021). They begin when a problem or question is initiated and end when marked as being finished by students or teachers, for instance, by going to the next task. For the two tasks in view of this paper, we analyzed in total 64 sense-making moments in six classes. In 20 of these 64 sense-making moments, no navigation practice was identified, because (a) students only presented their solutions without a teacher’s steering or with the teacher only eliciting further contributions or (b) connecting representations were not focused on in the sense-making moment. For the 44 other sense-making moments, the next steps were conducted to identify the navigation practices in detail.

  2. Step 2.

    Within a sense-making moment, all students’ and teachers’ utterances (or several utterances together or a segment of an utterance) were coded according to what concept element is addressed (part, whole, part-whole relationship, or encapsulated ratio). The distinction between encapsulated ratio and unfolded part-whole relationship was often inferred from the succession, for instance, when students changed from “three fifths” to “five, thereof three” or by other verbal unfoldings. Part and whole were only coded when addressed separately, not in the more compacted part-whole relationship.

  3. Step 3.

    Each of those coded concept elements was then coded according to the representations in which it was explicitly or implicitly addressed (implicitly, e.g., by gestures). By so doing, all coded turns could be visually located in the navigation space. Blue marked the teachers’ utterances and red marked errors in students’ utterances.

  4. Step 4.

    The semiotic processes by which the representations are related (and the teachers’ moves eliciting these processes) were coded according to their degrees of realized/demanded integration (see Fig. 3). In the navigation space, the semiotic processes were denoted as follows: no connection was represented by a blank, switching by - - - , translating by _____, and explaining connections by ====.

  5. Step 5.

    The prompts of the teacher were coded depending on which concept element (as in Fig. 5) and semiotic process the teacher was steering at (derived from Fig. 3: prompts for naming, translation, explanation, and explaining connection in part-whole relationship). Depicting the succession of processes in the navigation space (as in Fig. 8) provided the analytic summary of the coding. For this, shifts in the rows (i.e., in more compacted or more unfolded concept elements) were marked by vertical arrows to make the compacting or unfolding processes in the interaction visible, and the prompts were noted on the arrows.

  6. Step 6.

    By systematically comparing and contrasting the teacher’s steering in the navigation spaces of different sense-making moments, typical navigation practices were identified. Their potential to really support students was inferred from the empirical findings on student learning (see Theory section) and within the data from the degree to which the teachers succeed in leveraging students’ ideas by their steering. The identified types also guided the selection of the two cases presented in the empirical part of this paper. A brief summary on other analyzed cases is given in the end of the Analysis section, showing that these four practices were indeed typical for the data corpus in view.

Fig. 8
figure 8

Co-constructed processes in the navigation space of case 1, part 1: teachers’ navigation to translate unfolded components and teacher input to translate the compacted concept ratio (no connection was represented by a blank, switching by—- -, translating by._____, and explaining connections by =  =  = =)

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By these detailed and highly controlled analytic steps, we achieved a high validity for capturing the essentials in the data. The steps were first conducted by the first author and then discussed with the second author and other research team members to achieve agreement on the coding. No formal interrater reliabilities were determined because the agreement was achieved in communication, and no frequencies were studied.

Empirical insights into two cases of teaching practices with multiple representations

For illustrating the analytic steps and the resulting types of teaching practices, we selected two cases: case 1 (in the first subsection) shows a sense-making moment strongly scaffolded by the task design where the teacher took over for translating the compacted concepts without involving students in explaining the connections. The comparison with case 2 (in the second subsection) shows the rich navigation practice of a teacher who supported his students in successfully processing the information by connecting representations for the relevant concept elements. By contrasting these cases with further cases (in the third subsection), we identify navigation practices typical for our data corpus.

Case 1: portioning and taking over in a successful connection process

Case 1 stems from a mathematically strong grade 10 class in cycle 2. Subtask (a) of the task in Fig. 7 (left) had already been solved in pair work and correctly presented by two students. The transcript starts when the class started to discuss Simon’s statement, with more detailed prompts in the task design to systematically unfold the complex ratio information into the part and the whole, to find the phrases in the given text that signify the part-whole relationship, and to connect several representations for these unfolded concept elements (subtask [b] in Fig. 7, left).

Transcript of case 1Footnote 1

Teacher practices in portioning and taking over in a successful connection process.

figure a

Starting from the highly compacted statement of the fictitious student Simon in the task, the teacher guides the students to address the whole and the part separately (Fig. 8, turn 1). He introduces the abstract notions of “whole group” and “subgroup” and initiates students translating them from this meaning-related language (ML) into the given text (GT) or contextual language of the teenagers in the survey (CL), and Kim follows this prompt for translating for the whole (turn 2a, translating: GT/CL \(-\) ML) and for the part (turn 2b, translating: GT \(-\) ML). While translating for the whole, she unconsciously reproduces the phrase “the teenagers” from the given text in her own words as “all teenagers” (CL). This corresponds to switching between the given text and contextual language (turn 2a, switching: GT—- CL) without visible awareness of change. The students do not yet meet the prompt for explanation. In turns 3–8, the teacher initiates the translation to the visual model for the part (turn 3/6–8, translating: VM \(-\) GT) and the whole (turn 3/4, translating: VM \(-\) GT/CL), yet without eliciting students’ exact explanations of how they are connected.

Having elicited the translation of representations for the unfolded concept elements “part” and “whole” in all relevant representations (turns 1–8, Fig. 8), the teacher jumps back up to the row of the compacted ratio and asks for naming the symbolic fraction by saying: “What would be the solution, then?” (Fig. 9, turn 9b, shown via arrows, answered immediately by Annabel in turn 10). Interestingly, he provides another translation of the compacted concept on his own, by translating from the symbolic representation “45/144” to the meaning-related notion “shares” and to the technical notion of “fractions” (turn 11, translating: SR \(-\) TL \(-\) ML).

Fig. 9
figure 9

Co-constructed processes in the navigation space of case 1, part 2: teachers’ navigation to translate unfolded components and teacher input to translate the compacted concept ratio (no connection was represented by a blank, switching by—- -, translating by._____, and explaining connections by =  =  = =)

Full size image

As the summary of the analysis of case 1 in the navigation space in Figs. 8 and 9 depicts, the teacher portions students’ processes of unfolding the given information into part and whole together with translation processes between different representations for these concept elements. Meanwhile, the teacher does not talk about the part-whole relationship and jumps directly to the compacted concept ratio. It might be an important observation that the translations for the compacted ratio are conducted only by the teacher alone.

For this mathematically strong class, the identified navigation practice with a focus on initiating translations for part and whole and only teachers’ translation for the ratio might have been sufficient even without a more explicit explanation of connections. At least those students who were speaking in the next sequences (not presented in this paper) showed that they understood how to process the information. However, many students might have been lost in this navigation practice that does not engage students in critical steps through the navigation space back to the more compacted concepts.

Case 2: teacher’s navigation in a challenging situation

Case 2 shows a whole-class discussion from cycle 4, a grade 12 class with heterogeneous achievement. The whole-class discussion starts after the exercise phase when two students, Laura and Sascha, report conflicting interpretations of the given text in the statement.

Transcript of case 2

Teacher’s navigation for connecting representations for several concept elements.

 

Task

Statement “The share of boys among the teenagers who don’t watch funny clips is 1/2.”

 

90a

Laura

Yes, um, well, I have taken again all “teenagers” as the whole group, though the 1,200 asked people

90a Whole

Switch: GT-CL

Translate: ML-GT/CL-SR (false)

90b

 

And the share, oh, no, the part, is, then, um, again, this half, “who don’t watch funny clips.” […]

90b Part/Ratio

Translate: ML-GT (false)

…

   

92a

Sascha

Yes, well, “the share of boys among the teenagers.” Then, there are first only the 600 boys. Um, and not all, because, of the girls, nothing is asked

92a Whole

Translate: GT-CL-SR (false)

92b

 

And then, of these 600, 300 “don’t watch funny clips”

92ab PWR

Explain connection: CL = GT = SR (false)

92 c

 

Thus, the statement is correct because it is the half

92c Ratio

No connection: SR (false)

93

Paul

I would do that differently

 

94

Teacher

[…] Uh, complicated. What is the first thought that we have to make?

 

95

Sina

What is the whole?

Prompt for translation

95–97 Whole

Switch: GT-CL

Translate: ML-GT/CL (false)

96/100 Whole

Translate: ML-CL (false)

96

Teacher

[…] So, now everybody look at it. What is the whole? Which teenagers are addressed, here? Do they have particular attributes?

97

Mara

Well, the “boys”?

98

Paul

No

99

Teacher

Sophie?

100

Sophie

All?

101

Teacher

All? These are two different [ideas]? Who has another idea?

101/102 Whole

Switch: GT-CL

Translate: ML-GT/CL

102

Alex

In general, the whole part “who don’t” watch “funny clips,” girls and boys

103

Teacher

Ah. How do you recognize that?

Prompt for explanation of connection (Whole)

104

Alex

Because, here, it is written “The share of boys among the teenagers who don’t” watch “funny clips.” So, I think, all who don’t watch funny clips. Then, thereof this share, thus, the boys

104 PWR

Translate: GT-CL

105

Teacher

Yes exactly, this is how it is meant

 

106

Laura

Though, not the complete whole, but only #

 

107

Paul

#only those “who don’t” watch “funny clips”

 

108

Teacher

Correct! The “share of boys among” THOSE “teenagers who don’t watch funny clips.” Hence only those who don’t watch funny clips. The details are decisive. […]

108 Whole

Translate: GT-CL (embedded in PWR)

109

Teacher

The whole comprises how many in this task? […]

Prompt for explaining connection (PWR)

109–112 PWR

Explain connection: SR \(=\) GT/CL \(=\) ML

110

Tarik

300? 700

111

Teacher

Yes exactly, and among these 700, who is addressed? […]

112

Johann

The 300 males “who don’t watch funny clips”

113

Teacher

Yes, thus, what is the share?

Prompt for translation

113/114 PWR

Translate: ML-SR

114

Johann

300 per 700

115

Teacher

Exactly. And how many, in a simplified fraction?

Prompt for naming fraction

115–116 Ratio

No connection: SR

116

Johann

3/7?

117

Teacher

3/7, great, and that’s not the half, so the statement is wrong

117 Ratio

Translate: SR-GT

Laura and Sascha first present their ideas (turns 90–92, not included in the summary in Fig. 10): Laura identifies a wrong whole and does not complete her explanation for the part. Sascha addresses all relevant concept elements including the part-whole relationship, but with a wrong translation due to mixing the attributes describing the part and the whole.

Fig. 10
figure 10

Co-constructed processes in the navigation space of case 2, part 1: teachers’ navigation to connect representations for the critical concept element, the part-whole relationship, before compacting it to the ratio

Full size image

Our analysis starts with the first part of the discussion (turns 94–108), presented in Fig. 10. In turn 94, the teacher starts the moderation and navigates the class’s focus of attention first towards unfolding the whole. Over six turns, the class collectively searches for adequate translation, and, in the end, Alex gives a correct translation into the contextual language (turn 101/102; translating: ML \(-\) GT/CL). While translating, Alex and Mara implicitly switch between the given text and the contextual language (turn 97/102, switching: GT—- CL) by unconsciously reproducing and varying phrases from the given text. The third student Sophie does not link to the given text at all (turn 100).

When the teacher elicits an explanation for Alex’s translation (in turn 103), Alex refers to the part-whole relationship by translating the given share statement into her own words (turn 104; translating: GT \(-\) CL), which is then confirmed by other students (turns 106/107). The teacher completes the explanation on his own by locating the key phrases in the given text (turn 108: translating: GT \(-\) CL), embedded in a part-whole relationship. By emphasizing “those,” he goes slightly beyond translating and makes explicit the connection, yet without explaining it.

In contrast to turns 97–102, both not only switch but translate consciously between the given text and the contextual language and both involve the part-whole relationship (expression of the relationship in given text) into these translation processes (solid lines in the lines “part-whole relationship” and “whole” in Fig. 10).

In turn 109 (Fig. 11), the teacher navigates the focus of attention again towards the whole and elicits a translation from the meaning-related language into the symbolic representation “700” (turn 109–110, translating: ML–SR). When turning to identifying the part, the teacher does not address the part separately (turns 109–112: explaining connection for part-whole relationship: SR \(=\) GT/CL \(=\) ML): by “among these 700, who is addressed?” (turn 111), he embeds the search for the whole (turns 109 and 110) and the part (turns 111 and 112) in the thinking about the part-whole relationship. Johann contributes to the collective explanation of connections between representations for the part-whole relationship by translating the part from the symbolic number 300 to his own contextual language including phrases of the given text (turn 112, switching: GT—- CL; translating for part: GT/CL \(-\) SR). Considered together via these prompts and contributions in turns 109–112, the connection of the representations for the part-whole relationship is explained (presented as a double line in Fig. 11). No technical language is addressed in this strong sense-making moment.

Fig. 11
figure 11

Co-constructed processes in the navigation space of case 2, part 2: teachers’ navigation to connect representations for the critical concept element, the part-whole relationship, before compacting it to the ratio

Full size image

In the last turns (shown in the upper half of Fig. 11), the teacher navigates from the part-whole relationship back to the compacted fraction, the symbolic representation of the ratio. He does this by interpreting Johann’s symbolic (but still slightly unfolded) expression “300 per 700” (turn 114) as a fraction and calls for a simplified fraction (turn 115). In total, the teacher navigates the class from false translations of the whole to a careful connection of representations of the part-whole relationship and from there to the compacted ratio and the evaluation of the given text (turn 117). The part-whole relationship is first articulated in the structure of “whole, thereof part” (“among these 700, who is addressed?” turn 111), before it is encapsulated via “part per whole” (Johann in turn 114) to finally the ratio as a fraction (turns 116 and 117). The key work of connecting the representations by explicitly assigning the refined concept elements is conducted for the part-whole relationship.

Comparing cases to decompose teaching practices for dealing with multiple representations into typical navigation practices

When comparing the transcripts of cases 1 and 2 (and many further examples not presented here), we can derive typical patterns of relevant actions that teachers have to consider in their navigation practices when working with multiple representations. We first examine commonalities and differences between both cases, first with regard to the semiotic processes:

  • Switching between representations occurs in both classes, for example, when students unconsciously vary phrases given in the text in their own contextual language.

  • Beyond switching, both teachers easily succeed in engaging students in semiotic processes of consciously translating in an explicit translation discourse. This consciousness is achieved in both cases, for instance, by teacher moves such as “… would you … color it in red and blue in the area model…” (case 1, turn 3) or “The whole are how many in this task?” (case 2, turn 109). During the negotiations of these translations, the technical language seems to be rarely used by students and is not elicited by the teacher, as it does not help in grasping the meaning. Also, the contextual language alone (which is often very near to the given text) is insufficient and must be accompanied by a meaning-related language to explicitly address the mathematical structure of the reasoning, in this case, the repeatedly used language of parts and wholes (or subgroups and whole groups). Both teachers support the translation into the meaning-related language by moves such as “What is the whole group here, and what is the whole subgroup?” (case 1, turn 1) and “What is the whole? Which teenagers are addressed here?” (case 2, turn 96).

  • Both cases differ in how explicitly they connect the given text to other representations (in particular ML and CL): in contrast to case 1, the teacher in case 2 tries to elicit explanations of the connection, for example, by “How do you recognize that?” (turn 103). The semiotic process of explaining becomes relevant at another point: when students first answer with translating, the teacher navigates towards an orchestrated explanation of the part-whole relationship by unfolding into the concept elements: the collective explanation of connections between representations for the part-whole relationship is produced by translating representations for the part and the whole and embedding both in the part-whole relationship: “The whole are how many in this task? […] and among these 700, who is addressed?” (case 2, turns 109–112).

The cases also reveal commonalities and differences with regard to the unfolding and compacting of concept elements:

  • Both teachers aim at orchestrating an explanation of the overall connection of representations by articulating how the given statement about the ratio can be found in the other representations, and both teachers follow the scaffolds provided in the curriculum material to do this by unfolding the given text into the involved parts and wholes (see above).

  • A big difference between both cases is that teacher 2 invests substantially in connecting the representations not only for part and whole separately but for the part-whole relationship together. This is done, for example, by “The whole are how many in this task? […] Yes exactly, and among these 700, who is addressed?” (case 2, turns 109 and 111). In contrast, teacher 1 does not relate both components.

  • In both transcripts, there is also a process back from the unfolded concept elements parts and whole upwards to the compacted ratio. However, there is a big difference in the navigation of this compacting upwards. Whereas teacher 1 makes an unprepared jump upwards to the compacted ratio and states some translations of representations of the ratio by himself without involving the class, teacher 2 invests a lot in involving the class into this important process of compacting the separate information on the whole and the part into the part-whole relationship, for instance, by three successive moves: (1) “The whole are how many in this task? […] Yes, exactly, and among these 700, who is addressed?” (case 2, turns 109 and 111), (2) “Yes, thus, what is the share?” (turn 113), and (3) “And how many, in a simplified fraction?” (turn 115).

From the observations of these two cases and the further analyzed cases that could not be presented here, we infer four typical navigation practices used by teachers in our data corpus in different priorities (see Fig. 12). Each case illustrated one practice for the unfolding process and one for the compacting process. These practices are combined in different ways in other cases.

  • Navigation Practice “Addressing Part and Whole” can be characterized by teachers’ orchestration of explanations of connections, usually by steering towards unfolding into more refined concept elements (here part and whole). At the level of the compacted ratio, the semiotic process of translating is mainly conducted by simply naming the correspondence between different representations (in many parts of the transcripts not presented in this paper). In contrast, explaining the connections is frequently orchestrated by the teachers’ navigation throughout several turns, and sometimes conducted by the students alone. To enable the students to contribute to this orchestrated explanation, the teachers steer towards unfolding the compacted given information into its concept elements (here mainly part and whole), together with connecting processes between further representations for these concept elements (here mostly translating). These repeatedly observed navigation practices are visualized by a graphical navigation pattern from the top left to rich connections in lower rows of the navigation spaces of case 1 (only turns 1–8 in Fig. 8) and of case 2 (turns 94–103 in Fig. 10).

  • Navigation Practice “Explaining Connections for Part-whole Relationship” can be characterized by explications of crucial connections through reflection of the part-whole relationship. During the orchestrated unfolding process of the teacher in case 2, his prompts not only address the part and the whole (see Navigation Practice “Addressing Part and Whole”) but also very explicitly the part-whole relationship for which the connectives are investigated in the given text, as illustrated in the navigation space in Fig. 10 for turns 103–108. Additionally, this navigation practice is characterized by teachers’ prompts eliciting explanation of connections between representations for the part-whole relationship, so not only translating between representations for the part and the whole but also articulating explicitly how part and whole are related (as illustrated in Fig. 11 for turns 109–112). In both cases, the conscious connection of relevant representations is crucial.

Fig. 12
figure 12

Overview of the navigation practices

Full size image

After these unfolding practices, both teachers initiate the compaction back upwards to the ratio, but they do it with two different practices:

  • Navigation Practice “Without Transition” can be characterized by taking over for compacting the ratio without explicating the part-whole relationship. Many teachers initiate the compaction process back from the unfolded concept elements upwards to the compacted ratio. For example, teacher 1 starts his navigation from part and whole directly to the compacted concept ratio without transition or preparation, as the second part of case 1 in Fig. 9 (turns 9a–10) visualizes. In other cases, this compaction moving back is conducted solely by the teacher without engaging students in this critical step. Usually, hardly any representations are connected in this compaction step.

  • Navigation Practice “Compacting Connections for Part-whole Relationships” can be characterized by the orchestration of compaction towards compacted ratio via explicating the part-whole relationship: teacher 2 orchestrates the compaction process back from the unfolded concept elements upwards to the compacted ratio, together with the whole class. In this navigation practice, the explanation of the connection of representations for the whole part-whole relationship is often an intermediate step, expressed by the meaning-related connective “among,” by which the meaning of the compacted ratio can be constructed more actively by the students, as the second part of case 2 in Fig. 11 visualizes (turns 109–116).

These four navigation practices were identified not only in cases 1 and 2, but also in the other four analyzed classes, although they occurred with different degrees of student involvement: within the 44 sense-making moments analyzed for this paper, we identified nine sense-making moments in which the teachers used the most supportive navigation practice of Explaining Connections for Part-Whole Relationship for the unfolding process, but also seven moments with Addressing Part and Whole, which turned out to be less supportive for students’ process of building understanding. Although the tasks provided scaffolding also for the part-whole relationship, it only happened in some cases. In seven more cases, the part-whole relationship was addressed by naming the relation, without drawing further connections. For the compacting processes, the most supportive navigation practice of Compacting Connections for Part-whole Relationships was identified four times, and the less supportive practice of Without Transition also four times. (Some further navigation practices were identified in other sense-making moments, but as they occurred only in three or fewer moments, they are not presented here.) Within the 44 moments, the six teachers have different priorities and different combinations of the identified practices.

Discussion

Results in decomposing the teaching practices of dealing with multiple representations

There has been wide practical consensus about the relevance of using and connecting multiple representations (Calor et al., 2020; Lesh, 1979; Moschkovich, 2013; NCTM, 2014), and a strong research tradition focusing on students’ processes in dealing with representations (Ainsworth, 2006; Falcade et al., 2007; Renkl et al., 2013), including interactionist classroom studies showing that new visual representations require a careful negotiation of meanings (Meira, 1998; Steinbring, 2005). However, teaching with multiple representations has received much less research attention (Bossé et al., 2011; Dreher & Kuntze, 2015; Stylianou, 2010), so our aim was to decompose the teaching practices in detail.

Decomposition of teaching practices has been described as a practically important teacher educator task to substantiate teacher education (Grossman et al., 2009). It is also an empirical task for researchers that has successfully been conducted for other teaching practices such as responding to students’ ideas (Jacobs & Empson, 2016; Makar et al., 2015; Schwarz et al., 2021), although it has not yet been conducted for teaching practices on connecting representations.

Whereas some research on teaching practices has restricted its focus of attention to the small analytic unit of teacher moves, we follow the call of Schwarz et al. (2021) to consider units of analysis of medium-grain size and conceptualize navigation practices as the teachers’ steering activities for navigating orchestrated cognitive and semiotic processes co-constructed over several turns in teacher-student interaction. Within this conceptual framework, a topic-specific navigation space can serve as an analytic tool. In our topic of conditional probability, the navigation space (in Fig. 6) comprises the relevant representations and concept elements needed for unfolding the complex and highly compacted ratios, namely, part, whole, and part-whole relationship (Leuders & Loibl, submitted; Pfannkuch & Budgett, 2017; Post & Prediger, 2020), so this navigation space builds upon the rich state of topic-specific research on processing complex conditional probability information. It extends the current state of research by shifting the research attention from students’ information processing to teachers’ contribution in engaging students in deep learning processes.

When scrutinizing the teachers’ steering activities across the navigation space in the transcribed videos of the teacher-student interactions, the distinction of the four semiotic processes (see Fig. 3) turned out to be crucial for capturing the degree of integration that the interaction co-constructively achieves. And with respect to the epistemic processes of collective knowledge construction, crucial differences in teachers’ navigation concerned the unfolding and compacting of concept elements. Whereas teachers’ moves for switching, translating, and explaining connections for representations have also been analyzed elsewhere (e.g., Jacobs & Empson, 2016), these co-occurring vertical navigations of unfolding and compacting are an interesting discovery that deserves more research attention in the future. In our study, it turned out to be relevant into which concept elements the teachers and student unfold the highly compacted concepts, as part and whole often occur but the part-whole relationship is shown to be the concept element that is not automatically addressed deeply.

The two focus cases presented in this analysis both aim at orchestrating the connection of representations by successively articulating how the underlying concept elements occur in the addressed representations. While in many moments, teachers steer towards unfolding the compacted given information into the whole and the part and towards translating representations for these concept elements (navigation practice of Addressing Part and Whole), the teachers set different emphases on addressing the part-whole relationships. In the navigation practice of Explaining the Connection of the Part-Whole Relationship, the focus is set on reflecting and explaining the connection of the representations for the part-whole relationship with high degree of integration, which would cover the support that researchers identified as crucial in the research on students’ processes (Leuders & Loibl, submitted; Pfannkuch & Budgett, 2017; Renkl et al., 2013). Similarly, two different navigation practices were identified for the compacting process from more refined concept elements of whole and part back to the more compacted part-whole relationship and ratio: in some moments, teachers make unprepared jumps Without Transition to the compacted ratio, without students’ active participation and without further connections of representations. In other moments, we identified the navigation practice of Compacting the Part-Whole Relationship, in other words, teachers steered towards compacting the separate information on the whole and/or the part into the part-whole relationship as an intermediate step that contains the important semiotic process of also explaining connections on the way back upwards in the navigation space.

Of course, our sample size of 44 analyzed sense-making moments does not allow a stable quantification and can only reveal first humble indications of frequencies of occurrences that are tied to the particular data corpus without any claim of universality. Given this caveat, it is still interesting to see that the part-whole relationship was explicitly addressed only in nearly half of the sense-making moments in which they would have been suitable and supportive. This finding is compelling because the teachers in our design research study had a strong support for focusing on the part-whole relationship: it was included by written scaffolds in the curriculum material, and they were sensitized to it (too briefly) in preparatory sessions. For non-interventionist classroom data from teachers without support, we would thereby expect even less occurrences of these supportive navigation practices. This underlines that “typical” navigation practices do not necessarily occur often and without strong teacher support.

Even if the number of analyzed cases is still small, our findings indicate a further research need to look deeper into decomposing teaching practices on the orchestrated negotiations of meanings and connections of representations.

Methodological limitations and future research

For every empirical study, the methodological limitations must be reflected while interpreting the findings. An important limitation is that video-recorded classroom interaction always allows only limited insight into individual students’ thinking processes, so it is difficult to relate the identified navigation practices directly to students’ progress within the situation. Together with existing findings on necessary conditions for support (Pfannkuch & Budgett, 2017; Renkl et al., 2013), the deep analysis of interactions might nevertheless contribute to explaining how different teacher moves best facilitated the interactions between, and understandings of, the students.

A second limitation is that the complex analysis with its six steps resulted in such extensive documentation in navigation spaces that only two cases could be presented within the normal space restrictions of the paper, while analysis of our complete data corpus can only be summarized in some rough quantifications. The data corpus is here restricted to the particular tasks analyzed from six classrooms. Future research should therefore extend the data corpus. This larger data corpus would ideally sample across different cultures, as teaching practices are always tied to particular teaching cultures with particular interaction patterns and sociocultural norms, so we cannot claim any universality of the identified teaching practices. Cross-cultural analysis would allow to identify many differences and presumably some commonalities.

Future studies should not only extend the sample size and cultural contexts, but should also span to further kinds of tasks and further mathematical topics to study whether the identified navigation practices are comparable for other kinds of tasks. Given that our tasks have already provided much support to teachers for the explicit unfolding processes, it will be interesting in the future to compare with teaching practices based upon other curriculum material. Additionally, other mathematical topics might require other kinds of unfolding and compacting processes that should be systematically investigated and compared.

And finally, given the interventionist nature of the current design research study, the data cannot say anything about regular everyday teaching in our country: this would require non-interventionist studies of uninfluenced classrooms (like in Drollinger-Vetter, 2011; Steinbring, 2005).

Beyond probabilities: consequences for mathematics teachers’ professional development

Although the findings are still tied to the particular topic in view, we can infer some practical consequences that might apply far beyond probability. The study reveals that even with curriculum material that promotes students’ conceptual development by connecting multiple representations (Marshall et al., 2010; Renkl et al., 2013), the role of teachers’ steering in the classroom discussions is crucially shaping the students’ learning opportunities as really provided in the classrooms (Herbert & Bragg, 2021; Jacobs & Empson, 2016).

Many teachers still consider using multiple representations (without explicitly addressing the connections) sufficient (Bossé et al., 2011; Dreher & Kuntze, 2015), so teacher professional development should sensitize teachers for.

  • the need to explicitly explain the connection of representations (Renkl et al., 2013),

  • the involved processes of unfolding and later compacting the involved concept elements (Drollinger-Vetter, 2011), and

  • the teachers’ possibilities of enacting supportive navigation processes that engage students in explaining the relevant mathematical structures (like the part-whole relationships in our case) rather than only the involved elements (like part and whole in our case).

This can be done by first analyzing students’ (written or oral) explanations with respect to typical potentials and shortcomings, and then classroom videos focusing on the pedagogies (Herbert & Bragg, 2021), here with respect to teachers’ navigation practices. In our professional development programs, we have started to discuss videos of teachers’ navigation concerning the distinction of the four semiotic processes and the processes of unfolding and compacting. Our first (still unsystematic) experiences of the discussions with the participating teachers first reveal promising indications that teachers can become aware when their translation practices risk staying on the superficial level and then promote their unfolding of the compacted concepts into more refined concept elements. These joint discussions seem to enable teachers to reflect systematically and provide a first step towards navigating in the navigation space much more deliberately (similar to Jacobs & Empson, 2016). In the future, we will also experiment with making the topic-specific navigation space explicit for them.