1 Introduction

At least 50% of all humans speak more than one language (see Bialystock et al. 2012), thus multilingualism is a widespread phenomenon. This at first glance banal statement leads to the important question, how multilingualism relates to the thinking-and-speaking of the students and how this affects their participation in the educational system (see García and Lin 2016). The conceptual understanding of multilinguals in general and the influence of multilingualism on the cognition- and learning-processes in particular (see Barac et al. 2014) comprises a highly complex research field and relates to the question, how these processes do affect conceptual learning processes, for example in a highly cognition-related subject as mathematics. Such a research interest on the effects of multilingualism on students’ learning processes in mathematics education demands an interdisciplinary approach utilizing linguistic as well as subject-didactical perspectives (see Barwell et al. 2007; Barwell 2020). In this study, insights from the German context are presented regarding the conceptual development of multilingual students: The role of multilingualism in conceptual understanding processes is analysed by combining linguistic perspectives on multilingual processes of conceptual understanding, mainly by utilizing the Theory of translanguaging and the linguistic relativity hypothesis (see Sect. 2.1), with genuinely mathematical educational perspectives concerning the conceptual understanding of fractions (see Sect. 2.2). For the latter, the conceptual development is reconstructed by analysing learning-trajectories with the framework of ‘conceptual fields’ (see Vergnaud 2009).

In the last two decades, the research on multilingual learners’ use of their linguistic resources in mathematics learning has increased and international research on the role of multilingualism in mathematics education utilizes different linguistic, sociological and psychological approaches in different multilingualism-contexts (see Planas and Setati 2009; Moschkovich 2010; Dominguez 2011; Barwell 2016; Prediger et al. 2016). A common paradigm in these approaches is the competence-oriented view on multilingualism (see Moschkovich 2002), being rooted in the linguistic paradigm-shift on the perception of multilinguals’ language processing and their language competencies. Due to methodological and theoretical weaknesses, the deficitary approach on modelling multilingualism, meaning a simplistic focus on (negative) causes and effects of being multilingual by comparing it to monolingual language processing—as it is inherent for example in the separate testing of multilinguals’ language competencies in their individual languages by ignoring cross-linguistic competencies (see Hinnenkamp 2005)—, is broadly rejected, not only in linguistics but also in mathematics education (see Planas and Setati 2009; Moschkovich 2010; Planas and Civil 2013; Planas 2021). Especially a language-as-resource perspective on multilinguals in mathematics education is thus emphasized, meaning an inclusion of all linguistic and cognitive resources into mathematical learning processes instead of a fixation on deficits (see Brandt and Schütte 2010; Planas and Civil 2013; Klose 2015; Meyer and Tiedemann 2017; Planas and Chronaki 2021). Following this paradigm shift, this study contributes to this research field by analysing the form and function of multilingual learners’ linguistic and cognitive resources.

2 Theoretical Background: Interdisciplinary Perspectives On Multilingual Learning Processes

2.1 Linguistic Perspective: the Interrelation of Thinking-and-speaking in Multilinguals’ Minds

Speaking and thinking are two highly interrelated aspects since “the relation of thought to word is not a thing but a process, a continual movement back and forth from thought to word” (Vygotsky 1986, p. 218). Following up on Vygotskys postulate, the so-called linguistic relativity hypothesis (also called Sapir-Whorf-Hypothesis) was formulated and is of high relevance for speakers of multiple languages: Between different languages, nuanced differences in verbalizing and thinking the same concepts can occur. These different nuances mostly affect smaller categories such as ‘associations with colors’, ‘gender-associations to words’ or focused elements and recent research in psycholinguistics has substantiated the relevance of the linguistic relativity hypothesis (see Fausey and Boroditsky 2011; Pavlenko 2011). For example, the Turkish word ‘üçgen’ could be translated as ‘threekinded’, whereas the German word ‘Dreieck’ could be translated as ‘threecorner’, and in English, one speaks of ‘triangles’. All terms focus on different aspects of triangles—corners, angles or more abstractly the property (of threes)—, which becomes visible only through direct translations.

Research from a sociolinguistic perspective comes to complementary results: languaging-processes, a term Swain (2006) uses to describe the interrelation between thinking-and-speaking when “making meaning and shaping knowledge and experience through language” (Swain 2006, p. 98), are highly relevant for language learners’ cognitive processes. When a “person is producing language, what he or she is engaging in is a cognitive activity; an activity of the mind […] language to mediate cognition (thinking) […] it is too simplistic to think of language as being only a conveyer of meaning. Rather we need to think of language as being an agent in the making of meaning.” (Swain 2006, p. 95). Pavlenko (2011) explicitly emphasizes this important research gap when doing research on multilinguals by stating that it is necessary “to understand the implications of the Sapir’s and Whorf’s ideas for speakers of multiple languages.” (Pavlenko 2011, p. 2). Continuing this perspective, Garcia and Wei (2014) develop the term of ‘translanguaging’ to describe the interwoven and functionally combined language use of multilinguals, defining it as a theoretical perspective on multilinguals, where not only the visible and used languages, but also the cross-lingual thinking processes should be taken into account (see Garcia and Wei 2014). According to the translanguaging theory, multilinguals’ “languages are used in a dynamic and functionally integrated manner to organise and mediate mental processes in understanding, speaking, literacy, and, not least, learning. Translanguaging concerns effective communication, function rather than form, cognitive activity, as well as language production.” (Lewis et al. 2012, p. 641). Thus, the translanguaging theory focuses on thinking processes across all language repertoires more than on the mere surface of language use, meaning that a viable analysis of multilingual learners’ linguistic capacity needs a functional perspective on the way multilinguals’ use their languages for thinking processes, even if language switching is not observable directly. This means that, being possibly hidden on the language surface, language-related thinking processes have to be focused for which Wei (2011) gives the example of English-Chinese multilinguals, who think in Chinese even when speaking only English and vice versa (see Wei 2011). Furthermore, for analysing multilingual discourses, factors such as a changing linguistic behaviour depending on the presence of mono- and multilinguals have to be considered. For example, multilinguals might change their language depending on the presence of monolinguals: The presence of a monolingual, institutional person might lead to a ‘hiding’ of the marked language, meaning the standing out or non-allowed language (see Myers-Scotton 1993), and this might falsely be perceived as missing competence in the marked language (see Garcia and Wei 2014).

The relevance of the linguistic relativity hypothesis for multilinguals’ thinking-and-speaking processes leads to the question, if there are different nuances or associations in or between languages regarding concepts. Contrastive studies have shown nuanced effects of languages, f.e. on mathematical concepts such as ‘numbers’ or ‘time’, meaning not too strong but perceivable differences with regard to the processing of conceptual aspects (see Wagner and Davis 2010). Leung (2017) exemplifies such a nuanced difference by describing how “[…] ideas are organised differently in the two languages. […] in English, one starts with the smallest unit (the room, namely Room 312), then […] finally the largest unit, the country (China) […] In contrast, in Chinese, one starts with the largest unit, the country (China [中國]), […] and finally the room (Room 312 [312室]).” (Leung 2017, p. 211). These nuanced differences can be described through the terms ‘synthetical’ and ‘analytical’: A thinking from the ‘whole’ to the ‘part’ can be seen as analytical, meaning a perception of the ‘whole’ first, followed by a thinking of the ‘part’ within and a thinking from the ‘part’ to the ‘whole’ can be seen as synthetical, meaning a perception of the ‘part’ first and a thinking of the ‘whole’ afterwards (see Kuzu and Prediger 2017). Between Turkish and German, the languages of relevance for this study, there are similar differences: They differ in the sequence and relation of the ‘part’ in a ‘whole’ insofar as that in Turkish, an agglutinating language, the grammatical cases of locative and ablative are used in form of suffixes to express a thinking from an analytical perspective by localizing or taking-away the ‘part’ after thinking the ‘whole’ first, whereas in German, an inflected language, the ‘part’ is focused first and then located in a ‘whole’, matching the synthetical perspective. This difference does not only affect this particular case: The Turkish thinking from a ‘whole’ to the ‘part’ affects every thinking of objects-in-objects due to the right-branching, meaning a use of head-initial phrases, whereas in German, it is realized from ‘part’ to ‘whole’ because of a left-branchig (or sometimes left-right-branching), meaning a frequent use of head-final-phrases with prepositions etc. to express relations (see Fig. 1).

Fig. 1
figure 1

Comparison of the morphological structure. The heads of the phrases are highlighted

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What affects the organization of ideas is the position of the ‘head’ (thus of the ‘whole’) at first or final position: Placing it first leads to all further information coming later, placing it at final position leads to all further information coming before. In Turkish, speakers tend to focus the ‘wholes’ of objects first, whereas in German, speakers tend to focus ‘parts’ first (‘Winkel des Dreiecks’ versus ‘Üçgenin acıları’). Another issue is that the Turkish suffix ‘‑den’ can be translated into German with different prepositions, depending on the context of use (see Fig. 1). Since the German and Turkish spoken by the multilinguals in the context of this study is a contact-induced variety, the linguistic construction as well as translation might even be more variable (see Küppers et al. 2015). These differences in the languaging process and its influence on the organization of ideas in German versus Turkish are examples for the linguistic relativity hypothesis from a typological perspective. They give no insights into the question how multilinguals activate, combine or skip these different nuances, but they illustrate the complexity multilinguals have to coordinate mentally between languages. Such a coordination process demands a higher cognitive effort with regard to the activation, inhibition and combination of languages, but this might lead to a higher cognitive flexibility, self-regulation and to a better integration of information from different sources (see Kroll and Bialystok 2013).

2.2 Mathematics-Educational Perspective: Reconstructing Individual Notions of Fractions

2.2.1 A Constructivistic View On Learning Processes and Individual Notions in ‘Conceptual Fields’

According to a constructivistic view on learning processes, the acquisition of knowledge in general and learners’ individual notions in specific are not seen as ‘genuine truths’ depicting an ontological reality, but as intraindividually interpreted schemes with different nuances from an interindividual perspective. These interindividual differences in interpreting concepts are not seen as a deviation from the norm, but as being typical for complex mathematical learning processes (see Piaget 1977; von Glaserfeld 1991). Learning in the constructivistic sense demands the organization of a learning-environment leading to so-called ‘cognitive conflicts’ by being cognitively challenging and thus, designing an adequate, successively discussed and optimized learning environment plays a crucial role (see Waxer and Morton 2012). Ideally, a process of conceptual change then emerges, meaning that the “network of prior knowledge is restructured […] [in] various forms […] [like] the differentiation of concepts […] coalescence of concepts […] and changes in a concept’s ontological status” (Schneider et al. 2012, p. 736). An important aspect of relevance for the construction of knowledge involves the role of interactions: Learning is not solely an act of (predominantly) individual scheme-building, but rather socially situated and constructed through interaction with others. From such an interactionistic perspective, learning is viewed as a social act, where teachers, learning-environments and peers play a crucial role (see Schütte et al. 2019).

Individual notions can be understood as relational mental models or schemes by which the individuals can capture the meaning of mathematical objects and phenomena in everyday- or innermathematical-situations (see Fischbein 1989). These individual notions go beyond a procedural knowledge and more importantly, they contain a conceptual side concerning the understanding of a mathematical content (see Skemp 1976). Prediger (2008) emphasizes that these individual notions of learners’ may not be fully compatible with normative categories or concepts, making it necessary to reconstruct the learners’ differently nuanced individual notions—ranging from non-viable to partially-viable, intuitive concepts (see Fischbein 1975)—and to give the learners opportunities to re-shape them to viable concepts (see Prediger 2008). Furthermore, according to the model of ‘domains of subjective experiences’ (see Bauersfeld 1980), learners’ individual notions are not purely abstract, non-situational mental objects, but rather linked with concrete situations, for example with everyday experiences, didactical material or graphical representations (see Bauersfeld 1980).

Being related to this view on learning processes, Vergnauds (1996) ‘Theory of Conceptual Fields’ describes the emergence of learners’ individual notions by emphasizing individually different, but co-constructed interpretations of mathematical concepts as well as the situatedness of these concepts in ‘actions’, meaning that the individual notions may be bound to operations in concrete situations and to the material used (e.g., concrete material like wooden objects or graphical representations etc.) (see Vergnaud 1996). This means that notions are not only reconstructable when learners explicitly verbalize their thinking by saying phrases like “I think it as a …”, but also when they show conceptually relevant actions like counting the amount of objects, drawing a fraction etc., even if they do not say anything (see Vergnaud 2009). This does not mean that a concept can be reconstructed by only looking at actions: For a precise analysis of the concepts of the learners, both, verbalizations and actions with (mental or real) objects, have to be analysed and as analytic tools, Vergnaud (2009) differentiates between two forms of situated actions: The so-called ||concepts-in-action|| and the <<theorems-in-action>>. Concepts-in-action mean structural elements or categories allowing the subject to divide the world into distinct aspects and to choose the most adequate piece of information—for example when having “to count a large quantity of marbles, […] children learn how to separate smaller collections, count them, and add the cardinals” (see Vergnaud 1996, S. 221), thus activating a notion of cardinality—, whereas theorems-in-action consist of propositional elements, meaning true-or-false-sentences, rules or relations, which are held to be true by the subject in a range of situational variables—in the given example that would be to know that when a ‘whole’ is split, it still has the same amount of objects. Concepts- and theorems-in-action are interrelated, meaning that they complementarily exist with regard to a mathematical situation, both building the base for the ‘conceptual fields’ and their development (see Vergnaud 2009), but such a development cannot be assumed as a linear process of (vertically) changing notions: A non-viable notion might (still) be existent, even if a viable notion seems developed (see Prediger & Schnell 2012), thus the conceptual development can only be assumed hypothetically, being based on frequencies of occurrence in specific contexts without claiming to describe ‘final’ changes (see Sect. 3.2).

2.2.2 Consolidation of Mathematical Concepts

‘Consolidation’ is a specific phase in the learning process and means the solidification of individual notions through a recognizing and reorganizing activity, by which the individuals link and deepen an initial understanding of a mathematical object or concept (see Tabach et al. 2006). ‘Consolidation’ is characterized by the “reorganization of previous constructs, which are recognized, with higher confidence while capitalizing on previous constructs in the course of a new similar activity, or by a further elaboration.” (Tabach et al. 2006, p. 240). In this sense, a simple and identical repetition of an initial understanding is not enough, moreover a broadening of the conceptual understanding in a further activity, for example with new means of representations, is necessary, leading to a differently nuanced or deepened elaboration of the initial understanding (see Tabach et al. 2006). In other words, consolidation only happens when learners do not simply repeat but reflect upon an earlier understood mathematical concept in a new task, a new language, with new graphical representations etc.

Thus, the term ‘consolidation’ can be understood as a reflective, elaborating languaging process by reorganizing an initial understanding of a specific concept- or theorem-in-action. This means that for assuming the initiation of a ‘consolidation’ within analysed sequences, two steps are important: Firstly, a prior individual notion regarding the concept ‘part-of-whole’ has to emerge in the languaging-process of the learners (see Sect. 2.2.3); secondly, a similar but differently nuanced individual notion has to be reactivated in a further activity. These steps may happen between different tasks or within a task as long as it is reconstructable in the languaging process, but a consolidation cannot be inferred if it is the first time of interpreting a mathematical object or concept. Such a ‘consolidational pattern’, meaning a passing through both steps, thus occurs, if a differently-nuanced and non-identical individual notion regarding a mathematical object or concept can be reconstructed (e.g., the ‘part-of-whole’ concept), for example in form of the four language-relatend concepts-in-action (see Sect. 3.1) following up on a phase or discourse being related to an initial conceptual understanding, where the students discuss the meaning of a fraction, share, the ‘part’ etc. Furthermore, for assuming a consolidational pattern regarding a viable conceptual development, a lower amount of non- or partially-viable concepts- and theorems-in-action has to be reconstructed after a consolidational discourse.

2.2.3 Facets of Understanding Fractions: the Mathematical Object ‘fraction’

‘Fractions’ are highly relational mathematical objects symbolizing a new type of numbers (the rational numbers ℚ) to learners: A ‘part’ has to be interpreted in relation to a ‘whole’ and is—in the symbolical representation—divided by a fraction line, thus two components have to be seen as a new kind of merged, connected number (see Freudenthal 1983). International studies about students’ fraction-understanding show that one of the biggest issues in understanding fractions is the relational thinking of the ‘part’ and the ‘whole’ (for an overview, see Tunç-Pekkan 2015): When first encountered with fractions, students tend to interpret the ‘part’ and the ‘whole’ as two separate natural numbers, which may, for example, lead to a non-viable comparison of fractions (see Newstead and Murray 1998). Empirical insights indicate that students have to be given opportunities to re-shape a (possible) non-relational thinking of fractions as two separate numbers into a relational, viable concept of fractions as a ‘part-of-whole’ in adequate learning-environments focusing the relation between the ‘part’ and the ‘whole’ (see Tunç-Pekkan 2015). The ‘part-of-whole’ concept is an important starting point for understandig fractions and a viable construction of the conceptual fields is crucial for concepts building upon it, like the ‘part-of-part’ concept (see Glade and Prediger 2017), which is why the ‘part-of-whole’ concept is focused in this study. Further concepts of rational numbers are the quotient-, operator-, ratio- and measure-concept (see Behr et al. 1983; Prediger and Wessel 2013; Kollhoff 2022).

Understanding the ‘part-of-whole’ concept does not only include a relational, abstract understanding of the subcomponents of one fraction (‘part’ in relation to the ‘whole’), but also the understanding of structural relations and changes becoming visible when interpreting graphical representations of fractions: Learners should also reflect upon the relation between the (colored) ‘part’ and the (not colored) ‘rest’ as well as on the effects of operatively increasing the (numerical or graphical) size of the ‘part’ or the ‘whole’ (see Prediger and Wessel 2013). This means that one has to consider more than just the symbolic register of fractions with the ‘nominator’ and the ‘denominator’ when teaching fractions: it is important to let the learners reflect upon the meaning and relation of the ‘part’ and ‘whole’, and for that, languaging processes of the learners have to be considered as crucial. For such a conceptual understanding of the ‘part-of-whole’ concept in different representations, further subcomponents like the ‘pieces’, the ‘rest’, the equally sized ‘wholes’ etc. have to be considered as well and these subcomponents can be reflected either in everyday contexts or in more formal contexts (see Kuzu 2019). For fostering such a conceptual, meaning-based understanding of a ‘part-of-whole’ with regard to a relational understanding of the subcomponents of a fraction, the use of the fraction bar model has proved as an important graphical representation in empirical research since it supports the building of viable individual notions when thinking of a ‘part-of-whole’ and can later be extended to a fraction bar board (see Fig. 5) for the purpose of understanding the structural relations between multiple shares (see van den Heuvel-Panhuizen 2003; Wessel 2020). In such a fraction bar board, students can see the interrelation between the ‘whole’, the ‘part’ and the ‘pieces’ since several fraction bars are arranged vertically.

3 Context and Method of the Study

3.1 Context of the Study: Multilingual Conceptual Fields in a Bilingual Learning Environment

This study was conducted within the framework of ‘Design-based Research’, meaning an approach where learning processes in self-designed learning environments are analysed (see Bakker 2018; Kuzu 2019). It was an appropriate methodological framework for the study because the topic demands an interdisciplinary approach, an analysis of learning processes and the designing of tasks due to the lack of adequate bilingual learning environments. For the latter, four design principles were developed and implemented (see Fig. 2).

Fig. 2
figure 2

The four design-principles of the learning-arrangement with exemplary design-elements (see Kuzu 2019)

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These four principles represent important conditions for activating multilingual learners’ language resources as they are illustrated in this article. For a detailed discussion of these design principles, see Kuzu (2019).

The main research interest for the study was the analysis of the relation between students’ conceptual development and translanguaging processes, thus, their conceptual fields were reconstructed with regard to the ‘part-of-whole’ interpretation as well as the German and Turkish language means being used to realize their verbal interpretations. In total, 13 non-viable, partially-viable and viable concepts-in-action were reconstructed hypothetically (see Kuzu and Prediger 2017; Kuzu 2019). Non-viable concepts-in-action were mostly notions with a non-relational interpretation of fractions in the sense of two separate, natural numbers or with interferences from contextual experiences like the speed of download in a download bar. Partially-viable concepts-in-action showed first forms of relational thinking, but with a non-viable (too) strong emphasis of either the ‘part’ or the ‘whole’, still leading to wrong solutions for example in the comparison of shares. The viable concepts-in-action include four language-related nuances and will be the main concepts-in-action focused in the analyses in this article (see Fig. 3).

Fig. 3
figure 3

Indication of the four language-related concepts-in-action (see Kuzu and Prediger 2017; Kuzu 2019)

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These four reconstructed concepts-in-action are highly language-related (see Fig. 3): Using Turkish, the learners showed a thinking as localization of the ‘part’ in the ‘whole’ (||localizing nuance||) or as the taking-away of the ‘part’ from the ‘whole’ (||taking nuance||). The ||localizing nuance|| matched a more formal way of talking about fractions, whereas the ||taking nuance|| matched a more everyday-related way of talking about fractions. In German, both grammatical cases are missing and instead, the learners used various prepositions to describe a thinking with an emphasis on the part-stripe, either in an ordinal sense (||quasi-card. nuance||) or by focusing the part-stripe en bloc (||share nuance||) (see Kuzu and Prediger 2017; Kuzu 2019). The German nuances were more synthetical and abstract in comparison to the Turkish-related nuances. All four language-related nuances were reconstructed according to empirical indications (see Fig. 3). German utterances were only coded as ||localizing nuance|| or ||taking nuance||, if a Turkish utterance with a use of the locative or ablative case was visible priorly and with a close proximity; vice versa for the German nuances.

3.2 Method of the Study and Sample Selection

Being based on the qualitative method of the so-called ‘Interaction analysis’ (Krummheuer & Naujok 1999), a sequential and interpretative analysis approach was chosen (see Cobb and Bauersfeld 1995; Meyer 2010; Brandt and Schütte 2010; Schütte et al. 2019). These analyses were interpretative because of two reasons: First, they did not claim to be reconstructions of ‘truths’ but hypothetical and possible explanations and second, mathematical knowledge in learners’ interactions was seen as being constituted and constructed by social interaction (see Schütte et al. 2019). In a turn-by-turn approach, at first, multiple possible explaning hypotheses regarding the research interest were inferred abductively, meaning the generation of plausible explaining hypotheses regarding newly observed phenomena from within the data (see Meyer 2010; Paavola 2011). These interpretatively reconstructed explaining hypotheses were discussed and conducted in groups of researchers to reach the highest possible degree of intersubjective plausibility (see Steinke 2004; Kunsteller 2018; Schütte et al. 2019). After abductively generating these explaning hypotheses—for example after observing that students were focusing the aspects of a share from different directions when using Turkish or German—, these hypotheses were further enrichened with or rather compared to theoretical aspects (“Theorisation”), in case of the example with mathematical and linguistic knowledge being of relevance to the observed phenomena (see Sect. 2.1). In a last step, the occurrence of these explaining hypotheses was tested in all groups through a comparative approach (“Comparison”), meaning the creation and analysis of comparison groups with the aim “to depict the specificity of […] segment[s] of reality […] in their relation to each other.” (Schütte et al. 2019, p. 125).

The abductions regarding students’ conceptual fields and translanguaging processes were also conducted in (sub‑)steps. In total, three interwoven steps were necessary due to the interdisciplinarity of the research question. In a first open, mathematical substep, turns were analysed with the intention of reconstructing explaining hypotheses about learners’ individual notions and conceptual fields about the ‘part-of-whole’ concept, including a collective conduction and discussion of the viability of these reconstructions. In a second, linguistic substep, the turns with relevance to the conceptual fields were re-analysed with regard to possibly language-related nuances, meaning that the reconstructed individual notions were further analysed in terms of language means expressing relations, such as prepositions or suffixes. Especially the concepts-in-action were focused on in step 2 since they showed highly language-related differences in early analyses (see Kuzu and Prediger 2017). In a third substep, the individual notions were compared with other learners’ notions, meaning a broadening and detailed comparison of all 14 learners’ notions for making again abductions about quantities and interindividual similarities and differences (see Fig. 4).

Fig. 4
figure 4

Steps and substeps of the utilized interpretative approach (the dashed line indicates analogous facets)

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The data analysis was conducted until a local theoretical saturation could be assumed, thus until no further explaining hypotheses could be reconstructed (see Glaser and Strauss 1967; Schütte et al. 2019).

The focused interaction units for this article are the four tasks regarding the conceptual understanding of fractions as part-of-whole, including the subtasks (see Krummheuer & Naujok 1999). These four tasks were chosen due to two reasons: They were designed with a focus on the conceptual understanding of the concept—mirroring a developmental process within a self-contained unit and being designed according to the four design principles (see Sect. 3.1)—and prior analyses in the piloting data showed that especially in moments of conceptual understanding, a yet to be understood ‘surprising’ phenomena in form of a divergence in the learners’ language use was observable. The sample of learners (n = 14) was chosen following a criteria-led sample selection process with regard to the criteria ‘full participation in all sessions’ and ‘full participation in the pre-, post- and follow-up test’. For this article, the learners Emir, Osman and Ismail were chosen because they fully participated in the sessions as well as in the tests and they were a group of learners, who showed an active usage of both languages, but did not represent the extremes (e.g., the highest/lowest number of translanguaging, viable concepts etc.). The transcript sequences in this article were chosen with the intention to illustrate teacher-student interactions in concept-related explanatory sequences since student-student-interactions were analysed in prior publications (e.g., in Kuzu and Prediger 2017).

3.3 Research Questions for the Present Case Study

In prior analyses (see Kuzu and Prediger 2017; Prediger et al. 2019; Kuzu 2019), the empirical relevance and viability of the four language related concepts-in-action was verified as a result to the research question, how “students in a bilingual teaching intervention adopt and combine language-related conceptualisations of the part-whole concept in or across both languages” (Prediger et al. 2019, p. 193). The main result of the study regarding this research question is that in translanguaging discourses, students might use language-related nuances (see Sect. 3.1) in a bilingual-connective mode, meaning that Turkish-related nuances might occur in German utterances and vice versa (see Kuzu and Prediger 2017; Kuzu 2019). In Prediger et al. (2019), an outlook was given to the possibility of a multi-perspectivistic and consolidational function of the use of these four concepts-in-action (see Prediger et al. 2019), but left open for a further empirical discussion.

The latter, a discussion of a specific functions of the reconstructed creative language use with regard to the learning processes, is of importance since such an interwovenness of multilingual thinking-processes (see Sect. 2.1) with the development of differently nuanced individual notions (see Sect. 2.2) indicates not only a research need with regard to the form of language use but also with regard to a possibly learning-related function of such a highly cognition-related translanguaging process. Following up on this, this article focuses on a research question with a twofold differentiation:

(Q1)

Which functions with regard to the learning process can be reconstructed, if learners activate their multilingual resources within their conceptual fields of the mathematical concept ‘part-of-whole’?

(Q2)

To which extent are these functions in the learning processes explaining an experience shared by learners?

4 Empirical Insights into Students’ Multilingual Learning Trajectories

4.1 Students’ Learning Trajectories for Thinking the Fraction-Concept ‘Part-of-Whole’

In this sequence, a content-related multilingual discourse occurs. In task a), after coloring the share 2/6 in the fraction bar board, the students Emir, Osman and Ismail have to find other shares being equal to 2/6 (as well as for the share 3/5, in Task b), but it is not a mere process of transferring and finding, but also of explaining, why the shares are equal. That was an instructional aspect and the teachers of the intervention were trained beforehand to ask the students for justifications in such tasks. The students were used to the graphical representation of a fraction as a fraction bar as well as to the fraction bar board (see Fig. 5) from the prior tasks.

Fig. 5
figure 5

Task 4, work sheet of Ismail. Translation: a “Find three shares being equal to 2/6 in the fraction bar board”; b “Find three shares being equal to 3/5 in the fraction bar board”

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In the transcript sequence, Emir, Ismail and the teacher discuss how to find shares being equal to 2/6 (Osman is also present, but does not participate yet). Utterances in Turkish (T) are in grey and utterances in German (G) are in black. (Figure 6).

Fig. 6
figure 6

Transcript sequence 1

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The transcript sequence starts with a task-related question concerning the problem of how to look at the fraction for finding equal shares (Turn 22) and is followed by the explanation of the learners’ individual strategies. Diverse nuances of interpreting the share can be reconstructed for the learners Ismail, Emir and Osman: In Turn 22, Ismail’s task-related question might already bear a possible, intuitive emphasis on the part-stripe or numerator. He focuses the numerator “two” and asks if that has to be focused. Since it is a task-related question, a conceptual nuance cannot be stated here directly, but the emphasis may be related to a way of looking at the fraction. Ismail may perceive specific aspects such as the part-stripe or numerator first, which would match the German language-related nuance ||share nuance||, although here it is more a focusing of the numerator, not yet of the part-stripe. Later, in Turn 38, this hypothesis seems to be substantiated by another focusing of the part-stripe, again by switching into German, and this time with a deictical reference to the part-stripe, not only to the numerator “two” or “three”.

In the course of the sequence, Emir also activates the ||share nuance|| by focusing the part-stripe with deictical and verbal means in Turn 26 (“this one is longer… this one is smaller”) while at the same time looking at the right fraction bar (so the ‘whole’ is perceived implicitly), but he uses Turkish instead of German, indicating the bilingual-connective mode. Especially Emir shows this specific pattern in earlier sequences, where he starts using specific language-related nuances in the matching languages, but later he uses the same nuances in the other language (e.g., the German-related nuance in Turkish in Turn 26). What can be noticed is that right after Emir, Ismail justifies himself in Turn 28 (being indicated by the use of defensive language means, “I wanted to make …”), but for that purpose he a) switches into Turkish, maybe as a reaction to Emir’s switch to Turkish, but at the same time, he activates the Turkish-related ||localizing nuance||: His way of articulating his thinking matches the localizing process (“… ninth-bar, therein three pieces”) and in his Turkish utterance, he uses the locative case, but he also uses matching deictical means by patting on the part-stripe right after his “therein three pieces” utterance. It is noticeable, how all learners of this sequence use deictical means in the same way, they point or pat on specific ‘pieces’ and ‘parts’, mostly on the part-stripe. It is well-known for multilingual students that they use deictical means with a high frequency (see Ng 2016), but the deictical pattern leads to another possible assumption: Especially the patting- and striking-away-deixis (visible also in other sequences, see Kuzu 2019) may be related to both Turkish grammatical cases missing in German: The locative case as the localizing case and the ablative case as the taking-away case. If so, this means that the deictical means are not just means to express visually perceived elements but also language- as well as meaning-related actions, a hypothesis which could be affirmed in similar cases: Students used a specific, different form of deixis when activating the Turkish locative-case—by tapping on the ‘part’ stripe more vertically—or the Turkish ablative-case—by indicating a process of pushing out the part-stripe more horizontally (see Wagner et al. 2018; Kuzu 2019). This hypothesis would match the translanguaging perspective on multilingual learners since the (artificial) separation of languages is dismissed, which would mean that even in no language-switching moments and with (just) deictical means, a wholistic resource use of the learners may be inferred as in Turn 26, where his localizing deixis might be part of his multilinguality and due to the locative case in Turkish, which in this case seems to be logical since Emir also switches to Turkish, although he activates the German-related nuance. Due to methodological limitations, this hypothesis cannot be affirmed sufficently (since there is no possibility to “look” into implicit intentions behind deixis), but further linguistic analyses confirm the possibility of a language-related use of deictical means: It seems to be a pattern matching the grammatical structures of German and Turkish (see Redder et al. 2018).

After Turn 29, the teacher again leaves the group to visit another group and the students start marking in the fraction bar board. Coming back to the group, the teacher tries to activate the students by asking them to write down the fractions, which the learners had forgotten (Turn 37). (Figure 7).

Fig. 7
figure 7

Transcript sequence 2

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When replying to the teachers’ impulse, Ismail again focuses the numerators and changes his language use and nuance activation if compared with Turn 28: In Turn 38, he seems to activate the German-related ||share nuance|| by switching to German as presumed in Turn 22: Typically, a high emphasis of the part-stripe or numerator is visible when using German (“two and then three”), while at the same time the ‘whole’ is referred to implicitly (by looking at the matching fraction bar) or afterwards.

After the short sequence up until Turn 41, the teacher again leavers the group to visit another small group and the students start writing down the fractions on the right side of the fraction bar, being accompanied by off-topic conversation. Coming back in Turn 61, the teacher ties upon his last utterance in Turn 39, where he asked what the share was, but now tries to activate everday resources as they were part of the design-principles by hinting at the baklava context, which was a design-element used in prior tasks. (Figure 8).

Fig. 8
figure 8

Transcript sequence 3

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In this sequence, another multilingual discourse emerges, following the linguistic patterns being visible earlier: Ismails switching to German in Turn 38 might be forced through the language switching of the teacher, which is also possible from Turn 62–68, where Ismail now again switches to Turkish (after a Turkish utterance of the teacher) to explain his conceptual understanding and this time, he seems to activate the second Turkish nuance, the ||taking nuance||, instead of the earlier activated ||localizing nuance|| (Turn 28) or ||share nuance|| (Turn 38): In Turn 62, he now references to the ‘whole’ explicitly (“six”) and right after that, in Turn 64, he emphasizes a specific taking-away notion by verbalizing “six pieces from it”, indicating the ‘whole’ with the language mean “it”. This hypothesis is affirmed in Turn 66–68, where Ismail again verbalizes a taking-away process by using the ablative case in Turkish.

In the last turn of the sequence (Turn 69), Osman, a learner staying silent up until now, participates by verbalizing a more formal utterance with reference to the part-stripe, which is why carefully, a more German-related ||share nuance|| may be inferred (similar to Ismail in Turn 22 or Emir in Turn 26). What is surprising at this point is that in this very short sequence, at least three nuances and five nuance-related language switching moments (in Turns 22, 26, 28, 38, 62, 69) could be reconstructed. Thus, the multi-perspectivistic language- and nuance-switching in this sequence is a phenoma indicating a specific functional pattern and highly interactional activation of multilingual resources: a repetition-like, but differently nuanced explanation of the ‘part-of-whole’ concept, indicating a process of multi-perspectivistic joint elaboration. These repetition-like, but interindividually different interpretations may be interpreted as a reorganizing process through adding, or rather combining different interpretational nuances based on linguistic differences since all of the reconstructed notions are viable notions: Starting with Ismails (possibly) German viable nuance ||share nuance|| in Turn 22 (and the repetition of it through Emir in Turn 26 by switching into Turkish), a change in viable conceptual nuances is visible in Turn 28, where Ismail switches his language as well as the nuance (||localizing nuance||), then in Turn 38, where again ||share nuance|| is activated, followed by another switch into a viable nuance in Turn 62–68, where the ||taking nuance|| is activated in multiple steps. At last, Osman activates the ||share nuance|| again in Turn 69, which is another change of the viable conceptual nuance. All in all, this sequences shows a (late) phase in the conceptual development, where all learners seem to have developed a viable conceptual understanding of fractions as a part-of-whole, while at the same time showing a need of repeating their understanding with different nuances, thus from different perspectives: From an emphasis of the part-stripe to the thinking from the ‘whole’ to the ‘part’, the change in nuances marks a difference in the way the learners focus integral parts of a fraction as a ‘part-of-whole’. Thus, a consolidational pattern may be inferred for this group since not identical, but differently nuanced concepts-in-action of the ‘part-of-whole’ concept are repeated, which the conceptual development from Task 1 to 4 confirms (see Fig. 9).

Fig. 9
figure 9

Conceptual development of Emir, Osman and Ismail (in bold: Language-related viable concepts-in-action)

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Figure 9 shows, how all learners start with non-viable or partially-viable conceptual nuances in Task 1, and how these nuances can be reconstructed until Task 3 in a diminishing number. In Task 4, none of these non-viable or partially-viable nuances can be reconstructed. Task 3 seems to be a first changing point since a tendency to activate more viable nuances mixed with some partially-viable nuances is visible. Thus, a pattern of emerging multiperspectivity and a reorganizing momentum may be inferred in Task 3, whereas in Task 4, a multilingual consolidational discourse is inferable. Generally, the learners show a fluent and conceptually relevant use of their multilingual resources across all four tasks: As Grosjean (1985) states for bilingual learners, the switches between languages seem to be highly flexible, fast and fluid (see Grosjean 1985).

4.2 Extending the Analysis to All 14 Students

The case study presented in Sect. 4.1. gives insights into the phenomena of consolidation through multi-perspectivistic mathematical meaning making in translanguaging processes: A cross-lingual multi-perspectivity emerged and the multilingual students elaborated their interpretation of a fraction as ‘part-of-whole’ by adding and combining different concept-related facets and relations in German and Turkish. However, from a methodological viewpoint, these insights are not enough to state a similar functionality of using multilingual resources for all learners, which is why an in-broadth analysis of all groups will be conducted for answering the research Question Q2—how often a consolidational pattern does occur for all n = 14 learners—by analysing the table of reconstructed concepts-in-action for all 14 focus learners (see Fig. 10).

Fig. 10
figure 10

The four language-related concepts-in-actions for all learners (n = 14), chronological from top to bottom; G is German (black), T is Turkish (grey), GT means mixed utterances, […] stands for not changing nuances

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At first glance, Fig. 10 shows a specific tendency in the conceptual-development: All fourteen learners show at least one of the four concepts-in-actions throughout the four focus tasks, although a difference is also visible: For some learners, only up to three concept-in-action could be reconstructed (see Sevda, Rükiye and Mediha) and for others, more than six concepts-in-action could be reconstructed (see Emir, Ismail and Deniz). The quantity of the concepts-in-action does not mean that the understanding is better however, it only shows, how often the learners verbalized their thinking. Nearly half of the learners (6 out of 14 learners) show viable concepts-in-action after Task 3 (with non- or partially-viable concepts-in-actions being dominant prior to Task 3) and further qualitative analysis in Kuzu (2019) confirms this pattern: Non- or partially-viable concepts are dominant in the Tasks 1 and 2 and after Task 3, a significantly higher amount of viable concepts-in-action is visible. The concept-in-action, which could be reconstructed in the most cases is the ||share nuance||. Furthermore, all conceptual developmental processes start with the ||share nuance|| (14 of 14 learners), which may be linked to the fact that German is the language the students use when learning fractions for the first time or it could be related to the presence of the teacher (assumed as expected behavior).

With regard to the language-related concepts-in-action, Fig. 10 shows that not all learners activate multiple language-related nuances, but a high number does: Six from fourteen learners made utterances, where at least two language-related, synthetical and analytical concepts-in-action could be reconstructed, which means that nearly half of all learners showed a complex, cross-lingual conceptual development as it was illustrated in the case study in Sect. 4.1. This may seem like a small number of activation at first glance, but it should be considered that in the study, all 14 learners used their Turkish language in grade 6 for the first time—especially for talking about their mathematical understanding—, which is a very late starting point for activating the multilinguals’ resources for understanding concepts, and that this happened in a school system where the use of first language is often forbidden in school contexts—especially if the students speak low-status languages like Turkish, Arabic or Romanes (see Busch 2013)—, which leads to the problem that students might not dare to use their languages in front of institutional employees (see Fürstenau 2016). Under these circumstances, it is remarkable that 30 of 73 viable concepts-in-action (including the synthetical nuances) are uttered in Turkish or in Mixed-Language and it is even more remarkable that the different, language-related nuances were activated by nearly half of the students. What is more is that the reconstructed concepts-in-action only show the reconstructable utterances, not all processes in minds, which means that even the “silent” students might use their multilingualism in a receptive way (see Ribbert and ten Thije 2007): As it was visible in the case study, the responses of the students—even if they were only in German—were connected to utterances in Turkish and vice versa and furthermore, even monolingual utterances could contain cross-lingual deixis.

Cross-lingual consolidational processes, which include a multi-perspectivistic variation in the thinking of the order of share-facets as well as a language-related interpretation of the relation between the ‘whole’ and the ‘part’, can be inferred in those cases, where the students use one language-related nuance and activate another language-related nuance in the following Turns or Tasks since a reorganizing, multi-perspectivistic view on an share can be reconstructed in those cases as the analysis of the case study indicated. This applies to 6 out of 14 learners, but affects nearly all learners since these 6 learners were part of all groups and the learning environment was designed with the aim of fostering small group interactions, in other words: In all sequences, where the four viable concepts-in-action could be reconstructed, the learners were part of small group interactions. One cannot state that consolidational processes only happen when multilinguals activate both of the languages, but there seems to be a high potential for consolidating the conceptual understanding, if there are differently nuanced perspectives on conceptual elements like share-facets or the thinking of the relation between those facets in the sense of the linguistic relativity hypothesis.

5 Discussion of Results and Limitations

5.1 Results of the Study

The analysis of multilinguals’ conceptual development points out to the fact that one may—carefully—assume that multilingualism is linked to different nuances in thinking mathematical concepts and interpretations. These different, language-related ways of thinking a concept may be used as cognitive resources across the languages for understanding highly relational mathematical concepts like fractions and, with regard to research question Q1, may lead to a repetition-like reorganization and enrichening of the individual notions from different perspectives. An activation of multilingual resources may thus have a highly learning-related function by leading to consolidational processes insofar as that differently nuanced ways of thinking a concept in different languages may give opportunities to deepen and reflect the conceptual understanding from different perspectives and with different emphases: a change in the ‘thinking direction’ could be reconstructed as well as a slightly different, more context-related way of thinking the relation between the ‘part’ and the ‘whole’ when activating different languages (see Sect. 4.1). In Turkish, a more localizing or moving-out way of thinking the ‘part’ in the ‘whole’ could be reconstructed and in German, a thinking from the ‘part’ to the ‘whole’ with a stronger emphasis of the ‘part’ was visible. This leads to the question, if such a creative, cognitive multi-perspectivity was only a ‘special case’ or if it was a phenomenon being visible more broadly in the conceptual development of all n = 14 learners (research question Q2). Regarding the research question Q2, the in-broadth-analyses in Sect. 4.2 indicated that the consolidational pattern was not only a ‘special case’ but that at least half of the learners (7 out of 14) showed an analogous, highly multi-perspectivistic conceptual development by activating multiple language-related nuances across the languages. Furthermore, 12 out of 14 learners used another language than the original concept-nuance-language, the latter being a ‘lighter’ form of translanguaging (see Kuzu and Prediger 2017). Beyond these individual notions, these multilingual discourses occured in most of the groups (in four out of six groups) and considering receptive multilingualism, the number of students ‘thinking’ beyond one language is presumably higher. Thus, the insights from the case study presented in this article show, how a multilingual conceptual understanding consists of more than simple ‘language switches’: Language related differences in form of nuances have to be expected between every language, being in accordance with the linguistic relativity hypothesis as well as Vygotskys (1986) postulate about the interwovenness between language, cognition and culture/context. Furthermore, the multilingual learners from the study showed a self-driven motivation in using, activating, discussing and linking these nuanced differences (see Kuzu 2019). In this article, language switching moments were partly led by the teacher (being an important, interactional design-element for fostering translanguaging processes) and partly initiated by the learners and this shows, that a) a multilingual space (see Wei 2011) could be established with the design principles, meaning an acceptance of multilingualism in plural forms (with monolingual or mixed utterances, creatively mixed utterances, self-initiated switches versus teacher-initiated switches etc.), and that b) in such a space, multilingual learners may activate differently nuanced individual notions as part of a multi-perspectivistic conceptual development regarding concepts such as the ‘part-of-whole’ concept. In these highly creative processes of meaning-making, “language is the place to imagine and make possible equitable and quality ways of mathematics learning and teaching.” (Planas 2018, p. 13). More generally, one must assume that for multilingual students, a reflection of possible linguistic differences in form of conceptual nuances is important and ‘fruitful’ to deepen the conceptual understanding since it might lead to multi-perspectivistic individual notions. Linguistic research also confirms this hypothesis: Multilinguals should reflect, compare and link their languages in educational contexts and not only with regard to european (high-prestige‑)languages such as English or French, but also for languages from different language families such as Chinese, Catalan, Romanes Arabic or Turkish since especially languages from different language families might bear different linguistic conceptualizations due to highly different grammatical structures and language-related characteristics (see Sect. 2.1).

An important finding with regard to the language use is that in this study, the learners did not only activate an ‘everyday language’ in their (marked) first languages, in this case Turkish, or rather did not only use these means solely for profane ‘everyday purposes’: Instead, they used all of their language resources and even everyday language means in a highly cognition-related way by activating and combining both of their languages effectively to express and explain relations between share-related aspects such as the ‘part’ or the ‘whole’. This is surprising since they were only used to Turkish as a spoken language, but even if so, this does not mean that they were limited in their multilingual meaning making in or with Turkish as (marked) part of their multilinguality. These insights into the translanguaging processes of multilinguals indicate that an assignment of the languages of the multilinguals to specific registers or purposes, for example by allowing first languages only for everyday-related discourses, is not viable, and would by argumentum e contrario lead to the false and deficitary assumpution that it is only ‘fruitful’ to allow the institutional language for mathematical, cognitively demanding discourses (see Moschkovich 2002).

The insights given in this article substantiate the phenomena of consolidation through multi-perspectivistic mathematical meaning making in translanguaging processes and this adds up to the range of possibilities through the activation of multilingual resources in mathematics education: Multilingual learners might be more motivated and participative in mathematical, discoursive activities (see Planas and Setati 2009; Zahner and Moschkovich 2011), might activate more everyday associations as cognitive resources (see Dominguez 2011) or—as the main result of the study and the analyses in this article—it might happen that the reflection of language-related, nuanced differences in the mathematical concepts leads to more consolidated learning processes, for example through an emerging multi-perspectivity of thinking orders or the relations between conceptual facets.

5.2 Limitations and Future Research

This article indicates that future studies should consider multilinguals’ resources as an important resource for discussions about possibly different conceptual meanings and nuances. Language-related differences in conceptual facets have to be expected in multilingual mathematical discourses and since mathematics in general and multilingual mathematics in specific is a highly language-related topic (see Meyer and Tiedemann 2017; Planas 2018), future studies about multilingualism should consider the possibility of differently nuanced conceptual facets as a resource for deepening the conceptual understanding and consolidation.

An important limitation of the study is that in the school context, it might be challenging to orchestrate reflection discourses, if there are multiple groups of multilingual learners, and it might also happen that students do not have peers sharing their multilingualism or that the teacher is not multilingual. Here, further analyses of multilinguals’ learning processes in the context of pre-algebraic thinking indicate that a) age-mixed learner groups with emergent and more advanced multilingual learners might help in finding dialogic opportunities for fostering multilingual learners in the school context and b) digital tools like explanation videos might help in giving multilingual learners opportunities for using and reflecting their languages in the sense of the ‘Pushed Output’ while at the same time ‘conserving’ their productions (see Kuzu 2022; 2023). The activation of multilingual resources is thus not only important, especially if, as this study shows, concept-related nuances between languages can be expected, but also possible, if a student-oriented or dialogic approach is used (see Barwell 2016; Planas and Chronaki 2021) and if an extended view on multilingualism, including varieties, is used so that even ‘monolingual students’ might reflect conceptual nuances between different forms/varieties of their languages (Paul et al. 2009; Barwell 2020). A further limitation is related to methodological issues: It might be that there are not directly visible meaning-related differences regarding the conceptualization of mathematical concepts between the languages spoken by multilinguals, although from a translatological viewpoint, nearly every translational process is accompanied by either small or bigger differences in form of nuances (see Pavlenko 2011). A descriptive, careful analysis of students’ multilingual utterances is thus important, especially since translanguaging might lead to a divergent or rather creative way of using language means—for example by using atypical words, combining words or playing with the meaning or sound of language (see Garcia and Wei 2014). The different share-nuances in Turkish versus German and its creative connection in the multilinguals’ utterances are only one example for translanguaging, but further examples and various degrees of translanguaging-processes are possible, making it necessary to gain further insights into the conceptual translanguaging of multilinguals in further studies.

But—beside all limitational aspects—it is urgent to do explorative research about multilinguals’ learning-and-thinking-processes, especially in school systems, where a high proportion of all learners are multilinguals with a rising trend due to demographic changes (i.e., through climate change or wars) (see Planas and Chronaki 2021). It is highly relevant to understand how multilingualism and learning processes are interwoven and how to allow multilinguals to use their multilingualism as a cognitive resource in these processes.