1 Introduction

As the research on language and its roles in teaching and learning increases in the field of mathematics education, so does its influence on educational policies and school practice. Hence, we need to be aware that such decisions are influenced by ideologies underlying the research and that it is vital that these ideologies are constantly re-evaluated, explicated, and re-considered—“[…] what we think that language can do (ideology) is related to what we do with language (practice)” (Kusters et al., 2020, p. 3; emphases added). Often, the process of re-evaluating the influence and limitations of research starts in adjacent fields from which ideas and concepts are adopted for mathematics education, such as—in the case of language—linguistics and only slowly spreads into (mathematics) education research. In the past, such reflections on the nature and role of language resulted in the differentiations of notions around multilingualism and language use, such as code-switching (Gumperz & Hymes, 1972) and translanguaging (Blackledge & Creese, 2010; García & Wei, 2014). These notions conceptualize language and the language user in slightly different ways, grounded in differences in theoretical perspectives of psycholinguistics and sociolinguistics and in the underlying ideologies about agency, ability and purpose of the use of linguistic resources available (see MacSwan & Faltis, 2020). The development in the field was initiated and is catalyzed by acknowledging a deficit monolinguist perspective in which changing between languages indicates a struggle of expression in one language that has to be overcome through expression in another available language. Instead, bilinguals are increasingly considered as language users with a repertoire of linguistic resources that are used fluidly based on a number of social, cognitive and interactional motives (Faltis, 2020). Mathematics education research has gone a long way to employ and integrate a more asset-based perspective on languaging (Moschkovich, 2007; Planas, 2018) and to acknowledge linguistic discrimination within mathematics education research (Barwell, 2003, 2018). However, research on language that shapes the field and the view on mathematical concepts still largely sidelines and even excludes important categories of languages, linguistic resources and students with certain linguistic backgrounds by not truly engaging with languages that do not fall into western- and modal-normativityFootnote 1. For example, sign languages as primary means of communication of Deaf students and teachers are carried out in a modality that affords mathematics thinking, learning and discourse in specific ways that are not often considered in research on language in mathematics education (Krause, 2018, 2019; Kurz & Pagliaro, 2020; Pagliaro, 2010). These languages will be the focus of this paper that offers a path to a more inclusive perspective on research on the role of language in mathematics education.

1.1 Attitudes and ideologies towards sign languages

Kusters et al. (2020) provide a definition of language ideologies as “thoughts and beliefs about languages, varieties, modalities, and the people who use them. Attitudes about what language is (or is not), how and where languages are used, their value, and their origins are expressed as language ideologies” (p. 5). The authors’ perspective aligns with a wider view on language considered in discourse analytic approaches in mathematics education (Morgan, 2020), when indicating how language and language use is concerned in the focus of attention beyond issues of national language: “Language use is a way of enacting social identities and belonging to certain intersecting categories such as ethnicity gender, disability, sexuality, social class, and nationality. In this way, ideologies about everyday language practices can create shifting categories of sameness and difference” (Kusters et al., 2020, p. 5).

Krausneker (2015) points out several ideological attitudes towards sign language that ground the hegemony of spoken/written languages:

  • the devaluating ideology of sign languages not being ‘real’ languages;

  • the audist ideology of deafness not being the ‘normal’ such that sign languages are seen as an auxiliary means of communication until the ‘deaf deficit’ can be fixed by developments in the medical field;

  • the disability paradigm of understanding the Deaf as people with a disability in tension with understanding the Deaf signers as a linguistic minority—both perspectives coming with their own political and societal consequences;

  • the economic paradigm puts forth the pragmatic viewpoint of cost and benefit, arguing that additional access through sign language interpreters and other additional means might not be worth accommodating a linguistic and cultural minority, or discussing whether it might be more practical to unify all sign languages to a signed Esperanto.

Lastly, Krausneker describes one ideology for attitudes towards sign languages that counterpoints these deficit perspectives in that it “values human diversity and understands that the idea of human ‘normalcy’ is an illusion” (p. 421). The idea of Deaf Gain (Baumann & Murray, 2014) acknowledges Deafness and the cognitive, creative, and cultural diversity that we (as a society, but as I argue here, also as a field) can value and learn from when distancing ourselves from a deficit perspective.

However, Krausneker (2015) also acknowledges this shift of perspective not being an easy one, but points out the general benefit for society gained eventually:

At first glance, the Deaf Gain approach might seem to only superficially contradict the logic of other ideologies that generally see ‘the problem’. However, if we focus on our common interest (let us call it ‘a good life’), we see that most of the things that might seriously get in the way of a ‘good life’ are created by humans […]. As we know, maintaining inequality, unjust distribution, or discriminatory access to commodities always ends up posing a danger to social peace—the foundation of a good life for each and every one of us. So one could argue in short that maintaining devaluating, audistic, or stereotyping ideologies with regard to sign languages will in the end be more of an effort than seeing, understanding, and appreciating Deaf Gain. (p. 422)

1.2 Aims and structure

Following the goal of this special issue, this conceptual paper has a twofold aim. First, it takes the opportunity to direct attention to how modality of language influences mathematics learning to acknowledge Deaf learners’ language in mathematics education research. For deaf learners using sign languages, recognizing their linguistic background and not only deafness as a disability has crucial implications for educational policies (Krausneker, 2015). On the one hand, it touches aspects of culture and identity of the learners and places them under educational premises aligning with current topics in mathematics education otherwise not considered of importance with respect to Deaf students’ needs. On the other hand, it also requires a reconsideration of a distinction between information access and epistemological access (Krause, 2019; Tancredi et al., 2021, 2022), as sign language and its modal affordances influence cognitive processing and the formation of (mathematical) concepts (Grote, 2013; Krause, 2019; Krause & Wille, 2021; Marschark & Hauser, 2008).

Second, this paper is meant to contribute to reconceptualizing the relationship between language, mathematics and the learner in mathematics education research by raising the issue of modalism—“the practice of privileging certain modalities over others and ignoring other possible modal constitutions” (Tancredi et al., 2022, p. 211)—in that sign language so far does have close to no place in the development of theories on the role of language in mathematics education. In this paper I propose an approach that, so I submit, has the potential to promote dialogue to inform and enrich theory on the role of language in mathematical thinking and learning.

In Sect. 2, I will problematize engaging with Deaf students of mathematics exclusively through the disability paradigm rather than also as learners of mathematics with a specific linguistic background. I will counter common misconceptions about sign languages and provide information about the language as well as signers as linguistic community—in general and also in relation to teaching and learning mathematics—and the role of sign language in cognition (especially with respect to mathematics). I will then frame the relevance of acknowledging the linguistic background of Deaf learners to serve them equitably in mathematics education. Section 3 will outline aspects of the philosophical approach of Francois Jullien to suggest a more inclusive perspective on mathematics education research by engaging with contexts not typically considered in research on teaching and learning mathematics—in our case specifically research related to the role of language in mathematics education. These contexts might foreground aspects less salient in contexts focused on so far and might thereby challenge and enrich discussions in current mathematics education research. In Sect. 4, this perspective will serve us to make a case how research on sign language in the mathematics classroom—exemplified by a case study—can shed light on aspects of language that might not become as salient at first sight when focusing on spoken languages. Importantly, aiming at dialogue instead of mere integration, this perspective acknowledges concepts and theories already established in the field in a constant back and forth of reconsidering the ‘normal’. Related to our specific case, we will reconsider how iconicity of language—as aspect being foregrounded in signed mathematical discourse and specifically its role as reconstructed in the case study—might enrich the discussion about the role of language in mathematics education. Section 5 will summarize the main ideas and take-aways from this paper.

2 Invisible needs and linguistic resources of Deaf learners of mathematics

2.1 Background on sign language and why it matters for mathematics education

This section provides baseline facts about sign languages to counter common misbeliefs and misconceptions. Subsequently, modal specificities of this gestural-somatic language will be discussed with emphasis on their relevance for mathematics thinking and learning.

Sign languages are not simply verbatim translations of spoken languages into gestural signs. They are complex languages with their own syntactic rules and structure, developed and shaped in and by Deaf communities through the practice of signing (Vermeerbergen, 2006). The articulation and grammar are dynamic and spatial (Liddell, 2003), for example in assigning meaning to locations for spatial referencing, establishing pronouns within context or structure of arguments (Kimmelmann, 2022). The language signs are more or less conventionalized with their meaning varying in modifications of mainly five components: handshape, the location of performance in space and with respect to the body, the movement in aspects of trajectory, direction and speed, the orientation of the hands and the body orientation, and facial expression (see, e.g., Sandler & Lillo-Martin, 2012). Variations in any one of these five parameters can give rise to distinct meanings to the signed expression. For example, in American Sign Language (ASL) the same handshape is used to sign ‘mother’ or ‘father’, differing in location of chin or forehead (see https://www.spreadthesign.com/ for examples from a variety of sign languages). In Austrian sign language, the signs for ‘formula’, ‘complicated’ and ‘crafting’ are similar in handshape and location of performance, but differ in the corresponding facial expression (see Krause & Wille, 2021, p. 364). Also, signers can use non-conventionalized gesture in tandem with their conventionalized signs, similar to gestures being used with spoken languages (see Goldin-Meadow & Brentari, 2017).

Importantly and related, sign languages need to be distinguished from signed languages, which include Manually Coded Language and Sign-Supported Speech. These have been developed from a hearing perspective; their use foregrounds spoken language as superior means of communication and sees accompanying signs as an additional vehicle for information. As that, they follow structure and grammar of spoken languages.

As has already been mentioned in the ‘economic paradigm’, there is not one universal signed Lingua Franca but there are many sign languages around the world. However, all these languages share some features and modal affordances of gestural-somatic language, leading to peculiarities in cognitive processing and conceptual organizationFootnote 2 related to modality effects of language (Meier, 2012). For example, visual-spatial and simultaneous articulation affords concept organization differently than the auditory-sequential articulation of spoken languages. This is considered to influence the ability for serial recall of information in Deaf signers, making simultaneously presented information more easily accessible for cognitive processing while information presented linearly (as in spoken language and written text) is more difficult to follow and process. Moreover, these peculiarities in cognitive processing afforded by the use of sign language are also seen as linked to enhancing visuospatial skills and a preference for spatial coding (see Hall & Bavelier, 2010).

Also, sign languages generally exhibit a higher level of iconicity than spoken languages in that signs often reflect a concept by representing actions or objects in some kind of similarity. Since not all features of the concept can be conveyed in this resemblance, certain aspects are foregrounded in representing the concept. According to psycholinguistics research, these foregrounded features become associated more strongly associated with the overall concept than those that are not represented.Footnote 3 Articulation in sign language is furthermore inherently dynamic, allowing to capture not only objects but also actions and processes in iconicity of language signs.

Overall, this section highlighted that sign language is not just a trivial extension of the associated spoken language and acknowledges that its modality influences Deaf cognition in ways that impact concept organization and cognitive processing, both substantial features of learning mathematics.

2.2 The underestimated impact of deaf students’ linguistic background in learning mathematics

Mathematics is just one of the disciplines that are covered under the larger umbrella of deaf education, with the field focusing more strongly on language and fostering literacy of deaf students. At the same time, mathematics education includes deaf students merely as special cases, deviating from what is considered the normal learner and not included in the development of general theories on learning. Research investigating the learning processes of deaf learners of mathematics, the resources they draw on and how successful mathematics teaching and learning takes place is scarce (ibid., see e.g., Healy et al., 2016; Hyde et al., 2003; Nunes, 2004; Titus, 1995, for exemptions).

Moreover, deaf education as a field itself faces a large diversity among the population it intends to serve as it covers a range from ‘hard-of-hearing’ to ‘profoundly deaf’, and deafness developed at different stages in life. Relatedly, the language abilities of signers encompass a wide range: Deaf children born into a Deaf family usually learn sign languages early on (about 10%), while others born into hearing families with no prior knowledge of sign language might be delayed in language development and/or might be less exposed to everyday ‘mathematical conversation’ (Gregory, 1998). These circumstances have been shown to become problematic for gaining informal mathematical knowledge in early childhood (Nunes & Moreno, 1999). Yet others turn deaf when spoken language has already been acquired and might or might not learn sign language as an additional language. There is a general consensus in the field of deaf education that the use of sign language benefits the learning of mathematics (see Kurz & Pagliaro, 2020; Pagliaro, 2010) and sign language competency appears to be an indirect predictor of mathematical skills (Hyde et al., 2003; Nunes, 2004). However, with the lack of theoretical knowledge about signed mathematical discourse and how it relates to learning mathematics in the classroom, this potential cannot be leveraged.

Also, reading competency becomes an issue for Deaf learners, with sign language not corresponding with the associated written language as spoken languages do—neither in syntactic structures nor in perceptual correspondence between articulation and perception, important for coding and encoding written language (Mayer & Trezek, 2014). Written language therefore becomes an additional language for Deaf learners (Traxler, 2000) which—together with the necessity to process written information linearly—results in generally lower achievement levels for reading (e.g., Qi & Mitchell, 2012).

Considering the Deaf learners merely within the disability paradigm and not as learners of mathematics with a specific linguistic background can easily result in reducing the effect of sign language in the mathematics classroom to an issue of information access and the Deaf learners of mathematics to “hearing students that cannot hear” (Marschark et al., 2011, p. 4), dismissing their language as learning background. That is, within the disability paradigm it is easy to overlook cognitive components that indeed matter for Deaf students’ learning of mathematics. Simply adding sign language interpreters to the mathematics classroom does not accommodate Deaf learners of mathematics adequately, as they do not provide epistemologically equitable access (Caselli et al., 2020; Krause, 2019; Tancredi et al., 2021).

For a while now, similar discussions have been raised in mathematics education for students that learn mathematics in a second spoken language (e.g., Austin & Howson, 1979; Krause & Farsani, 2022; Prediger et al., 2019), considering that language is more than words and shapes our thinking in conceptual and structural aspects (Boroditsky, 2011; Lucy, 1996). However, not having considered sign language as a language in another modality—with peculiar implications for learning—has excluded Deaf students and their education from this discussion in the field.Footnote 4

2.3 Considering the affordances of sign language in Deaf students’ mathematics instruction

Albeit sparsely, aspects of the different ways of thinking of Deaf students related to their language background and the modal affordances of sign language (as described in Sect. 2.1) have been recognized in mathematics education as relevant when considering how to design instructional material for Deaf learners (e.g., Healy et al., 2016; Krause & Wille, 2021; Kurz & Pagliaro, 2020; Nunes, 2004). For example, taking into account signers’ preference for simultaneous processing of information, Nunes and Moreno (2002) suggested spatial simultaneous representation of multiplication tasks and tested the influence of this approach on performance of Deaf students in the UK, using British Sign Language (also Nunes, 2004). In a study with four Deaf signers of Mexican Sign Healy et al. (2016) embraced that “the way in which movement is incorporated into the signs expressed by deaf mathematics learners embodies variation in a way that is different to algebraic symbolism but perhaps serves as an effective means of enacting meanings for it” (p. 153) and designed mathematical learning scenarios that foregrounded potential preferences for dynamic understandings of concepts by introducing visual and dynamic proofs of the Pythagorean Theorem. Kurz and Pagliaro (2020) focus in particular on how using sign language as language of instruction can enhance the teaching of mathematics of native signers, discussing pedagogical practices, potentialities of signed mathematics instruction and issues that might arise in the transition from theory to praxis. Iconicity appears to become of specific relevance in many of these considerations for Deaf students’ instruction, as it might influence how mathematics ideas are understood, and how this might become an obstacle but also a resource for learning mathematics (Krause, 2019; Krause & Wille, 2021; Kurz & Pagliaro, 2020).

As conventionalized mathematical signs do often not exist—especially for more advanced mathematical ideas—teachers and learners often invent and negotiate their own signs as embedded in situation and context. In a study on arithmetic and algebraic thinking observing signed mathematical discourse of Deaf students using LIBRAS (Brazilian sign language), Fernandes and Healy (2014) describe the students’ encounter of the idea of the variable as dealt with when working on sequences. In their analyses, they found how the signed expression of SECRET NUMBER (see Fig. 1) arises as suggested by a student in the context of talking about the variable n, and how it becomes the shared sign for the idea of a variable, used by the students in the further discourse.

Fig. 1
figure 1

Sign for “secret” in LIBRAS, reminiscent of hiding something in your sleeve (recreated from Fernandes & Healy, 2014, p. 55)

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This adds a social component to the influence of iconicity on conceptual understanding, considering their role in social meaning making: The student expresses her understanding of the variable in the context of the task as a specific, but unknown number—a number that is secret. By suggesting the signed expression of ‘secret number’ to refer to the variable, she offers this idea of a variable to her peers as well as a linguistic means to refer to it, contributing to the development of mathematical meaning in the social interaction. As a visual means of expression in social interaction, the signed discourse can show the peers what is meant in a hybrid of conventionalized signs and potentially non-conventional gestures (Goldin-Meadow et al., 2012), while at the same time shaping the mathematical language itself.

Developing a comprehensive understanding of a mathematical idea requires the coordination of different representations in different registers (Duval, 2006), each including some aspects and leaving out others. All of these forms of representation should be understood as being equally important to become able to flexibly switch between or combine them when necessary or appropriate. Mathematical signs hence become much more than just a signed name of a mathematical idea, with the iconicity of the mathematical sign potentially suggesting that one aspect or representation is more important for a concept than another.

3 Towards engaging in conversation with sign language in mathematics education: philosophical principles to reconsider the role of language

To value what looking at sign language can tell us about the role of language in mathematical thinking and learning, we need to avoid curtailing our view by falling back on the familiar and comfortable. That is, we have to refrain from spoken language as the ‘normal’ and from considering sign language only in comparison to this ‘normal’ as defining reference. Instead, we need to be open for considering learning mathematics in sign language as an alternative ‘normal’ in its own right. In a similar discussion in linguistics, Vigliocco et al. (2014) suggest a thought experiment prompted by the question ‘What if the study of language had started with the study of signed language rather than spoken language?’ Similarly, we might ask which aspects of the relationships between language, mathematics and the learner might have come to the fore if looking at teaching and learning of Deaf learners in sign languages would have been equally ‘normal’ as considering spoken language in the typical classroom. This is not to be seen as a competing approach to the research as it has been established, but rather as an alternative that engages with the perspective of Deaf learners of mathematics in their own merit and that can be put in conversation with established research. To actually do justice to an inclusive theory of language in mathematical thinking and learning, these perspectives need to get into dialogue, with each informing and feeding into the other.

To accomplish this, I propose a conceptual approach informed by the philosophical principles of the work of François Jullien (2012/2014). His work is originally grounded in his studies of European and Chinese thought and will be adapted here for opening avenues for fruitful reflections on the role of language in mathematics education.

3.1 Reconsidering the role of language in learning mathematics: the distance and the in-between as the core of dialogue

Jullien’s work is characterized by rejecting a comparison in terms of differences—which would assume a hierarchical standpoint from which the differences are perceived—but instead organizing a vis-á-vis that foregrounds dialogue to allow for mutual reflection. Through this conceptualization, he seeks to avoid a philosophical perspective that favors the Western thought as the starting point of comparison, just like we seek to avoid the role of spoken mathematical discourse as point of origin from which we look at signed mathematical discourse when investigating its role in mathematical education:

Speaking of the diversity of cultures in terms of difference thus defuses in advance what the other of the other culture might contain in terms of the external and the unexpected, the surprising and the disturbing, the confusing and the non-conformist. Right from the start, the concept of difference is situated in a logic of integration—of classification and precise designation—not in a logic of discovery. (2014, pp. 28–29, translated by CK, emphases in the original)

Instead, he uses the concept of ‘écarts’—ofdistance—that allows to look at what he calls the in-between as perceivable tension of what is brought into dialogue:

The distance rises solely from the generation of space and does not go beyond it anywhere, hence does not lead to any postulates. Furthermore this concerns, because it does not depend on a positing of differentness but emerges from a distance, a concept which, insofar it reveals the act[/process] of separation from which it was born, is less analytical than dynamic, which springs from self-realization and extends it. It is a peculiarity of distance—and this is the essential thing for me—that it is therefore not intrinsically illustrative or descriptive, like difference, but productive - and this to the very extent that it can create a tension between what it has separated. Creating a tension relationship: this is the actual function of distance. (pp. 33–34, translation by CK, emphases in the original)

Importantly, reflecting on this tension forces a ‘deconstruction from without’ (du dehors) of the established assumptions that are underlying our thinking and doing. This way, what the tension brings to the fore is supposed to elucidate the ‘unthought-of’ (l’impensé) in this thinking and doing. The tension then puts a focus on resources (ressources) or fecundities (fécondités) of languages and cultures, rather than considering them from the perspective of their ‘differentness’. In this sense, Jullien considers the concept of the distance as productive, while the difference is not.

Stepping back and looking at the tensions that distance creates when bringing in dialogue relationships between language, mathematics and the learner in the scenarios of spoken/written language and sign language can allow us to detect buried biases that we tacitly and implicitly postulate when considering the role of language in the mathematics classroom. The in-betweens—the tensions created by the distances—then become the actual place of dialogue, not the similarities. ‘The similar’ (‘le semblable’) produces only what is uniform, which we then mistake for ‘universals’. Jullien (2014) points out that “it needs an other, and consequently simultaneously distance and in-between, to encourage a common. Because the common is not the similar: it is not the repetition and the uniform, but the opposite” (p. 72, translation by CK, emphases in the original). The tensions—the places of dialogue we seek to engage in—lie in “the external and the unexpected, the surprising and the disturbing, the confusing and the non-conformist” (Jullien, 2014, pp. 33–34).

In an endeavor of taking a more wholistic stance on the relationships between language, mathematics and the learner, I am thus less interested in describing differentness and sameness of learning mathematics in languages across modalities, as it does little to inform my goal. My goal rather requires exploring what can be understood as distances when putting in conversation modal language scenarios in terms of their relationships to mathematics and the learner, what tensions arise and how they surprise and disturb us, and how engaging in the emerging dialogue can elucidate the ‘unthought-of’ in our practices and research involving language in the mathematics classroom.

4 Becoming sensitized to language in the Deaf classroom—the case of iconicity

Jullien’s philosophical principles captured in the notions of ‘distance’, the ‘in-between’ and the ‘unthought-of’ provides paradigms for engaging with sign language in the research on language in mathematics education and can guide methodological decisions in this research. This concerns especially—but not exclusively—approaches to analysis and interpretation of data and the conclusions that can be drawn.

Exploring the potential tensions created by the distances and the unthought-of might elucidate calls for an open approach to analyzing and interpreting data of Deaf students mathematical learning. Furthermore, sensitizing concepts (Blumer, 1970) become of specific importance as they can “suggest direction along which to look” (p. 91). More concretely, they can foreground potential aspects of relevance in signed mathematical discourse as related to mathematical education that have not been salient in prior research on the role of language so far. Being of direct relevance with respect to learning mathematics, sensitizing concepts can then fuel the research process: The empirical grounding and further framing of a particular sensitizing concept based on data can nourish further investigations of (signed) mathematical discourse and its role in teaching and learning. It can then become a starting point for re-sensitizing us with respect to potentialities and roles of language in mathematics teaching, thinking and learning. Relating it to Jullien’s ideas, a sensitizing concept offers a lens to consider distance and to frame a tension that is created by the distance. The following sections will illustrate these claims focusing on the sensitizing concept of ‘iconicity’, as it has become a focus of attention from various theoretical perspectives on mathematical thinking and learning, such as different semiotic perspectives as well as embodiment (see Krause & Wille, 2021). Some research developing further the concept of iconicity related to mathematical discourse will be described to then reflect on these ideas to re-sensitize us to other possibilities when considering the role of language in mathematics education.

4.1 An eye on iconicity in mathematical signed discourse

Motivated by the unavailability of conventionalized mathematical signs and sensitized to the modal peculiarity of iconicity of sign language and its influence on conceptualization of the signed idea (see Sect. 2.2 and 2.3), Krause (2019) carried out an exploratory study on aspects of iconicity in signed mathematical discourse in the mathematics classroom. Based on data collected in interviews with Deaf 6th graders talking about concepts related to fraction arithmetics, Krause developed a distinction of three different kinds of iconicity in which the signs used by the students referred to the mathematical idea it denotes (see also Krause, 2017). Two notions are borrowed from the description of iconicity of speakers’ gestures in mathematical talk (Edwards, 2009): iconic-physically referring to a physical activity carried out in the context of the ideas; iconic-symbolically referring to a symbolic or graphical representation related to the idea (see Fig. 2 for an example for both).

Fig. 2
figure 2

Example for ‘simplifying’: a sign featuring two kinds of iconic relationship to the mathematical idea of ‘simplifying a fraction’ (recreated from Krause, 2019, p. 90). The sign shows iconic-physical features in referring to the action of ‘crossing out’ when simplifying a fraction by common division of numerator and denominator. Referring to the symbolic top-bottom notation of fractions, it also bears iconic-symbolic features

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A third kind of iconicity picks up linguistic features like handshape and/or location of articulation to give rise to what has been called ‘innerlinguistic iconicity’ (Krause, 2017). In these cases, signs—or features of the signs—are reminiscent of another sign that might be borrowed from signs used in mathematical or everyday contexts. The iconic link to another sign can be stronger or weaker—depending on the number of articulation features that constitute this iconicity—and is considered to be causing associative semantic links to the concept (Krause, 2017; Kurz & Pagliaro, 2020). An example for this kind of iconicity has already been mentioned in Sect. 2.1 with reference to Krause and Wille (2021) as the signs for ‘formula’, ‘complicated’ and ‘crafting’ (p. 364). Here, innerlinguistic iconicity manifests in the same handshape and location of manual articulation.

Importantly, the students did not necessarily use the same signs when referring to the same written mathematical notation. Their mathematical signs and mathematical signed discourse reflected iconic features referring to different aspects of the mathematical concept, indicating a plurality of signs and understandings accepted in the signed mathematical discourse in this classroom.

4.2 Grounding iconic meaning of mathematical language—iconization and modal continuity

Iconicity of sign language signs can also be drawn on to ground meaning of mathematical terminology introduced by the teacher. In a study in a 5th grade geometry classroom in Germany, in which all students as well as the teacher were proficient signers of German sign language (DGS), Krause (2018, 2019) traced how the development of mathematical meaning of mathematical ideas can develop side by side with the iconic meaning of the respective mathematical signs in a teacher’s instruction. Such processes of iconization—of establishing iconic meaning in the sign—allow the students to understand the mathematical signs as related to the teacher’s introduction of the mathematical concept, grounding them in meaningful action (iconic-physical), notations (iconic-symbolic), and/or related concepts and their signs (innerlinguistic iconicity). In particular, within an embodied paradigm to mathematical thinking and learning, the ontological innovation (diSessa & Cobb, 2004)—a newly identified phenomenon of educational relevance—of ‘modal continuity’ from action to discourse developed from the data as a peculiarity of signed mathematical discourse, afforded by the manual modality of sign language (Krause & Abrahamson, 2020).

To illustrate these ideas, the following case (Krause, 2019) will present how the Deaf teacher guided the students from the action of rotating, via his gestures that he integrated in his signed explanations, to introduce his sign for point symmetry.

To start his explanations, the teacher indicates a concrete point (Fig. 3b) in a diagram on the blackboard (Fig. 3a) and singles out those entities of the diagram that will be central to his subsequent activity and multimodal explanation—a point and the lower left area of the diagram as it concerns one half of the rectangle separated from the congruent upper right half by the diagonal (Fig. 3c).

Fig. 3
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a Inscriptive diagram underlying the teacher’s initial explanation, b and c use of the gestural modality to prepare the ground for a following explanation of point symmetry

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He then loosely frames the upper right half with both hands and simulates its rotation into the lower left part (Fig. 4). The teacher explicates his action as “half-circling” (halb kreisen) around a point in his sign (Fig. 5). In this sign, a new feature of the rotation becomes integrated: with the thumb placed in the palm of the open hand, it indicates that it is rotated around a certain point.

Fig. 4
figure 4

Simulating the rotation of one part of the diagram into another one

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Fig. 5
figure 5

Linking his action to a gestural sign for “half circling” (also “part circling”)

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Introducing a second example, he simulates another rotation, this time integrating the handshape of the sign used just before to explain the action (Figs. 5 and 6) in the simulated gestural rotation. (For spatial reasons I refer to Krause (2018) for extended description).

Fig. 6
figure 6

Simulating the rotation of another figure

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Eventually, the teacher’s explanation accumulates in introducing his mathematical sign for “Punktsymmetrie” (point symmetry) (Fig. 7). This sign is a compound of the sign for “Punkt” (point) and the gesture that has been used prior to indicate the circling/rotating-actions. However, taking a closer look at how he established common reference to the diagram in Fig. 3b, we see that the handshape indicating the point is the same as for the sign for ‘Punkt’; the sign the teacher conventionalizes as signed mathematical terminology takes form and is grounded in his multimodal explanation.

Fig. 7
figure 7

The teacher’s sign for “Punktsymmetry” (point symmetry)

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The example shows a specific method of iconization, potentially unconsciously employed and situationally based on the teacher’s own linguistic background, experience and expertise. It makes use of the unique affordance of sign language to facilitate modal continuity from enacted experience into signed mathematical discourse. The language used to talk about the mathematical concept stays very close to the experiences—enacted or observed—engaged with when exploring the concept itself.

5 Reflecting on tensions and the unthought-of to re-sensitize us to the nature of mathematical language

The following section will point out selected tensionsFootnote 5 arising from putting in conversation modal language scenarios in terms of their relationships to mathematics and the learner. It engages with a perspective and practices of learning mathematics in sign language to raise new questions about the roles of language in mathematical thinking and learning more generally to prompt thought and stir discourse within the field.

5.1 Detecting tensions and unthought-of in mathematical language

Putting iconicity center stage in research on language in the Deaf classroom gave rise to understandings of the role of language in mathematics learning that have not been salient so far—that have been unthought-of.

The ways students refer to mathematical ideas they are just learning about, as presented in Sect. 4.1, reminds of the student’s situated idiosyncratic sign for ‘secret number’ as referring to the mathematical idea of ‘variable’, described in Sect. 2.3 (Fernandes & Healy, 2014). The need for developing signs to name mathematical ideas makes mathematical language partly idiosyncratic and ambiguous, making the invention or choice of terminology to refer to mathematics and a potential plurality of signs to adapt to part of the process of developing conceptual understanding. Expressing through sign languages in mathematics entails being comfortable with negotiating signs and meaning and with them being contextual, dynamic, and relational. It allows—and potentially even requires—to shape mathematical language based on (situated) understanding and flexibly adjust to other’s understandings. This raises the questions: Can we imagine shaping the mathematical language itself in spoken and written languages as well? Might it be already something we do but overlook in our theories of language in mathematics education?

One tension arising from the consideration of iconicity of mathematical signed language hence concerns the acceptance of ambiguity and fluidity of mathematical language. Dominant spoken and written mathematical discourses are usually uncomfortable with ambiguity and eventually insist on ‘universal’ conventions and terminology. This tension can force a ‘deconstruction from without’ of established assumptions underlying our thinking and doing. It can potentially re-sensitize us to imagining a more open and flexible mathematics and understanding of language in mathematics education that is not focused on universality. Could sign languages help our field re-encounter mathematical language, to feel comfortable with situatedness and fluidity of contextual mathematical terminology, of continuously inventing situated mathematical terminology, rather than insist on a single ‘universal’ one? Might we be overlooking the fact that fluidity and situatedness of spoken/written mathematics has always been present but ignored in dominant theories of language in mathematics education? Could this oversight be harmful? Can the tension between situatedness in sign languages and dominant claims of ‘universality’ in mathematical discourse guide us to re-evaluate and re-consider how we perceive language in mathematics education towards more equitable possibilities?

Another tension is perceivable from looking at a distance opened up by the role of meaning of mathematical terminology, as mathematical signs can be motivated and grounded in the process of learning about the mathematical idea they refer to. More specifically, modal continuity allows to carry the action into the mathematical discourse, along a continuum of situatedness from action via gestures into language. Meanwhile, conventionalized spoken/written terminology is often disembodied and while it might carry its etymological origins, these are often withheld from the students. Questions about potential ‘unthought-of’s arising in this context re-sensitize us to the potential embodiment of mathematical language. Is there some form of modal continuity already present in our relationship with languages but overlooked by language theories in mathematics education? Can this notion of modal continuity re-sensitize our bodies to the relationship between the body, language and mathematics that is often sidelined in predominant theories of language in mathematics education but that is very present within sign languages? Could iconicity help us remember that dominant mathematical terminology also comes from specific contexts, relations, and bodies—which has been obscured through normalizing hegemonic understandings of language in mathematics education?

5.2 Main fields of attention brought to the fore: re-thinking the role of the gestural modality in mathematical language

The questions raised above are non-trivial and need further reflection, research (and space) in order to be addressed further. They however offer at least two fields of attention that might be worth exploring:

  • What place does ambiguous and idiosyncratic language have in the mathematics classroom?

  • How embodied is mathematical language – and how embodied can it be?

Both these questions point in the direction of giving gestures a more prominent role as part of language and as linguistic resource in mathematical discourse (Krause & Farsani, 2022) beyond merely being considered a part of communication and interaction (e.g., Planas & Pimm, 2023). While there has been a stronger curiosity for gestures as part of the multimodal interaction in the classroom in research in mathematics education over the last two decades (e.g., Arzarello et al., 2009; Robutti et al., 2022 for an overview), they are still mainly seen as complimentary means, subordinate to spoken language. Reconsidering questions raised in Sect. 5.1, this might be related to the prevalent discomfort with ambiguous, contextual, dynamic, and relational meaning of mathematical language as gestures show all of these features.

Special potential might lie in the multilingual classroom, where mathematical terminology can be an obstacle for learners who learn mathematics in an additional language. Research acknowledges the benefit of integrating multimodal resources in engaging and supporting multilingual students (e.g., Castellon & Enyedy, 2006; Domínguez, 2005; Moschkovich, 2007; Shein, 2012; see Robutti et al., 2022 for an overview). The more surprisingly, methods for using gestures as linguistic resource coupled with spoken and written language to support multilinguals’ learning of mathematics as well as their language skills have not yet been systematically elaborated. As a consequence, not much is known about how they might contribute as a resource in the language-responsive classroom (Erath & Prediger, 2021).

Meanwhile, the process of iconisation of mathematical terminology can be considered a case of scaffolding language in the mathematics classroom, with gesture bridging from experience to language in modal continuity. Bringing in dialogue the ideas of scaffolding language and the role of gestures in making meaning of mathematical terminology observed in the Deaf classroom offers potential for cross-pollination between the respective scenarios: It opens up new avenues for research on gestures’ role in scaffolding language as well as reconsidering the notion of language-responsiveness for the Deaf classroom.

The first field of attention furthermore links to a necessity in the Deaf classroom of being comfortable with mathematical language being contextual and relational. Focusing on conceptually meaningful manual actions, situationally conventionalized gestures can be highlighted as linguistic means beyond words for all learners. This might be linked to the idea of ‘associated gestures’ (Krause, 2016)—gestures that become ‘fueled’ with mathematical meaning by students in the process of engaging with a mathematical problem, thereby becoming a gestural linguistic resource shared by the students as situationally conventionalized.

The second field of attention sets a stronger focus on the ‘doing’ of mathematics as becoming reflected in language. With a stronger focus on the ‘doing’ of mathematics as becoming reflected in language, mathematical practice and language can stay closer to the embodied ways of experiencing and acting.

6 Summary and concluding remarks

Considering the Deaf learner as learner of mathematics as well as considering that their linguistic background affords thinking and learning in peculiar ways, this paper highlighted the example of iconicity as feature of signed mathematical discourse impacting Deaf learners’ mathematical cognition as well as processes of meaning making in social settings of learning mathematics. The theories of Francois Jullien then served as guiding principles for a more inclusive understanding of language: By directing our attention away from what we would perceive as ‘differentness’ (or difference), and towards the distance generated by the relevance of a concept like ‘iconicity of mathematical discourse’ to make us aware of what lies ‘in between’ certain aspects of sign languages and what is predominantly considered the ‘core’ of mathematical language. With this approach, new dialogue can unfold that serves to reconceptualize our understanding of the nature and roles of language and linguistic resources in teaching and learning mathematics.

Let us come back to our thought experiment of ‘What if the study of the roles of language in mathematics thinking and learning had started with classrooms communicating in sign language rather than spoken language?’ The gestural modality would have perhaps taken on a much more prominent role in research as part of language in the mathematics classroom in both its conventionalized and non-conventionalized form and also in the ways both of these play together. Transmodality might have become a naturally developing focus in research on language in the mathematics classroom, giving room for ambiguity and non-uniform ways of mathematical discourse.

With this, the unthought-of’s brought to our attention by elaborating on a previously invisible feature of language, namely iconicity, do not only concern the gestural modality as related to language. More generally, it might open our perspective to situated spoken and written linguistic ways of referring to mathematical ideas as valid and valuable in their own right rather than always something to be directed towards ‘universally conventionalized’ mathematical discourse.

This conceptual paper aims at shedding a new perspective on our understanding of language in mathematical discourse, how it can relate to the learner of mathematics, and to the mathematics they learn through this discourse. By raising awareness of modalism in research on language in mathematics education—in particular, privileging the spoken modality as language modality—the contribution calls for a stronger and more equitable consideration of research on language in mathematics education.