Discursive dimension
The discursive dimension refers to the prevalence of episodes of the interaction in which rich discourse practices (i.e. explaining and arguing in contrast to telling and reporting) are made relevant. This includes that students are demanded to contribute with more than single words or half sentences, and their utterances are qualified according to their discursive complexity.Thus, this dimension does not directly refer to the discussed mathematical content, but relies on a linguistic analysis and characterization of the interaction since it is incorporated from Interactional Discourse Analysis (Erath et al., 2018). However, in the present study, the notion-rich discourse practices always imply a mathematical focus of the conversation.
Nevertheless, the hypothesized importance of the discursive dimension is deduced from the conceptual dimension that is in the heart of mathematics education and points out that meaningful learning of mathematics cannot be separated from and is intertwined with rich discourse (Erath, 2017; Lampert & Cobb, 2003).
The quality of classroom discourse has been the focus of classroom studies for many years. As early as 2001, Brophy characterized quality teaching by teachers who invite students to “develop explanations, make predictions, debate alternative approaches to problems ... [and] clarify or justify their assertions” (Brophy, 2001, p. 13), hence in a supply perspective, and emphasized the need to involve all students.
Erath (2017) highlights that on the one hand especially explaining and arguing are linguistically more demanding than reports or descriptions, but on the other hand these challenging discourse practices are connected to talking about conceptual knowledge (Hiebert & Lefevre, 1986) and thus are important discourse practices for a meaningful learning of mathematics. This conceptualization of the discursive dimension is based on the understanding that discourse practices always serve to solve a communication-related problem in a speech community like math classrooms (Erath et al., 2018, displaying the perspective of Interactional Discourse Analysis): For instance, explaining serves to convey or construct knowledge, argumentation serves to negotiate divergent validity claims and reporting serves to retrospectively articulate actions and processes (ibid., p. 168). Thus, the definition of the discursive dimension is in line with research in mathematics education that can be framed as a participationist perspective in mathematics classrooms (Krummheuer, 2011; Sfard, 2008; referring to Vygotsky, 1978). In this perspective, learning mathematics is intertwined with “a process of enculturation into mathematical practices, including discursive practices (e.g. ways of explaining, proving, or defining mathematical concepts)” (Barwell, 2014, p. 332).
Quantitative studies (Hiebert et al., 2003; Pauli & Lipowsky, 2007) cannot easily evaluate the discursive quality as this requires a highly inferential rating. As a compensation, these large video studies have included the length of students’ utterances as a first proxy to the level of students’ participation in discourse practices. As many teachers do not provide many opportunities for long utterances in the observed lessons, there is only a limited probability that students actively participate in richer discourse practices (irrespective of its demands) that exceed filling smaller slots. Furthermore, Pauli and Reusser (2015) analyze video data from the PYTHAGORAS study and show that teachers’ questions and elicitations were mainly simple ‘recall questions’ and that only a very small number of student statements included justifications. Otherwise, Erath et al. (2018) show, in their small quantitative investigation, that in the observed mathematics classrooms, teacher demands for explanations and argumentations were most frequent. Nevertheless, students’shares of active participation in whole-class discourse practices varied considerably between classrooms.
Qualitative studies in the discursive dimension emphasize the importance of rich discourse practices for fostering mathematical understanding (e.g. Barwell, 2012; Erath et al., 2018; Moschkovich, 2015), especially for language learners. Mostly, this issue is addressed in multilingual contexts (e.g. Moschkovich, 2015; Zahner et al., 2012), but increasingly expanded to also include monolingual children with low language proficiency (Prediger & Wessel, 2018). In each case, students’ passive (by listening to the conversation) or active (by talking) participation in rich discourse practices have been shown to be a condition for deep mathematics learning. This resonates with results from qualitative research on ‘argumentation’ in mathematics classrooms (for an overview for the supply perspective, see Walshaw & Anthony, 2008, p. 535; examples for more interaction-based studies are Krummheuer, 2011; Yackel & Cobb, 1996) that use different definitions of argumentation that can be subsumed under the notions of explaining and arguing as defined above. All of these qualitative studies emphasize that the teacher’s steering alone is not sufficient to characterize discourse quality, the interactionist nature of discourse requires to include the interactive mechanisms of co-construction. As a consequence, the supply perspective refers not only to teachers’ moves, but also to students’ participation of all students together.
Summing up, the discursive dimension of classroom interaction quality refers to individual students’ active participation in rich discourse practices (use perspective). The supply perspective can be operationalized, on the one hand, by the discursive demands of tasks and teacher moves and, on the other, by the average students’ participation. This is included as a variable for supply quality since discourse is always co-constructed in the interaction, and listening to rich discourse practices can be considered as a supplied learning opportunity.
Lexical dimension
The lexical dimension of classroom interaction quality refers to the ways in which language learners are supported on the lexical level and to what extent students take on the offered lexical means with respect to developing their academic and technical vocabulary in discursively rich situations.
The lexical dimension has been shown to be particularly relevant for students with still limited proficiency in the language of instruction, because when chosen wisely and introduced in sustainable ways, the vocabulary can serve as a tool for participating in discourse practices like explaining, arguing, describing, etc. (e.g. Gibbons, 2002; Prediger & Wessel, 2013; Smit et al., 2013).
As language education research has identified, the quality in the lexical dimension depends on the degree of discursive embeddedness of all vocabulary focus. That means that vocabulary should not be offered and demanded for their own sake and in isolated ways, but by lexical support that is integrated in meaningful communicative settings such as authentic discussions (Lawrence, Crosson, Pare-Blagoev, & Snow, 2015).These quality criteria have also been reconfirmed in qualitative mathematics education research on language learners (e.g. Smit et al., 2013).
Additionally, the selection of vocabulary is crucial: It should contain not only technical vocabulary, but also the meaning-related vocabulary required by discourse practices in interactive meaning constructions. Vocabulary must be offered by rich input and good opportunities of constructing meanings (supply perspective), but their learning by the students also depends on repeated opportunities for probing and practicing them (Lawrence et al., 2015). Students have different intensities in which they uptake the vocabulary; this has been made visible by so-called trace analyses which explored the traces of students’ individual and collective vocabulary inventions and uptakes (Prediger & Pohler, 2015).
The supply perspective of the lexical dimension is thus captured by teachers’ moves to introduce, demand and support the lexical means (e.g. investigated by Smit et al., 2013), and the use perspective by the degree to which the students adopt the newly introduced lexical means (e.g. studied by Prediger & Pohler, 2015).
