ZDM – Mathematics Education https://doi.org/10.1007/s11858-023-01497-0 ORIGINAL PAPER Mathematics education research on language and on communication including some distinctions: Where are we now? Núria Planas1 · David Pimm1 Accepted: 18 May 2023 © The Author(s) 2023 Abstract In this article, we present a narrative review of mathematics education research on language and on communication over 2019–2022, but also look ahead by addressing challenges posed by the lack of distinction between language and commu- nication. The persistence and significance of the problem of the distinction between language and communication are thus outlined in a historical moment of celebration of growth of research in the domain. Informed by the analysis of a selection of research journal articles and by our trajectories, we discuss influential topics in the recent discourse: multilingual mathematics classrooms; mathematics teacher education on language in mathematics teaching; multimodal mathematical communica- tion; interaction and mathematics learning; mathematical language and discourse. We connect this with new emerging or old revisited concepts: instructional designing, gesturing, argumenting and languaging. We finish by further reflecting on multimodal mathematical communication and gesturing, and on the potential of expanding the notion of mathematics register towards a notion of mathematics communication register. 1 Introduction et al. (2018)—and two research articles—Morgan et al. (2014) and Planas and Schütte (2018). From this point on, Since the late 1980s with the publication of Speaking math- our review work was required to be focused solely on journal ematically: Communication in mathematics classrooms articles (as opposed to book chapters or conference papers). (Pimm, 1987/2017), research on aspects of language and, Those previous review studies reflected on the scope and increasingly, of communication in mathematics teaching achievements of the field research on language and on com- and learning, as well as in mathematical pedagogies, has munication, and on newer approaches in the domain. They continued to grow. A question then is: Where are we now in showed together that language remains a contested focus of this domain? We elaborate a response by combining lessons research, and that work in the domain is being developed on learned in our trajectories as researchers in the domain, and how to conceptualise and discuss language, as well as on by discussing academic articles published over the period how to enable a more complex and visible notion of commu- 2019–2022 in a selection of leading international journals nication in research and educational processes. The diverse of mathematics education research and neighbouring ones meanings of language and communication are still a driving at the intersection of educational and language disciplines. force and an expression of vitality. This newer, short-period Despite a proposal of articles is neither global nor neutral, review is similar to the above-mentioned reviews in its nar- we have traced works for the benefit of citational and epis- rative method, yet is different in the value given to specific temic justice. An Anglo-centric bias and absences arrive, topics and concepts, including the discussion of the fact that however, with the choice of international journals. language and communication are not the same, so we do not Referring to four reviews of the same research domain study them as the same. The nature of this distinction often before initiating this current review, we took over from two becomes blurred at the intersection of the broadest views of book chapters—Radford and Barwell (2016) and Planas mathematics education research on language—i.e., studies in which linguistic communication is one of the many semiotic * Núria Planas modes attended—and the narrowest views of mathematics Nuria.Planas@uab.cat education research on communication—i.e., studies solely attending to linguistic communication. We are not just doing 1 Universitat Autònoma de Barcelona, Cerdanyola del Vallès, a review, but are also drawing on review work to address the Catalonia, Spain 13 Vol.:(0123456789) N. Planas, D. Pimm problem of the distinction between language and communi- and Education (LAE), Linguistics and Education (LIE), cation in mathematics education. Mind, Culture and Activity (MCA), Teachers College Record (TCR​) and Teaching and Teacher Education (TATE). The second step was initiated individually. The final over- 2 Method for mapping topics and concepts view of topics and articles came as a result of our research experiences and conversations during the months of prepara- Beyond the attention to different world regions, theoretical tion of this review. Given the special issue guidelines about traditions and groups of authors, any identification of inter- considering some 50 articles and annotating a small subset esting research articles on mathematics education, language of them, we estimated that 5 topics can be illustrated by and communication is not trivial. There must be more in means of some 10 articles each (and one of each topic is quantity than a total of articles chosen in a period of time (in annotated in the references). We were open to the arguments our case 50 in 2019–2022 considering the first online pub- of each other in favour of or against the choices proposed, lication data), if not limiting the search by citation records and looked at representing a variety of authors, regions and or other constructs upon which rankings are based. Either traditions. When one of us wanted to know more about what way, how we view language and communication has been an was in a topic, or in an article, and the reasons for proposing influence on our article identification. Interpreting Makoni its inclusion and annotation, we started a discussion which and Pennycook (2007), we mostly share a broad understand- could lead to changing the recommendation or to reinforcing ing that emphasises language as primarily a communication the reasons for keeping either on. process, and both language and communication as produc- The discussion of influential topics encouraged us to ers of meaning in thinking and in interaction with people, identify new emerging and old revisited concepts, which objects and practices. In all this, we challenge binary dis- are explored in the second section below here. While the tinctions, such as people versus objects or interaction versus topics reflect lines of research in the domain, the concepts thinking, and argue for a nuanced approach to the multiple help us to understand and illustrate a variety of current chal- aspects involved in language and in communication across lenges within and across topics. For example, at the end sites of mathematics education. Guided by this understand- of producing the respective 10-lists of articles in the topics ing, we address the following questions: of multilingual mathematics classrooms and mathematical language and discourse, we agreed on the importance of the Which are some influential topics of mathematics educa- languaging concept, which is today central in applied lin- tion research on language and on communication in the guistics and emerging in mathematics education in reference recent discourse (2019–2022), and why are they influ- to the many challenges of understanding the potential of ential? moves between languages and between everyday and formal Which are some new emerging or old revisited concepts registers in mathematics teaching and learning. For this, and over this period, and what do they add to or imply for the three more concepts, we chose publications to be focused domain? upon in the later part of this article, again considering our How do some topics and concepts of the domain at pre- citational practices and acknowledging work that is not ours sent contribute to the discussion about the distinction nor like ours. between language and communication? As first step, amongst prestigious research journals in 3 Zooming in on influential topics mathematics education we chose the following: Digital Experiences in Mathematics Education (DEME), Educa- The questions guiding the first part of our review are as tional Studies in Mathematics (ESM), For the Learning of follows: Which are some influential topics of mathematics Mathematics (FLM), International Journal of Science and education research on language and on communication in Mathematics Education (IJSME), Journal of Mathematical the recent discourse (2019–2022)? Why are they influen- Behavior (JMB), Journal of Mathematics Teacher Education tial? Topics or lines of research in the mathematics educa- (JMTE), Journal for Research in Mathematics Education tion community are embedded into a timeline of ideas and (JRME), Mathematics Education Research Journal (MERJ), thoughts (a historical scope), and the collective practices Mathematical Thinking and Learning (MTL), Research in of specific groups (a sociocultural scope). We may thus use Mathematics Education (RME) and ZDM–Mathematics the same words for mentioning a topic as known in the past, Education (ZDM). This alphabetic-order list was extended but its meaning will have changed. The topics in this review with journals covering wider fields: Classroom Discourse are: i) multilingual mathematics classrooms; ii) mathemat- (CD), Comparative Education Review (CER), International ics teacher education on language in mathematics teaching; Journal for Lesson and Learning Studies (IJLLS), Language iii) multimodal mathematical communication; iv) interaction 13 Mathematics education research on language and on communication including some distinctions:… and mathematics learning; v) mathematical language and refocus language diversity on any site of mathematics teach- discourse. They differently reflect on sites of communica- ing and learning, and not just on school classrooms in con- tion, in which language is a process linked to other com- texts of poverty, migration, colonialism, ethnic difference, munication processes (e.g., gestures, voice intonation, …, nor as a feature exclusive of the learner. drawings) in the production of meaning in interaction with The articles in Table 1 provide examples of all this. Our people (e.g., teachers, teacher educators, students), objects review specific to the mathematics education journals shows (e.g., lesson plans, computers, manipulatives) and practices research informed by: classroom interaction theory in analy- (e.g., writing, explaining, argumenting). Hence, the topics ses of patterns of multilingual talk and of mathematical gen- are related to each other because together they represent the eralisation (El Mouhayar, 2022a); content meta-analysis of multiple aspects constituting language, communication and empirical evidence of successful teaching across classroom the relationship between them. studies with English language learners (Sharma & Sharma, 2022); learning activity theory in the discussion of how 3.1 Topic 1: Multilingual mathematics classrooms young multilingual learners’ reflective actions indicate ini- tial algebraic thinking (Eriksson & Eriksson, 2021); learning In the most recent period, we find a continued emphasis, as trajectory tradition for inclusion of linguistic diversity in well as advances, on research regarding multilingual math- multilingual teaching and learning of proportional reasoning ematics classrooms. Besides its historical significance, we and linear functions (Zahner & Wynn, 2023); sociolinguistic consider this topic to be strong, growing and even more and dialogic theory in the exploration of multilingual mathe- present now than in the past (e.g., Adler, 2001). Important matics learning in different second-language classroom con- attention remains in the study of flexible uses of languages texts (Barwell, 2020); sociolinguistic and second language that support mathematics learning and teaching. Part of acquisition theory in the study of exploratory mathematics what is different from past research is the socially more talk of multilingual learners (Robertson & Graven, 2019). complex thinking about language as a local practice, rather Various studies are published in neighbouring journals, as than a given static structure or system (Makoni & Penny- is the case with four articles in Table 1. These adopt ‘trans–’ cook, 2007). Social views of language have gained decisive approaches to illustrate creative language practices of math- momentum, and all-or-nothing views of the multilingual ematics teaching, assessment and learning, hence transcend- learner participation in mathematics are being replaced by ing bounded labels of languages, modes, registers and semi- views of mathematical participation through a plurality of otics. Gandara and Randall (2019) revisit code-switching languages, mathematics registers, communities and cultures. and language separation approaches that represent the mul- Moreover, today’s multilingual mathematics classroom tilingual learner as a speaker with an incomplete knowledge research is specifically influential, because it is helping to of the language of instruction. Tai and Wei (2021) reflect on Table 1  List of 10 Topic 1 studies Topic 1 Multilingual mathematics classrooms Article title, journal, year Location(s), authorship Triadic dialog in multilingual mathematics classrooms as a promoter of generalization during class- Lebanon; El Mouhayar room talk. MERJ 2022a Onto/epistemic violence and dialogicality in translanguaging practices across multilingual mathemat- Greece, Spain & Sweden; Chronaki et al. ics classrooms. TCR​2022 Learning actions indicating algebraic thinking in multilingual classrooms. ESM 2021 Sweden; Eriksson & Eriksson Successful teaching practices for English language learners in multilingual mathematics classrooms. New Zealand; Sharma & Sharma MERJ 2022 Rethinking learning trajectories in light of student linguistic diversity. MTL 2021 online/2023 USA; Zahner & Wynn Co-learning in Hong Kong English medium instruction mathematics secondary classrooms. LAE China; Tai & Wei 2021 Teachers’ beliefs and practices with respect to translanguaging university mathematics in Iraq. LIE Iraq; Alhasnawi 2021 Learning mathematics in a second language. JRME 2020 Canada; Barwell Exploratory mathematics talk in a second language. ESM 2019 South Africa; Robertson & Graven Assessing mathematics proficiency of multilingual students. CER 2019 Democratic Republic of the Congo; Gandara & Randall All 5 tables are ordered by published date 13 N. Planas, D. Pimm the role of language across languages, modes, registers and The articles in Table 2, with an almost entire focus on semiotic resources to co-learn and negotiate mathematical spoken and written language, show research in which the meaning. Chronaki et al. (2022) study translanguaging in participants are: future mathematics teachers on writing for relation to pedagogical practices designed to develop math- learning to teach mathematics through posing non-tradi- ematical and identity meaning across languages and school tional mathematical problems (Leavy & Hourigan, 2022); contexts. Alhasnawi (2021) addresses pedagogical practices future mathematics teachers on leading mathematical discus- with a focus on translanguaged mathematical discourse in sions by drawing on the linguistic resources of the learners university mathematics teaching. All these articles adopt a (Shaughnessy et al., 2019); novice primary-school math- notion of language as a social practice and a dynamic pro- ematics teachers on their use of mathematical vocabulary in cess that can be also creative. linguistically diverse classrooms (Turner et al., 2019); mid- dle-school mathematics teachers in a content-specific profes- 3.2 Topic 2: Mathematics teacher education sional development program aimed at promoting expertise in on language in mathematics teaching language-responsive teaching (Prediger, 2019); secondary- school mathematics teachers in developmental workshops Topic 2 refers to studies in mathematics teacher education on word use aimed at naming and explaining meaning in with pre-service and in-service mathematics teachers and a algebra teaching (Planas, 2021); secondary-school math- language focus which privileges the oral/written mode over ematics teachers in a geometry lesson study mediated by spatial, visual, gestural, embodied, … modes. These oth- a framework with a focus on word use and meaning (Adler ers are not necessarily neglected, but are often subordinated et al., 2023). to verbal accounts in teacher education and school teach- In line with notions of dialogue, some studies focus on ing. We concur that current mathematics teacher education developmental work with teachers on word use aimed at research on language in mathematics teaching—with some supporting teaching talk for mathematical discussions with researchers being the teacher educators in the developmental learners. In their theoretical article, Williams and Ryan site—is influential, because it is building a convincing case (2020) make a claim for the dialectic nature of dialogue, for justifying and disseminating innovative practice in initial and present data from a lesson study in which teachers made and continuous mathematics teacher education. As argued in progress in the understanding of features of mathematical Morgan et al. (2021), findings from classroom research on dialogue. Bergman et al. (2022 online/2023) approach dia- mathematics and language in the past have not much been logue through the analysis of the responses of pre-service reflected at school-teaching levels. It may well happen that teachers to a scripting task in which they were asked to react today’s collaborative research with mathematics teachers to a learner’s conjecture regarding fractions. Ng et al. (2021) on language in their or others’ classroom teaching becomes examine linguistic features and dialogic moves in the teach- reflected in progressive improvement of educational practice ing of a primary-school teacher involved in a developmental in the coming decades. teacher-intervention study. Drawing on noticing literature, Sjöblom et al. (2022) present work with four teachers in Table 2  List of 10 Topic 2 studies Topic 2 Mathematics teacher education on language in mathematics teaching Article title, journal, year Location(s), authorship Prospective teachers’ responses to students’ dialogue on fractions. RME 2022 online/2023 Canada; Bergman et al. Teachers’ noticing to promote students’ mathematical dialogue in group work. JMTE 2022 Sweden; Sjöblom et al. From defining as assertion to defining as explaining meaning. IJLLS 2022 online/2023 South Africa & Malawi; Adler et al. Balancing competing demands. JMTE 2022 Ireland; Leavy & Hourigan How linguistic features and patterns of discourse moves influence authority structures in the mathematics China; Ng et al. classroom. JMTE 2021 How specific can language as resource become for the teaching of algebraic concepts? ZDM 2021 Spain; Planas On the compatibility of dialogism and dialectics. MCA 2020 UK; Williams & Ryan A study of early career teachers’ practices related to language and language diversity during mathematics USA; Turner et al. instruction. MTL 2019 Investigating and promoting teachers’ expertise for language-responsive mathematics teaching. MERJ Germany; Prediger 2019 An investigation of supporting teacher learning in the context of a common decomposition for leading USA; Shaughnessy et al. mathematics discussions. TATE 2019 13 Mathematics education research on language and on communication including some distinctions:… design research cycles to analyse, design and evaluate teach- the setting of sign language), but are frequently growing ing for engaging learners in mathematical dialogues across connected with language and its research. curricular contents. With regard to the 10 articles in Table 3, it is interesting to note the representation of gestures in the communication 3.3 Topic 3: Multimodal mathematical of data by means of transcripts. Alibali et al. (2019) focus communication both on student and teacher gestures and, in their produc- tion of class transcripts, they juxtapose, although separated, Morgan (2021) wrote that she would generally refer to com- gestures and speech in comparable detail. Maffia and Sabena munication rather than language because, “mathematics (2020) attend to the teacher’s gestures and gestural repeti- teaching may make use of a wide range of communicational tion in class discussion, and describe gestures as a “semiotic modes” (p. 102). Jewitt (2006) saw that, “all modes of com- set”, one that links to others “(e.g. spoken words, mathemati- munication are attended to as part of meaning making” (p. cal symbols)” (p. 16). The gesture descriptions are marked 3). Sfard (2008) wrote, “thinking can be usefully defined as inside the speech transcript (italics and inside square brack- an individualized version of interpersonal communication” ets), which presents them as subordinate, a not-uncommon (p. 81; italics in original). In Pimm (1987/2017), however, circumstance. Oechsler and Borba (2020), by contrast, rely there was no distinction between language and communi- on a series of photographs to represent students’ gestures in cation (even at the title level), and it solely comprised an their mathematical videos. El Mouhayar (2022b) provides English monolinguality. an example of representing teacher and student gestures by The distinction was addressed much later. Pimm (2021) means of a combination of images and gesture descriptions disagreed with O’Halloran’s (2015) extension of Halliday’s inside the speech transcript, some of which are not associ- linguistic notion of the mathematics register to a multimodal ated with the images documented. form; not least because O’Halloran’s two additions (sym- Table 2 also shows research with: grade-three students bolic notation and geometric images), while not global, are in small-group work on combinatorial problems and mul- translingual in the sense of these written/drawn elements timodal mathematical reasoning, including pointing and transfer consistently and widely across languages, which sliding (Wathne & Carlson, 2022); pairs of undergraduate Halliday’s (1975) notion did not. For Halliday, each math- students working on calculus tasks and their embodied rea- ematics register was a functional subset of a single language: soning through talk, gestures and interactions in between “the mathematical use of natural language, that is: not math- (Yu & Uttal, 2022); grade-six students using hand gestures ematics itself” (p. 65). Neither of these additions are lingual. in their thinking of angles relationships and the laws of expo- We might see O’Halloran’s extension as being (part of) a nents (Yeo & Tzeng, 2020); young students who, without the multimodal mathematics communication register. Her multi- iPad present, would mimic what their contact gestures on modal list does not (yet) include gestures (and gazes and …), the iPad screen when using TouchTimes had been (Bakos & which would also be significant elements of a mathematics Pimm, 2020); young students whose ‘collaborative’ gestures communication register. These are not linguistic (unless in and ‘gesture moves’ (as an extension of ‘talk moves’) reveal Table 3  List of 10 Topic 3 studies Topic 3 Multimodal mathematical communication Article title, journal, year Location(s), authorship Third grade students’ multimodal mathematical reasoning when collaboratively solving combinatorial prob- Norway; Wathne & Carlsen lems in small groups. MTL 2022 Teacher’s and students’ use of gestures and home-language during classroom-talk to elicit a shared under- Lebanon; El Mouhayar standing of structure in figural pattern. LIE 2022b Gestures, embodiment, and learning the rate of change. MTL 2022 USA; Yu & Uttal Cognitive effect of tracing gesture in the learning from mathematics worked examples. IJSME 2020 Taiwan; Yeo & Tzeng Beginning to multiply (with) dynamic digits. DEME 2020 Canada; Bakos & Pimm Mathematical videos, social semiotics and the changing classroom. ZDM 2020 Brazil; Oechsler & Borba On the mathematics teacher’s use of gestures as pivot signs in semiotic chains. FLM 2020 Italy; Maffia & Sabena Collaborative gesture as a case of extended mathematical cognition. JMB USA; Walkington et al. 2019 Ritualisation in early number work. ESM 2019 UK & Canada; Coles & Sinclair Managing common ground in the classroom. ZDM 2019 USA; Alibali et al. 13 N. Planas, D. Pimm information out of talk—hence, once again, distinguishing mathematics learning as influential, because of its capacity language and communication (Walkington et al., 2019); one to frame the discussion of language and of communication in primary-school student involved in numeral-naming tasks in mathematical argumentation and reasoning in the classroom. which the ritual (but not rote through repetition) rhythmic Table 4 mentions 6 studies which approach learners’ rea- match of (researcher) gesturing and his responding is ana- soning through analyses of: interactional episodes with the lysed (Coles & Sinclair, 2019). All this multimodal research teacher on the assessment and refining of reasoning products continues to be important in moving the field forward. The developed in lesson collaboration (Zhuang & Conner, 2022); lessons learned may have a role in mathematics teaching cultural patterns arising in the human activity system around with linguistically disadvantaged students. mathematical problem solving in whole groups (Sekiguchi, 2021); shared construction of mathematical explanations 3.4 Topic 4: Interaction and mathematics learning and meanings while discussing the enlarging of plane fig- ures (Erath, 2021); collaboration group work in situations of Since Cobb and Bauersfeld (1995), a book where a group minimal teacher instruction and activities of linear functions of authors addressed the complexity of mathematics teach- (Kämäräinen et al., 2021); conversations in mathematics les- ing and learning in classroom interactions by sharing videos sons across different schools and teachers in which learners and transcripts of lessons in an elementary school, much has produce explanations (Ingram et al., 2019); small and whole- been studied about the mediation of interaction in math- group settings of lesson work focused on the development ematics learning. In the early years, a basic orientation was of epistemic claims and probability reasoning (Goizueta, around the function of the classroom language to allow 2019). mathematical discussion in interaction. Part of the research We note an empirical study in the university classroom that has intensified during the past few years relates to (Rasmussen et al., 2020) reporting the finding that math- reform contexts where mathematics learning is equated with ematical conceptual reasoning is supported when learners participation in mathematical reasoning. Later in this article, engage with another’s reasoning, as necessitated by inter- we will comment on the indistinct use of or common lack of active conversation. Also, 2 theoretical-oriented studies distinction between ‘argumentation’ and ‘reasoning’—which illustrate reflections through classroom data with younger parallels the general absence of distinction between language learners. One of these studies connects together theories in and communication. The fact of argumentation often being the interpretation of an interactional episode in the kinder- subsumed under the umbrella of reasoning is pushed out, garten school with the teacher and nine children working on particularly in European approaches to interaction and math- an addition problem (Breive et al., 2022). The other draws ematics learning, by the influence of the notion of collective on two primary school classrooms to explore patterns of argumentation in Krummheuer (2007) that exemplifies some dialogic and non-dialogic communication in the develop- inseparability between reasoning mathematically and argu- ment of the learners’ participation in mathematics (Faustino ing with others. We see today’s research on interaction and & Skovsmose, 2020). One last theoretical-oriented article Table 4  List of 10 Topic 4 studies Topic 4 Interaction and mathematics learning Article title, journal, year Location(s), authorship Secondary mathematics teachers’ use of students’ incorrect answers in supporting collective argumenta- USA; Zhuang & Conner tion. MTL 2022 Interpreting a kindergarten episode through three perspectives on agency. FLM 2022 Norway, Australia & UK; Breive et al. Enhancing students’ language in collective processes of knowledge construction in group work. ZDM Germany; Erath 2021 Activity systems analysis of classroom teaching and learning of mathematics. ESM 2021 Japan; Sekiguchi Initiation and decision-making of joint activities within peer interaction in student-centred mathematics Finland; Kämäräinen et al. lessons. CD 2021 Ways in which engaging with someone else’s reasoning is productive. JMB 2020 USA & Israel; Rasmussen et al. Dialogic and non–dialogic acts in learning mathematics. FLM 2020 Brazil; Faustino & Skovsmose The early history of the scaffolding metaphor. MCA 2019 Russia & The Netherlands; Shvarts & Bakker When students offer explanations without the teacher explicitly asking them to. ESM 2019 UK; Ingram et al. Epistemic issues in classroom mathematical activity. JMB 2019 Chile; Goizueta 13 Mathematics education research on language and on communication including some distinctions:… traverses school ages to discuss the scaffolding metaphor, communication, the view of the mathematical discourse to and how it is often associated with the Vygotskian zone of include oral and written mathematical language in combi- proximal development in the interpretations of mathematics nation with academic and everyday registers and specific learning in classroom interaction (Shvarts & Bakker, 2019). building practices (e.g., Moschkovich, 2021) is, for example, Aspects of all these articles cross aspects present in Topics strong. In this scenario, studies on reading, writing, speaking 1, 2 and 3. The scaffolding metaphor, for example, is posed and listening to mathematical language coexist with studies in relation to scaffolding in teaching, and how language and on participation in the mathematical discourse. communication operate in facilitating the child’s interaction Four of the selected articles in Table 5 suggests com- with the others. mognition is a collective project shared by authors all over the world, although with a geographic centre, with empirical 3.5 Topic 5: Mathematical language and discourse studies covering a range of mathematical topics and research participants, as well as differently focused on parts of the During the period covered in this review, we have found mathematical discourse characterised by this theory. These various articles, to add to the sets of articles in a JMB and articles examine: types of routines of the mathematical dis- an ESM special issue, all guided by the Vygotskian-based course and how these are transformed from rituals to explo- theory of commognition. With its foundations published rations in processes of learning (Lavie et al., 2019); emerg- in Sfard (2008), this theory sees mathematical activity as ing routines in the mathematical practices of describing practising a particular form of communication—namely, and defining geometrical solids by undergraduate students a discourse—with its own language, consisting of words (Fernández-León et al., 2021); discourse development of and grammar, but also of visual mediators, routines and young children in tasks of classifying odd and even numbers narratives. Moreover, we have found articles that report and of reasoning about their sums (Knox & Kontorovich, work guided by discursive psychology, positioning theory, 2022); teachers’ narratives about unknowns and variables ethnomethodology or functional linguistics in the study of and on mathematics as mutable (Moustapha-Corrêa et al., mathematical discourse or of one of its constitutive parts, 2021). Sinclair (2022) is a 5th article that interestingly com- that is mathematical language. The existence of simultane- ments on the advances of commognitive research, as well as ous theories makes today’s mathematical language and dis- on the challenges of contributing to the quandary of learning course research specifically influential, because it addresses disability and to a shift towards pluralising mathematical the challenge of exploiting ways of intellectual interaction discourse. and theoretical networking both inside and outside math- The other 5 articles in Table 5 illustrate the vivid moment ematics education. in the research on reading, writing, speaking and listening to As discussed in Planas and Schütte (2018) for ear- mathematical language in school or university mathematics. lier periods, this diversity of theories goes with a variety We include examples that report studies on: words, grammar of meanings for the notions of discourse and mathemati- and linguistic patterns of additive word problems and how cal discourse. Alongside the latter as a particular form of prospective elementary teachers analyse the mathematical Table 5  List of 10 Topic 5 studies Topic 5 Mathematical language and discourse Article title, journal, year Location(s), authorship Beyond keywords. IJSME 2022 USA; Kwok et al. A cumulative, coherent and convincing theory that is also seductive, singular and selective. JMB 2022 Canada; Sinclair Growing research groves to visualize young students’ learning in small groups. MERJ 2022 New Zealand; Knox & Kontorovich Analyzing the written discourse in calculus textbooks over 42 years. ESM 2022 online/2023 Iran; Haghjoo et al. Identifying routines in the discourse of undergraduate students when defining. MERJ 2021 Spain; Fernández-León et al. Problematizing mathematics and its pedagogy through teacher engagement with history-focused and Brazil & UK; Moustapha-Corrêa et al. classroom situation-specific tasks. JMB 2021 Using moment-by-moment reading protocols to understand students’ processes of reading mathematical USA; Dawkins & Zazkis proof. JRME 2021 Identifying vernacular language to use in mathematics teaching. LAE 2019 Australia & Papua New Guinea; Edmonds-Wathen et al. Routines we live by. ESM 2019 Israel; Lavie et al. Reading mathematics text. IJSME 2019 South Africa; Berger 13 N. Planas, D. Pimm language of these problems (Kwok et al., 2022); written the survey article, Erath et al. (2021) discuss language in mathematical languages and the related diagrammatic fea- mathematics teaching and learning, and state design prin- tures in Iranian calculus textbooks (Haghjoo et al., 2022 ciples of learning environments to enhance language for online/2023); linguistic challenges faced by student teach- mathematics learning, and of “teaching practices (including ers of mathematics when reading the calculus sections of a teacher moves and classroom norms) involved in the enact- mathematics textbook (Berger, 2019); cognitive and seman- ment of those designed learning environments” (p. 245). tic challenges faced by novice and experienced undergradu- While with no mention of the distinction between language ate students in their reading of mathematical proofs and and communication, in the focus on “discourse practices that noticing of the mathematical-linguistic features (Dawkins are means to communicate” (p. 246; italics in original), or & Zazkis, 2021); identification or development, in col- between argumenting and reasoning, in the focus on “dis- laboration with teachers, of mathematical terminology and course practices such as explaining, justifying, arguing, etc.” grammar for counting systems, measuring and comparing (p. 246), the idea of designing language and instruction for location and shapes in Indigenous mathematical languages inducing learners’ into mathematical communication and (Edmonds-Wathen et al., 2019). reasoning is present. Planas et al., (2022 online/2023) have expressed some caution in the consideration of instructional designing that 4 Zooming back in on new emerging/old does not address the design of mathematics teaching talk revisited concepts on its own as well, specifically the perils of suggesting this talk and its design as subordinated to, backgrounded or sub- In the prior section, we have differentiated 5 topics that illus- sumed into the mathematical discourse practices, the writ- trate part of the map of the current research domain. We now ten materials for supporting them, and their design. As the address commonalities by introducing some concepts that field of mathematics education has developed, and as our are important in two or more topics. Which are some new understanding of the tensions and the role of language in emerging or old revisited concepts over 2019–2022? and mathematics teaching has fostered further research and more What do they add to or imply for the domain? As it occurs attention to teacher education and professional development, with traditional topics, even if we write about old concepts, other tensions have arisen. In this respect, Planas et al. point we cannot expect them to mean the same today compared to tensions in arguing for the important place of mathematics with the earlier years, and hence they are ‘old’ concepts teaching talk at the level of word use within the discourse revisited in or infused with ‘new’ contexts and meanings. practices in language-responsive mathematics teaching, and In our response, we zoom in on: (i) instructional designing; in arguing for the important place of talk about word use (ii) gesturing; (iii) argumenting; (iv) languaging. In order to in mathematics teaching in teacher education pedagogies discuss each concept, we choose and draw on a few research aimed at valuing participation in the mathematical discourse. texts. We are leaving many concepts out, as well as posing differently the emphases by referring to gesturing instead of 4.2 Concept 2: Gesturing gestures, or to languaging instead of language, in order to suggest the practices that go with the concepts across sites Gesturing as the practice of using gestures is present in of mathematics education. Again, we intend to avoid binary studies on multimodal mathematical communication and distinctions that may suggest a separate role of the practices increasingly considered in research on interaction and math- from the objects, processes, people, materials, events, … ematics learning. Robutti et al. (2022), a chapter entitled undergoing. ‘Gesture in mathematics thinking and learning’, recognise the work of researchers investigating language, communica- 4.1 Concept 1: Instructional designing tion and interaction in mathematics thinking, teaching and learning, with a focus to include the roles that gestures like Practices of instructional designing are frequent in studies hand movements can play in these processes. These are in a of mathematics teacher education on language in mathemat- way ubiquitous co-speech gestures, receiving growing atten- ics teaching, but also of multilingual and multimodal math- tion in field research. Gestures in many mathematics educa- ematics teaching and classrooms. These practices generally tion articles are specified as hand movements that frequently include instructional materials, lesson plans and the teaching adjoin with speech, but there are some larger body gestures intervention followed by forms of validating effects of the that are also involved and may be deployed in silence and instruction. Since the promotion of mathematical discus- taking place in real time, as does speech. Pimm (2021) sions in the ‘old’ classroom experiments, a rising body of describes a university Analysis lecture with regard to the literature studies the design of language in content teaching. lecturer’s extensive and significant pedagogic gestures, with The 2021 ZDM issue 2 offers examples of this literature. In an interest between ‘conceptual’ and ‘pedagogic’ gestures. 13 Mathematics education research on language and on communication including some distinctions:… Concept(ual) gestures are more often than not generic—an A new term for us both, which has supported our conver- extension to Mason and Pimm (1984). sations about the recent discourse in the research domain In Pimm (1987/2017), a metaphoric adjectival exten- examined, is ‘hyponym’: a word of more specific mean- sion of the notion ‘triangle’ is indicated by means of the ing than a general or superordinate term applicable to it. A constructed term ‘spherical triangle’ (pp. 101–104), which hyponym is a term used to designate a particular member always seems to require the meaning of the noun to be of a broader class. It could even be a terminological actual altered (in this case, broadening to geodesic rather than subset (in relation to mathematics). But one of the things straight line sides of a ‘triangle’). Similarly, Sinclair and de that occurs regularly in school and university is that the same Freitas (2014) develop the term ‘digital gesture’, which can term is itself generalised, such as triangle (e.g., on a sphere) be seen as a significant contemporary metaphoric term as or multiplication (e.g., involving fractions or matrices). This well, in part because in order to affect an iPad item currently becomes the result that a (plane) triangle is a hyponym of requires physical touch and, for some researchers, gestures a (more general) triangle, once you have left the plane, as are viewed as distinct from actual contact (e.g., pointing mentioned in Concept 2. So, is proof a hyponym of argu- versus touching). In addition, Sinclair and de Freitas make ment? Is argumenting a hyponym of reasoning? Is language an explicit argument for digital gestures being ‘epistemic’ a hyponym of communication? Still, what if mathematics and not just ‘communicative’, following the idea of Jürgen education research use of terminology proved to be as seem- Streeck on ‘tangible gesture’—which again, for some, could ingly straightforward as this? be viewed as an oxymoron—and they bring together the technology world’s thinking about gesture and the math- 4.4 Concept 4: Languaging ematics education/linguistic one in a very profound way. It is also the case of increasing access to screen items by simply The practice of languaging points to one more commonality bringing one’s finger close to the screen, but without having across topics, specifically present in studies about multilin- to touch it: a new notion of ‘digital gesture’, seen now as gual mathematics classrooms (tied to specific languages and simply a specific adjective that does not require altering the translanguaging), mathematical language and discourse (tied noun ‘gesture’, may well be on its way. to specific registers and moves between them), and multi- modal mathematical communication (tied to specific modes and transmodal practices). The entrance of the languaging 4.3 Concept 3: Argumenting concept into mathematics education came hand in hand with Western sociolinguistics (e.g., Makoni & Pennycook, The practice of argumenting is common, although of varying 2007) and is today moving the focus on systems of rules research interest, in studies across the 5 topics discussed in and structures towards the communicative linguistic prac- this review. Due to our paying attention to this common- tices, with attention to social and interactional processes, ality, one more distinction came to light that is important rather than products of communication. This approach is for understanding the most recent period of research on contributing to revisiting mathematics teaching and learning mathematics education, language and communication. We as practices and processes in which language is a media- refer to the distinction between reasoning and argumenting, tor and an agent of meaning. The multilingual learners’ with several authors across the 50-list of articles seemingly languaging around ‘aircraft’ and ‘minecraft’, supported by equating both terms rather than associating them in one way hybrid everyday Hong Kong Cantonese Chinese, Manda- or another. In other research domains of mathematics edu- rin/Putonghua and academic English language in Tai and cation, the specificities of mathematical argumenting are Wei (2021, pp. 232–234), exemplifies the creative use of distinguished, not without multiplicity of distinctions from, the vivid languages of the learners mediating and producing for example, mathematical proving and mathematical con- mathematical meaning and reasoning. jecturing, and all these are viewed as ways of mathematical Some authors are importantly challenging the modes of reasoning, or even of mathematical thinking or understand- language and of communication used in representations of ing (see the curricular, disciplinary and research variety of data beyond the expectations of a certain academic register meanings for ‘reasoning’ in Reid, 2022, and for ‘argumen- mostly in the language of English. Staats (2021) presents tation’ in Reid & Knipping, 2010). Some researchers thus a variant of communication by challenging mathemat- seem to be arguing about argumenting (and argumentation). ics education research as a site of languaging around the For some, argumenting is viewed as a broadening of the personal voice, thinking metaphors, poetic lines, … She term ‘proving’ in relation to mathematics. For others, it is provides an example of opening up paths of languaging by available to and from learners of any age and site, and is recognising the poetic function of language in the analysis suggested as equivalent to mathematical reasoning, and at of a mathematical conversation between two undergradu- times to thinking and understanding. ate students on the algebra of polynomial, exponential and 13 N. Planas, D. Pimm logarithmic functions. In this way, she argues for alternative the term ‘communication’ more difficult and less immediate. forms of seeing and representing data in order to overcome For example, we have seen the exploration in Ingram et al. “the illusory distinction between mathematical discourse (2019) of lessons with the teachers not ‘explicitly asking’ and mathematical reasoning” (p. 1). We see languaging in learners for the mathematical explanations that they came Staats’ analyses in how she considers prose-looking data as to produce. The addition of the term ‘explicitly’ to ‘asking’ poetic forms of spoken repetition of phrases, informing how suggests the existence of some mathematical demands to the students understand the mathematical task around the learners being communicated at a non-linguistic level. To equation for the perimeter of a string of N advanced hexa- know more about how the linguistic and the non-linguistic gons. Part of the mathematical reasoning in the students’ features of this classroom communication relate to or cross collaboration would have been silenced, had the researcher each other, the development of a notion of mathematics com- not searched for linguistic patterns, rhythms, similarities or munication register seems, again, an important direction to dissimilarities in the pieces created by the conversational pursue in the coming period. repetitions of each other’s contributions to the task. Funding Open Access Funding provided by Universitat Autonoma de Barcelona. This work was funded by Ministerio de Ciencia e Inno- 5 Concluding about the distinction vación (Grant no. PID2019-104964GB-I00) and Govern de Catalunya, between language and communication AGAUR, 2021SGR-00159. Open Access This article is licensed under a Creative Commons Attri- The question in this concluding part is as follows: How do bution 4.0 International License, which permits use, sharing, adapta- some topics and concepts of the domain at present contrib- tion, distribution and reproduction in any medium or format, as long ute to the discussion about the distinction between language as you give appropriate credit to the original author(s) and the source, and communication? In the presentation of the topic of mul- provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are timodal mathematical communication and of the concept included in the article's Creative Commons licence, unless indicated of gesturing, we have particularly addressed the distinc- otherwise in a credit line to the material. If material is not included in tion between language and communication by emphasising the article's Creative Commons licence and your intended use is not studies and comments in which non-linguistic practices are permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a central to processes of communication in sites of mathemat- copy of this licence, visit http://​creat​iveco​mmons.​org/​licen​ses/​by/4.​0/. ics education. Several studies provide empirical evidence of communication that is not (only) language, and some of them clarify that language is not (always) communication— References not only because we can communicate without language, but also because we can use language in ways that are of difficult Adler, J. (2001). Teaching mathematics in multilingual classrooms. understanding in mathematics teaching and learning, and Springer. in teacher education. In all this, we have proposed zoom- Adler, J., Mwadzaangati, L., & Takker, S. (2023). From defining as assertion to defining as explaining meaning: Teachers’ learning ing forward by focusing on challenges such as the elabo- through theory-informed lesson study. International Journal for ration of a notion of mathematics communication register Lesson and Learning Studies, 12(1), 38–51. that necessarily expands and enriches the language-specific mathematics register. Research and developmental collaborative work Despite the advances in the research of different modes of with a group of Malawian secondary‑school communication and in the understanding of non-linguistic mathematics teachers engaged in a lesson study practices such as gesturing, the lack of distinction between project in plane geometry guided by a focus language and communication persists. A basic part of the on language responsive teaching practices. problem is that the study of this distinction does not seem to Qualitative content analyses of cyclical lesson be in the research agenda, at least it is not overtly discussed plans and transcripts and of reflective discussions in any of the 50 articles reviewed. Some of the articles reveal teachers’ learning of forms of word use reviewed do not treat language explicitly as distinct from that communicate mathematical meanings for communication, with findings that can be misleading in con- geometric properties and the exterior angle of a clusions about the scope and nature of the communication triangle. processes. The nature of the distinction between language and communication is, nevertheless, suggested in some Alhasnawi, S. (2021). Teachers’ beliefs and practices with respect to other articles. Some of the studies support their thinking translanguaging university mathematics in Iraq. 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