ZDM (2021) 53:277–288 https://doi.org/10.1007/s11858-020-01190-6 ORIGINAL PAPER How specific can language as resource become for the teaching of algebraic concepts? Núria Planas1 Accepted: 19 September 2020 / Published online: 28 September 2020 © The Author(s) 2020 Abstract Classroom research into mathematics and language has studied issues of context specificity such as cultures of explanation or the impact of language policies on practice. More recently, researchers in the domain have started to study issues of content specificity aimed at performing language-responsive mathematics teaching for the learning of precise mathematical con- tent. Progress in the conceptualization of language as resource for mathematics teaching and learning makes it necessary to strengthen the discussion of the contexts of culture and interaction along with the linguistic demands given by the specificity of the mathematical content at play. In this paper, I introduce a sociocultural framing for a mathematical-linguistic view of grammar as resource with the focus on explicitness in communication. I then report developmental work with two teachers on their teaching of algebraic concepts, and address the question of how to learn to communicate explicit meanings for these concepts in classroom mathematical talk. The structuring principle adopted for this work was to critically distinguish and choose or produce instances of teacher talk that overtly communicated conceptual meaning within the algebra of equations. I conclude with preliminary evidence of the effectiveness of the work with the teachers. Keywords Mathematics teacher professional development · Classroom teacher talk · Explicitness in communication · Content specificity · Algebra of equations 1 Introduction teaching that is based on findings from classroom research on mathematics and language (e.g., Adler and Ronda 2017; We cannot address issues of communication in and for Moschkovich and Zahner 2018; Prediger and Zindel 2017). school mathematics teaching and learning in full without In complementary ways inspired by a diversity of theoreti- working for the understanding and improvement of class- cal-analytical frames, these pioneering studies share views room teacher talk. Mathematics teacher education research of language as resource for mathematics teaching in class- particularly needs more systematically to examine features room talk. The background and quantity of studies in this of teacher talk with the potential of opening up the discus- line of research is on the rise, but more empirical evidence sion and learning of specific mathematical content. So far, of the didactic possibilities of teacher talk in school math- research on how to connect the language as resource orien- ematics is necessary. In the current study, I seek to make a tation to the exploration and refinement of the language of contribution by conducting developmental work with two the mathematics teacher in the classroom remains limited in secondary school teachers during their teaching of concep- terms of curricular content and educational levels. This is tual aspects of the algebra of equations. In this context of in part due to the prevalent focus, for some decades, on the practice, I research the question of how to learn to commu- languages of the diverse learners in this orientation in the nicate explicit meanings for algebraic concepts in classroom field (Planas 2014, 2018). An increasing body of studies is mathematical talk. reversing the current research gap with work on mathematics My regular work for more than a decade with mathematics teachers in Catalonian schools provides the institutional and * Núria Planas educational backdrop to this paper. In 2005 I started working Nuria.Planas@uab.cat with a group of mathematics teachers in what would become a ten-ýear project (Planas and Civil 2009). The major struc- 1 Universitat Autònoma de Barcelona, Cerdanyola del Vallès, turing principle of the practice with teachers at that time was Spain 13 Vol.:(0123456789) 278 N. Planas to construct, implement and evaluate classroom tasks—with concrete theory that substantiates the organisation of the a version for learners and a version for teachers—aimed at developmental work with Jana and Maia. In prior PDs, promoting explanatory talk in the sense of talk that supported aimed at creating and enacting classroom tasks as media- practices of explanation of mathematical concepts and rela- tors of explanatory talk, part of the mathematical content tionships between them in the lessons with the learners (Planas planned—that is, specified in the teacher written version and Civil 2013). At present this principle continues to mediate of the task—often remained hidden or out of focus in both my work with teachers along with a principle for explicit com- teacher talk and whole class discussion. While the tasks munication of specific mathematical content in teacher talk. had been designed carefully and practices of explanation In what follows I explain the steps towards the justification, were promoted in the lesson experiments, implicitness or adoption and shaping of this newer principle. We must go back ambiguity in content meaning seemed to impose limitations to the spontaneous conversation in 2018 with Maia and Jana, on student learning across lessons of different teachers. For two mathematics teachers from the same school in Barcelona example, in a 2015 lesson regarding plane isometries, the with several years of teaching experience. They took the ini- teacher said, “When mathematicians think of angles, they tiative to share some difficulties encountered in their teaching have more in mind than two segments and an end point.” of algebraic concepts, and I was then invited to spend time in In the lesson talk of that teacher, this sentence did not go the school and to visit their classrooms. Once there, I mostly with any explicit mention of the rotation concept, which observed pedagogies of whole class teaching in which Jana is essential for the construction of the angle concept in the or Maia presented a rule or procedure, and then routine exer- geometry of transformations. In the PD practice, we had cises from the textbook were done to consolidate the rule or created a task based on manipulating materials and having procedure. The planning of a progression of goals for teacher participants interacting with each other, but in its design learning and classroom transformation was conducive of a and enactment, the clear communication of important math- professional development intervention (PD) with the initial ematical meaning was not fulfilled. goal of improving the mathematical talk of the teachers. The There is a wide range of theoretical-analytical frame- structuring principle for the PD was to critically distinguish works for the study of language, meaning making and com- and choose or produce instances of teacher talk that explicitly munication in mathematics classrooms (see a survey of the communicate conceptual meaning within the algebra of equa- theoretical diversity in the domain by Planas and Schütte tions. This purpose was supported by practices of noticing (2018), but not all of them offer tools for the analysis of mathematically relevant content amongst sentences spoken in mathematical meaning communicated through the resource lessons, and of zooming in on issues of lexical elaboration for of grammar. The theory of Systemic Functional Grammar, the overt communication of the content in question. or SFG in brief (Halliday 1978, 1985), takes the realiza- Understanding how teacher talk unfolds for content-spe- tion of meaning as mediated in practice, and hence fits in cific teaching fits the topic of this Special Issue by expand- well with sociocultural views of teaching and of teacher talk ing the principle that instruction should include comparison that address appropriateness to the rules of language and of of language pieces for raising students’ language awareness the relevant social context. SFG consists of various linked (Erath, Ingram, Moschkovich and Prediger 2021) into a prin- aspects such as the ideational functioning of grammar in ciple that includes sites of mathematics teacher education. context to construct and communicate objects, relationships Comparing instances of the language in the classroom can and logic, which for the mathematics classroom includes be transferred at the level of work with teachers for raising the objects, relationships and logic of school mathematics. awareness of the didactic potential of their talk in teach- Although there is a research basis that suggests the means ing. While there is agreement on the didactic importance by which prompting practices of explanation in the teach- of teacher talk in our research field, much remains to be ing mediates school mathematics learning (e.g., Khisty and studied with regard to the instructional practices that create Chval 2002); Zahner et al. 2012), we do not know much opportunities for learning how to improve this talk in sites about how specific grammar choices in teacher talk (i.e., of professional development. choosing which words to use and how to connect them with other words) differently realize meaning, and can offer or fail to offer learners the opportunity to hear and engage with 2 Theoretical basis for designing learning specific mathematical content. A mathematical-linguistic practices on language and mathematics view in which grammar is a process of making mathematical teaching meaning in context can thus enhance sociocultural research on classroom teacher talk. Following a sociocultural view of teaching as socially medi- Halliday (1978, 1985) equates grammar with the indis- ated by cultural tools such as tasks, practices and languages soluble articulation of syntax and context. Accordingly, (Adler and Ronda 2017), in this section I present the more the language that functions to make and communicate 13 How specific can language as resource become for the teaching of algebraic concepts? 279 mathematical meaning is both syntactically and contextu- While clarity in meaning should prevail in teacher talk, ally ruled. In the more linguistically focused mathematics this is not always easy. The moment the teacher places words education research literature, some authors have drawn together in a sentence for teaching mathematical content, on SFG to study uses of language for a variety of curricu- this sentence enters into a relation with the context, specifi- lar contents and educational contexts. Herbel-Eisenmann cally including its relation with other sentences and what has and Otten (2011) tracked content-based ideational and or has not been communicated in precedent talk. Although interpersonal functions of grammar in lessons aimed at certain meanings can be left as tacit in some lessons, the teaching and learning plane areas; Morgan (2006) exam- use of talk in ways that determine the meanings pursued as ined learner written productions to elucidate ideational content of learning turns out to be problematic. To this end, and textual functions of grammar in the communication of clarity in meaning through lexical elaboration is challenging conceptual meaning for triangles and generalization pat- due to semantic vagueness of grammar and disassociation terns; and Pöhler and Prediger (2015) explored linguistic between syntactic precision and meaning in use (Halliday means in the realization of precise mathematical ideational 1978, 1985). Vagueness and disassociation are actually meaning in lessons on percentages. Taking the aspects of intrinsic to communication and essential in order to grasp Halliday’s theory related to the creation of content-based the complexity embedded in the realization of the didactic ideational meaning, rather than interpersonal meaning pro- potential of teacher talk. Clarifying the semantics of a sen- duced in the interaction or textual meaning more broadly tence requires words that can function to indicate objects, produced to organize coherent messages, Schleppegrell relations, attributes, variables… If the grammar is minimal (2007) pointed to the complexity of choosing and com- in the sense of lacking lexical complements for objects, rela- bining words for the enactment of meaning that can be tions…, the sentence stays open or vague with the subse- recognized as constitutive of school mathematics. She quent difficulty in noticing the meaning intended, and the called for research on the ways learners and teachers use risks of varied interpretations. All of this is connected with resources from the grammar, like the “dense noun phrases the questions as to how grammar in teacher talk can become that participate in relational processes” (p. 139), to pro- a resource for the communication of concrete mathematical duce academic meaning. meaning, and to which degree this can be planned in sites Halliday’s concept of grammar as both syntactic and con- of mathematics teacher education and then made effective textual is shaped by a number of analytical tools that con- in classroom teaching. nect linguistic forms with the realization of meanings in use; The elaboration of sentences that are precise in meaning these tools are however mostly developed for the case of the takes even greater significance for sentences made of rela- English language. When an investigation deals with sources tional verbs, since these are a feature of academic registers of data in spoken languages other than English, the adoption in contrast to sentences made of material verbs suggesting of parts of SFG requires some caution. In accordance with processes of doing and acting. As developed by Halliday Boas and Gonzálvez-García (2014), the SFG considerations (1985), relational verbs are those realized in grammar to for clause processes in spoken English generally apply well perform functions of explaining and connecting attributes to Catalan and to Spanish, the two languages of the data in and relations. ‘Be’ is the most typical relational verb and my research context. Although there are differences, such the most ambiguous as well. In the mathematics classroom, as the obligation in English of explicitly including the sub- it is common to find sentences with this verb conjugated, in ject of a sentence or clause, in these three languages most which an example is meant to exhaust the larger set: ‘A is often lexical complements to verb forms are not obligatory. B’ put as ‘3x2 + 4 = 2 is the equation’ expresses a one-to- This linguistic feature implies that syntactically correct sen- one relation of equivalence when what is mathematically tences may stay open or vague in meaning and function for involved is a one-to-many relation between the equation a diversity of contexts and registers. For example, if we take at play and one of its representatives. While the choice of the sentence ‘she orders the tickets’ and do not specify the more concrete verb words that function as relational, such as context of use, the meaning for ‘orders’ remains vague; it exemplify, name, classify, compare, define, and so forth may may variably function within out-of-school and school math- avoid some ambiguity, openness or vagueness regarding the ematics registers. Moreover, in English, Catalan or Span- meaning for the object equation may still remain due to poor ish, we find syntactically precise sentences that, in spite of lexical elaboration. ‘A exemplifies B’ put as ‘3x2 + 4 = 2 knowing the context in which they are said, may require exemplifies the equation’ keeps a wide range of possible lexical complements not to stay semantically open. Even in meanings for the equation concept. In the absence of more the mathematics classroom, ‘She orders the tickets’ may be lexical detail, it is unclear why 3x2 + 4 = 2 is an example of understood as requesting tickets. If the intended meaning is an equation and what this tells us about how examples of mathematical, lexical complements like ‘from the cheapest the same object should look. If we intend to communicate to the most expensive’ are useful. that 3x2 + 4 = 2 represents an equality reasoning, this might 13 280 N. Planas be further elaborated lexically, for example, as ‘3x2 + 4 = 2 be’—these tasks intend to mirror descriptive aspects—the exemplifies an equality that some numbers may satisfy.’ ‘what is’—of classroom data in the form of short instances Albeit timidly and not particularly centered on issues of of talk comprising mention of algebraic concepts. In the semantic clarity or precision within the algebra of equations, next subsections, I particularly discuss the basis and role of SFG approaches are also present in the study of language the structuring principles in the design and enactment of the in mathematics teacher education and professional devel- PD tasks. In the first subsection, I present the phase around opment research. In the reflection on language issues in the the empirical preparation and theoretical basis of the ses- domain, a major argument is the distinction between the sions for developmental practice. In the second subsection, I content of mathematics teaching and learning at the level of report details of the implementation and analysis of the work the classroom and this content at the level of mathematics with Jana and Maia in one of the sessions and with regard to teacher education and professional development. Prediger one task. At the time of this writing, the analysis of changes (2019) is attentive to the two levels of content in her argu- in the talk of the two teachers, that seem to be important for ment towards strengthening the collaborative research with the conceptual mathematics learning of the learners in their teachers in the construction of mathematical content-specific classrooms, is being done. Some of the changes identified knowledge about language for mathematics teaching and are discussed in the section following this report. learning. She proposes a language focus in design-research work towards the anticipation, discussion and evaluation 3.1 Preparing developmental practice grounded of explicit explanation and connection of content meaning. on lesson data and research findings Morgan (2014) specifically addresses the question of choice, which is central to the theory of Halliday. In the analysis In the introductory section, I anticipated the instructional of means of describing and judging learners’ mathematical principle that structured the developmental work with Jana texts of utility for ways of organizing the knowing, doing and and Maia: Critically distinguish and choose or produce teaching that are important to teachers and teacher educa- instances of teacher talk that explicitly communicate con- tors, Morgan interrogates, “what might be the differences ceptual meaning within the algebra of equations. In the if other choices had been or were to be made?” (p. 132). It theoretical section, I summarized the rationale for taking is noteworthy that the choices of teachers and teacher edu- the aspects of SFG that deal with content-based ideational cators about which language to use may not be deliberate, meaning. In line with all this, I now provide the details of but rather the result of common ways of speaking. In this how the tasks used in the PD practice were thought out and respect, choices that keep mathematical meaning implicit in prepared. While it had been possible to work with anony- the interaction with learners may not be seen as problematic mous instances of teacher talk released from past collabora- because implicitness and its impact on classroom learning tions with other teachers, my experience of developmental may not be perceived. It is thus very pertinent, in research work recommended building the newer collaboration with on developmental work with teachers, to guide the interro- insights and material from Jana and Maia’s own classrooms. gation and noticing of grammar choices for content-based Having in mind the difficulties explained by these teachers mathematics teaching. in our initial conversation, the decision of working with and from their classroom talk was specified for lessons aimed at formulating, manipulating and solving equations. 3 Methodology and methods The source data were collected in my visits to the for the developmental work and its study classrooms across the 2018 school year, and the PD was then planned to start in October 2019. I finally chose In this section, I explain the methodology and methods for two lessons, one per classroom and teacher, which had researching and developing task-based professional learn- been audiotaped and turned into transcripts limited to the ing practices with two secondary school mathematics teach- instances of teacher talk. None of the lessons were part of ers. Guided by the view of mathematics teaching as making a prepared design experiment or of a pedagogic innova- grammar choices, I used an interventionist methodology for tion. They reflected the everyday dynamics of whole class simultaneous research and developmental work organized teaching and controlled exercises from the textbook, and in around the performance of tasks that highlight issues of both instruction was centered on the routine of manipulat- language choice and content-specific talk. By this I mean a ing quadratic equations towards producing representative methodology that involved preparing and contextualizing the items for application of the formula. In the second year of emerging PD practice, from, and within, data drawn from the the PD, the reach of the project was expanded and moved participant teachers’ talk in lessons that I observed, analysed on to the purpose of also working on the choice or design and selected for the design and implementation of tasks. and enactment of innovative tasks for teaching algebraic For the discussion of normative aspects—‘what ought to concepts through combination of verbal and graphical 13 How specific can language as resource become for the teaching of algebraic concepts? 281 tools. We kept the same curricular topic and left for future teacher talk suggesting lack of clarity in meaning around the studies the intuition that professional deepening inside the following relational processes: language demands of teaching equations will support the learning of how to improve the teaching of other topics. i. Equating representatives. One-to-one processes in The following is an instance of Jana’s talk to the whole which the two objects are representatives of an equa- class in the lesson selected, followed by an English ver- tion, e.g., “you change each equation into the follow- sion (shifts between Catalan and Spanish in the original ing.” are not marked): ii. Exemplifying equations. One-to-many processes in which one of the objects is the equation and the other Podem resoldre una equació quadràtica amb fórmula. is a collection of representatives, e.g., “the equation Modificarem una mica l’equació escrita inicial. Anar looks a bit different after each transposition rule.” fent canvis pas a pas és bàsic. Canvieu cada equació per la següent i obteniu una seqüència. Cada ecuación The processes above were therefore taken as relevant con- la cambiáis un poco. Tenéis que utilizar las reglas de tent of teaching in lessons on the topic of equations, as well transposición. Vais asociando una forma escrita con as relevant content of professional development work on this otra hasta llegar a la fórmula general de la pizarra. topic with the teachers. Drawing on the reading of the lesson We can solve a quadratic equation with a formula. We transcripts, it was evident that the potential of relational pro- will modify the written initial equation a bit. Chang- cesses for fostering reasons and connections did not always ing step by step is key. You change each equation into imply the communication of precise mathematical mean- the following and get a sequence. Every equation, you ing. I found several sentences suggesting relational processes change it a bit. You have to use the transposition rules. that were not further elaborated towards the construction of You go mapping one written form to another up to the algebraic concepts but rather went together with sentences general formula on the board. functioning to make learners notice rules or procedural steps The final material for the PD came from a second round of the resolution method. In this way, the explanations ini- of data selection so that the units of attention that could tiated were not lexically elaborated to communicate what bring grammar choices into focus in the work with the teach- remained invariable with regard to the equation concept and, ers were relatively short. I looked into the transcripts of more generally, the criterion for belonging to a class of rep- teacher talk for sentences with the potential of constructing resentatives. The following are two examples of this finding: meaning for algebraic concepts. With the ultimate objective You change each equation into the following and get a of selecting material for the PD practice, I did not pretend sequence. Now pay attention to how far you are from to do an exhaustive tracking. I identified sentences of a rela- the general formula. tional type, and hence with the potential of supporting func- This final equation is the equation from the beginning. tions of explaining and connecting. Throughout this process, But now the equation equals zero as we wanted and relevant research literature provided me with criteria with can stop. which to identify some of the most significant concepts and relations constituting the semantics of school algebra. Her- As visible in the examples above, most relational sen- scovics and Kieran (1980) and Pournara, Sanders, Adler and tences appeared paired in the talk of Jana and Maia with Hodgen (2016) documented areas of difficulty regarding the the lexical elaboration of complements suggesting mate- algebra of equations, and indicated the relevance of teaching rial processes of doing and acting. The recurrent objects of equivalence (between expressions obtained by procedural explanation (‘explanandum’ in Erath, Prediger, Quasthoff manipulation) through expanded meanings for the equal sign and Heller 2018) were overall the procedural steps of the as a relation, and not just an operator. Kieran and Drijvers resolution method and the transposition rules. Across the (2006) specifically pointed to difficulties encountered by two lessons, there was not explicit talk about mathematical learners in understanding algebraic equivalence as well as reasons underlying the steps in the method or about connec- in dealing with structural properties of operations involved tions between these reasons and the transposition rules to be in connecting representatives or examples. Accordingly, I applied. As a result, exemplifying equations and equating considered that relational processes of equivalence and of representatives were relational processes whose communi- exemplification were crucial to be intended in classroom cation was initiated in teacher talk and interrupted when teacher talk aimed at prompting the learning of concepts paired with material processes of algebraic manipulation. such as algebraic equivalence and quadratic equations. Since It was not less revealing that very few teacher sentences in these processes cannot be omitted or left unclear at the level both lessons contained the words ‘equivalent’ or ‘equiva- of the classroom, I also considered its discussion to be cru- lence’: “These are equivalent, and these too”, “Not the same cial at the level of the PD. I compiled critical sentences from but are equivalent equations”, “Equivalence in fractions is 13 282 N. Planas similar”, and “Transformations that keep the equivalence.” communication of conceptual meaning within the algebra Collectively all these findings guided the selection of some of equations. As already explained, the tasks had emerged moments of teacher talk over others for the practice in the from my analysis of the teachers’ talk in two lessons the PD, but also confirmed the adequacy of the SFG approach year before. Since it was I who had identified aspects to be to the interpretation and study of the didactic potential of improved in their languages of teaching and hence in their the language of the mathematics teacher in the classroom. teaching, it was necessary to engage the teachers themselves In the next subsection, I present some events of the third in noticing the didactic content-specific possibilities embed- PD session centered on the discussion of the task in Fig. 1. ded in their talk. To accomplish this aspect, each workshop Options 1 and 2 in this task expose Jana and Maia to critical consisted of the iteration of three tasks: one of interroga- instances of their own talk in the lessons analysed. These tion of instances of these teachers’ talk (like the example in are instances in which the communication of conceptual Fig. 1), and two of lexical elaboration. The process for each mathematical meanings is initiated and then interrupted by session was tunnelling the attention of the teachers towards a change in focus from the identification of representatives the practice of producing linguistically small but mathemati- of a particular equation to the indication of the procedural cally significant changes in their talk when teaching. All actions to be taken. A consequence expected from the dis- this was to be done without much talk on my side though I cussion and comparison of Options 1 and 2 is the creation or flexibly played the role of facilitator. design of newer options of talk that more overtly explain and The five workshops always started with the reading and connect precise mathematical meanings. A total of fifteen interrogation of two short pieces of language, one from each tasks were developed as progressive towards the discussion, lesson transcript, to engage Jana and Maia in examining their comparison and production of diverse options in languages talk in the classrooms. The pieces occupied approximately of teaching that explicitly highlighted important concepts one page and represented about three of four minutes of and mathematically relevant meanings in the algebra of teacher talk. This was the material in focus from which to equations. find evidence in order to construct a shared argument on the importance of explicitness in mathematical communica- 3.2 Enacting and analysing teacher participation tion. The instruction for the five first tasks was as follows: and learning in developmental work Underline the sentences in which you talk about concepts of significance in the algebra of equations, and specify the The first year of the PD practice with Maia and Jana was concepts. Jana and Maia faced this instruction with no prior designed to develop and support professional learning of preparation as a way to simulate the way teachers typically features of teacher talk at the time of their participation in encounter their own talk or that of their colleagues in class- a single module of five 90-min workshops, one per month room teaching. In the discussion, both teachers struggled in the school site. All the workshop tasks were planned to to find sentences in which meanings for specific concepts be completed in collaboration during the time of the ses- were explicitly communicated, and hence came to notice that sions, and they were all centered on issues of explicitness in important mathematical content remained under-specified Fig. 1  Task given in the third Task. In a lesson aimed at teaching the method of solving quadratic equations with the formula, session of the 2019 module the teacher is writing on the board the expressions below. = 3 + 10 − 3 = 10 − 3 − 10 = 0 Then the teacher stops writing and talks to the whole group. Option 1. You change each equation into the following and get a sequence. Now pay attention to how far you are from the general formula. Option 2. This final equation is the equation from the beginning. But now the equation equals zero as we wanted and can stop. Is the talk in Options 1 and 2 communicating meanings of algebraic equivalence? Can these instances convince learners that the expressions on the board represent the same equation? Can you think of one or more Options 3 in which the teacher talk in the class explicitly communicates the relation of equivalence between the expressions of the equation? Options 3. _________________________________________________________________ __________________________________________________________________________ 13 How specific can language as resource become for the teaching of algebraic concepts? 283 within the algebraic register. Whilst in the first PD session consideration of disciplinary content knowledge, through Jana and Maia justified implicitness and ambiguity in mean- the unpacking and noticing of mathematical facts to be ing by raising some risks of using too long sentences in the made explicit in the lessons, such as: (1) each algebraic teaching, as the module evolved, they progressively associ- equation is a class of equivalent representatives; (2) repre- ated explicitness and exactness with the possibility of intro- sentatives are equivalent if they have the same solutions; ducing small changes in classroom talk. and (3) operating with the same numbers or expressions on To make teachers notice the range of didactic possi- both sides of an equation produces an equivalent equation, bilities embedded in talk, and working with the material except for the case of multiplication or division by zero. provided for each first task, the instruction for the five We also had to work on misconceptions such as the neces- second tasks was as follows: Propose changes in sentences sity of finding the numerical solutions of the equations in in order to communicate explicitly conceptual meanings of order to test their equivalence. significance within the algebra of equations. Table 1 shows All five sessions ended with a third task made of sen- extended texts jointly created by Jana and Maia in collabo- tences of teacher talk for which the inclusion of lexical com- ration with me over the course of the workshops. For each plements was sufficient to allow the explicit communication sentence, the curricular content to be made explicit was of conceptual meaning of relevance in the algebra of equa- discussed, and lexical elaborations that could contribute tions. Throughout the study of the lesson data, I found some to this end. Interestingly, most of the sentences used as sentences rather easy to turn into mathematically relevant material for the set of third tasks of the sessions were also talk for conceptual meaning making; these were the sen- in the group of sentences underlined or selected by Jana tences chosen as material for the third five tasks. Figure 1 and Maia as critical. The concept of algebraic equivalence reproduces an English version of the group of instructions and the related issue of how to teach the expressions that in one of these tasks, with the ultimate instruction to Think can and those that cannot be regarded as equivalent were of different ways of explicitly communicating the relation particularly brought up by the teachers in all the work- of equivalence between expressions of an equation. The shops. We agreed on the affordances of the procedural teachers worked together around the sentences in Options definition of algebraic equivalence in terms of equations 1 and 2. For example, in Option 1 “You change each equa- that have the same solutions, but concluded that learners tion into the following and get a sequence”, the criterion of should also be given conceptual reasons. To support the change and what remains unchanged are not overtly said. further elaboration of newer sentences, in my role of facili- Further lexical elaboration could mathematically reinforce tator I mentioned linguistic devices to be used such as: the sentence with respect to the semantics of algebra and ‘which means’, ‘which is to say’, ‘that is’, ‘in other words’, algebraic equivalence. Here, I facilitated the discussion of ‘X means Y’, ‘by X we/I mean Y’, or ‘X is the same as clarity in conceptual meaning with respect to the teaching Y if/because.’ The strategy was to make noticeable that and learning of quadratic equations, as well as the discus- some additional text would support the communication of sion of how small changes in talk can function for meaning meaning otherwise implicit. These devices functioned to disambiguation and explicitness. In the third PD session, the support explanations and connections while combining or Option 3 jointly produced by the teachers and thought of as reworking entire sentences, and changing or adding words. an improvement compared to Options 1 and 2 (see Fig. 1), Another fundamental outcome of the discussions was the was as follows: Table 1  Examples of lexical elaboration by the teachers Selected sentences Lexical elaborations We can solve a quadratic equation with formula We can solve a quadratic equation with formula. That is, we can obtain the numerical values for x that solve the equation We will modify the written initial equation We will modify the written initial equation. In other words, we will look for ways of writing the same equation for the final application of the formula Get a sequence Get a sequence, which is to say, get a sequence of equivalent equations, or equations with the same solutions Every equation, you change it a bit Every equation, you change it a bit. By changing it a bit, I mean adding, subtracting, multiply- ing or dividing both sides with the same numbers so that the solutions do not change You have to use the transposition rules You have to use the transposition rules. That is, the rules for the generation of equivalent equations You go mapping one written form to another up You go mapping one written form to another up to the general formula on the board. All the to the general formula on the board equations will be the same because the same numerical values solve all them 13 284 N. Planas You change the place of some terms for each equation, Maia are explaining solving methods through a language but the solutions do not change. x2 = 3x + 10 is the that makes conceptual mathematical meaning explicit in the same as x2 − 3x − 10 = 0 if we look at the numbers teaching. Jana overtly renames transposition rules into rules that solve them. of equivalent equations, and Maia clarifies that calling some- thing equivalent does not clarify the mathematical meaning By asking Jana and Maia to choose and produce improved of this word. In either case, teacher talk seems crucial in instances of teacher talk, I had progressive access to what offering learners access to explanations for mathematically counted for them as appropriate mathematical communica- relevant word meanings, reasons and connections. Still, we tion with the learners in the classroom and could reorient could discuss the fact that routine exercises remain central some of the later discussions as well as slightly modify the aspects of the teaching pedagogies in the two classrooms. In content of tasks prepared for the remaining workshops. All the 2020 PD module with the five mathematics teachers in this process opened opportunities for these teachers to learn the school, work on inquiry pedagogies has been incorpo- about language in mathematics teaching, but also opportuni- rated. One of the strengths of the initial module, and of how ties for mathematics learning by learners in their classrooms. the collaboration with the teachers was sensibly approached avoiding too many changes and goals at a time, is thus the successful role played in building continuation and expan- 4 Initiating methods for the evaluation sion of the developmental work project. of the developmental work 4.1 Lesson episode with Jana: “Why do I call them The first year of the PD was not intentionally designed with the rules of equivalent equations?” the goal of finding an impact of teacher talk on the school learning of algebraic concepts. I had the idea that the partici- Learner 1: Okay, then we stop here. Ready for the formula. pation in the PD practice would affect the teachers’ profes- Jana: What is the same with them [ x2 + 4 = 1 − 2x; sional growth and learning but would only reach learners x2 + 2x + 3 = 0]? in their classrooms indirectly. So, when three weeks after Learner 1: Same equation. the ending of the 2019 module I observed their teaching in Jana: How do you know this? They look different… lessons aimed at manipulating and solving equations, I was Learner 1: Well, not so different. They are quadratic and not looking for evidence of impact at the learner level but have two solutions. rather for little differences and improvements in teaching at Jana: x2 = 4 [on the board] is also quadratic, and also has the level of explicitness and precision in mathematical com- two solutions. munication. The intention was to see whether some profes- Learner 2: But this is different. sional learning about explicitness in content-based commu- Jana:How do you know this? nication was visible in the language choices of the teachers Learner 2:You cannot apply a rule and make two x once back in the classrooms. On this occasion, a research disappear. assistant videotaped and transcribed the totality of classroom Learner 1: You cannot find a rule that goes from here talk over one week of lessons because that would be source [ x2 + 4 = 1 − 2x ] to here [ x2 = 4]. material for the 2020 module. Despite the short exposure to Jana:So, if you can find a rule… What did I call transpo- the PD practice, the reiteration of the curricular topic as the sition rules? Rules of equivalent equations? Why do I call teaching and learning content was an occasion for teachers them rules of equivalent equations? to experiment, and for me to observe any innovation in their Learner 1:Because they are equivalent. talk. The observation of the individual teaching was thus the Jana: And how do you know this? What is the same with way in which I had planned the evaluation of the effective- them? ness of the PD. What I found were many opportunities for Learner 1: Same solutions? learners to hear sentences semantically situated within the Jana: Right! These two equations are equivalent because algebra of equations. This allows me to provide a prelimi- they have the same solutions. How can we check this? nary evaluation of the effectiveness of the PD from the dual In this example, the attention in the teacher talk to word perspective of professional learning and changes in teacher use (i.e., “transposition rules”, “rules of equivalent equa- talk, and opportunities for mathematics learning created in tions”) and to meaning explanation (i.e., “how do you know the classroom. this?”, “what is the same with them?”, “these two equa- What follows are two short lesson episodes, one from tions are equivalent because they have the same solutions”) each classroom, for illustrating some evidence of mathemati- is obvious. Learners are given the opportunity to hear and cally relevant choices in the teacher talk with an effect on the learn how the teacher talks about algebraic equivalence, responses of the learners in the interaction. Both Jana and which in turn generates mathematical talk about this concept 13 How specific can language as resource become for the teaching of algebraic concepts? 285 amongst learners. A theme of discussion in the PD was word Learner 3: The solutions! use in naming procedures, specifically the possibilities of This second example brings us back to discussions in supporting conceptual meaning by renaming terminology the PD sessions about the idea that the words “equivalent” related to procedures and drawing on it for elaboration of and “equivalence” were not often said in the teaching of explicit talk about the reasons of equivalence between alge- algebra, and about the implications of omitting not only braic expressions. We also discussed that making a concept taken-as-known words but also their explanation in the talk clear is not realized at the level of word use but requires with learners. In the workshops we concluded that with- the communication of relational processes of explanation out deliberate attention to the equivalence relation through and connection in the teaching. We precisely discussed the explicit grammar in teacher talk, opportunities for learn- open semantics of transposition in “transposition rules”, and ing the notion are limited. We can see connections between brought up the possibility of referring to them as “rules of this conclusion in the PD and what Maia says to her learn- equivalent equations.” In the episode above, Jana does not ers in the episode above. She explains that calling the word abandon the common name as generally termed in the local equivalent does not make its meaning clear. In a situation textbooks, but draws on the potential of “rules of equivalent in which learners may not have identified equivalent as a equations” to make learners notice what remains invariable mathematically relevant word, the teacher elaborates on its in the expressions across the application of such rules. meaning in the algebra of equations and does it by explicit contrast with the meaning within the arithmetic of fractions. 4.2 Lesson episode with Maia: “What makes them The talk of the teacher continues to connect the word to equivalent is not calling them equivalent” explanations immediately after the interrogation of meaning (“what makes them equivalent?”), which denotes the kind Maia: Let’s think of fractions. Some fractions are equivalent, of changes practised with the selected sentences in the PD okay? sessions. In a sense, when Maia elaborates on word use and Learner 3: Yes. content meaning, she offers what she learned in the work- Maia: What do we know about fractions? What makes shops to her learners. them equivalent? Can you tell me when two fractions are equivalent? 4.3 Classroom teacher talk as demonstrated Learner 3: One half and two fourths. evidence of professional learning Maia: Okay, one half and two fourths. What makes them equivalent? While being cautious and aware that no systematic or com- Learner 3: Yes, when you make the division, you get parative analyses have been done with the lesson data from point five. the participating teachers, good choices for mathematical Maia: Okay, so they are equivalent because of the deci- meaning making in the language of teaching can be appreci- mal. Let’s go back to equations. What do we know? Some ated. The year before, Maia had resorted to the analogy with equations are also equivalent. We call them equivalent, but fractions but had not explicitly addressed the construction why? What makes them equivalent? Two equations are of distinct meanings and structures for numerical equiva- equivalent if… lence and algebraic equivalence. The mention of equiva- Learner 3: Like fractions. lent fractions had not been elaborated in her classroom talk Maia: Now it’s about equations. What makes them equiv- to function for disambiguation and concretion of specific alent is not calling them equivalent. By equivalent equations, mathematical meaning with respect to algebraic equations. we mean… Although the data above allow us to be optimistic, it is not Learner 3: Can you divide equations? clear whether the participation in five workshops oriented Maia: Good question! Yes, you can divide them, but what the talk of Jana and Maia in the episodes shown. We cannot makes them equivalent is not about division and decimals. be certain that the (transformed) teaching practice illustrated It’s about having same solutions. is due or partially due to participation and learning in the Learner 3: Then you don’t know with the calculator. PD. To test the relationship between participation in such Maia: Well, it’s about finding the numerical values that a short PD site and changes in the teaching talk is rather solve them. complex, and this is also true and even more complex for Learner 4: Not finding the decimals. the relationship between teacher professional learning and Learner 3: No, that’s for fractions. She was just putting changes in the learning opportunities offered to the school this example first. learners. Regarding the first relationship, I asked Jana and Maia: Look at the board again. What about all these equa- Maia to reflect on their experiences in this regard. Both tions? Let’ see. What do all of them have in common? Which were clear about how much they have gained awareness of is to say what makes them equivalent? the importance of being explicit in the communication of 13 286 N. Planas mathematical meanings in classroom talk with learners. As in the teaching and in teacher education. This significance said by Jana: “Now I cannot stop thinking… did I explain particularly applies to classrooms with learners who are what I want them to learn or did it get taken for granted?” in the process of learning the language of instruction, and The PD practice thus seems to have mediated processes for who are more often than not disadvantaged by being offered teachers to become reflective on the importance of support- simplified, conceptually poor versions of mathematical lan- ing access to mathematical meaning through explicit talk. guage that hinder their understanding (Barwell et al. 2016). As I continue the analysis of the post-PD classroom teacher Throughout the paper I have briefly mentioned that the talk talk in collaboration with Jana and Maia, it becomes more of the teachers was bilingual, but I have not referred to the apparent that these teachers have much to say about their various home languages other than Catalan and Spanish language choices in teaching. in their classrooms. In spite of not having dealt with the multiple languages and the multilingual practices in the les- sons observed, or with the fluid bilingualism that operates 5 Possible paths for how to move in the PD context, it is important that the argument for lexi- on from here cal elaboration be further examined in articulation with the argument for flexible translanguaging in teacher talk. Most I have illustrated a very small-scale, unfinished project talk of Jana and Maia in their classrooms mixes Catalan of collaborative developmental work with two secondary and Spanish in ways that those holding linguistically purist school mathematics teachers aimed at improving their talk perspectives of language would not identify as precise or in teaching for the enhancement of content-based learning. adequate. We may thus need to be clearer about interpreting I do not have large-scale, long-term results as evidence of syntactic precision in teacher talk with regard to the math- teachers taking up practices in a wide, systematic way fol- ematics, not the language of instruction. But even so, the lowing participation in professional development programs unpacking of what precision implies for the use of language that are cyclical (e.g., the lesson study reported in Adler in the linguistically diverse mathematics classrooms is more and Alshwaikh 2019). Despite the limitations of the study at complex than it may seem. Just as for all learners and class- this stage, I have argued my point on how grammar features rooms, classroom researchers on mathematics and language of teacher talk can be interrogated, noticed and practised recommend not interpreting precision separately from the as effective ways of improving mathematics teaching and didactic potential of informal languages in communication of offering learners opportunities to hear mathematically and learning. While the use of precise talk in mathemat- relevant conceptual content. That said and in line with the ics teaching is key, studies in the field also tell us that low literature about the difficulties of lexical elaboration for levels of explicitness may function to connect the informal semantic accomplishment (Halliday 1978, 1985; Halliday languages of learners with the processes of making precise and Martin 1993), we cannot expect teachers to realize the mathematical meaning (Moschkovich 2008), and that some didactic potential of their classroom talk without sustained level of vagueness is inherent in language and communica- educational programs that allow them to develop this kind tion in all situations of mathematics teaching and learning of professional learning. Lexical elaboration for explicitness (Rowland 2000). Precision and explicitness are hence not in mathematical communication, as object of professional absolutes in the sense of being always desirable or reach- learning and kind of professional expertise, can be placed at able norms to be pursued or satisfied in classroom practice. the intersection of mathematical and pedagogic knowledge Important learning difficulties may arise due to low exposi- for mathematics teaching. tion to precise mathematical talk, but also due to restrictive Some opportunities of mathematics learning in the class- use of informal everyday meanings for the communication room rest upon a number of grammar choices in the teaching of the essential diverse ideas behind specific mathematical and the overt provision of mathematically precise sentences concepts. in teacher talk. In this respect, the attention to talk in teach- I cannot finish without a final expression of relativeness ing should become an object of explicit attention in sites of and caution. The body of research on knowledge for math- mathematics teacher education and professional develop- ematics teaching and mathematics teacher education tells ment. Practices of lexical elaboration through, for example, us that effective teaching for the learning of specific content adding explanations that give a clue as to the meanings of is not always informative if other contents are implied. This important concepts are being very productive also in the fact brings up several questions. Did/could the develop- second year of our PD with the participant teachers complet- mental work begun with school algebra have a side impact ing “you find an equivalent equation” with “which means on Jana and Maia’s everyday teaching? Did/could Jana and one with the same solutions.” Not only lexical elaboration Maia connect their developmental and teaching experi- of explanations, but also clarity in talk between the expla- ences around the concept of algebraic equivalence to the nation and what is explained are of outmost significance understanding of the teaching of other content? What we 13 How specific can language as resource become for the teaching of algebraic concepts? 287 know so far is that, along with the variability introduced Halliday, M. A. K. (1985). Introduction to functional grammar. London: by the context of culture and of interaction, content vari- Edward Arnold. Halliday, M. A. K., & Martin, J. R. (1993). Writing science: Literacy and ability increases the challenge of articulating language- discursive power. London: Falmer. responsive content-based studies on mathematics teaching Herbel-Eisenmann, B., & Otten, S. (2011). Mapping mathematics and mathematics teacher education. 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The co-emergence of machine tech- how to learn to communicate explicit meanings in class- niques, paper-and-pencil techniques, and theoretical reflection: A room teacher talk that articulate mathematical content and study of CAS use in secondary school algebra. International Journal contexts of practice. of Computers for Mathematical Learning, 11(2), 205–263. Morgan, C. (2006). What does social semiotics have to offer mathematics education research? Educational Studies in Mathematics, 61(1/2), Acknowledgements I thank the Catalan and Spanish Agencies for 219–245. https​://doi.org/10.1007/s1064​9-006-5477-x. funding SGR-2017-101, EIN2019-103213 and PID2019-104964GB- Morgan, C. (2014). Understanding practices in mathematics education: I00. My gratitude also goes to the teachers, learners and school that Structure and text. Educational Studies in Mathematics, 87(2), participated in this study. 129–143. https​://doi.org/10.1007/s1064​9-013-9482-6. Moschkovich, J. 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