Schleppegrell, M. (2012). Linguistic tools for exploring issues of equity. In B. HerbelEisenmann, J. Choppin, D. Wagner & D. Pimm (Eds.), Equity in Discourse for Mathematics Education: Theories, Practices, and Policies (pp. 109124). New York: Springer.
 Methods: systemic functional linguistics1 (SFL), a theory of language that recognizes that language var ies according to the context of use (Halliday 1978; Halliday and Matthiessen 2004)…A key notion in SFL is that language is a powerful means of construing our social reality and of enacting social relationships. Speakers and writers are constantly mak ing choices as they use language.
 Analysis using linguistic tools helps us see how mathematics is pre sented to students, how learning is accomplished in mathematics classroom discourse and how different kinds of participation result in different kinds of learning.(110)
 What is the nature of the mathematics being offered to students through class room discourse and pedagogical materials? What views of mathematics and of mathematical activity do students construct as they participate in learning? These questions address equity issues related to the nature of the mathematics being offered to and taken up by students, illuminating issues of access and achievement. They can be explored by analyzing the thematic patterns in the ways mathematics is construed by teachers, students and texts and by exploring the process–participant configurations and modality in the language used in mathematics classrooms.(110111)
 What are the processes through which knowledge is developed? How are students positioned as learners through classroom and pedagogical discourses? How does teacher interaction with students mediate their learning? (111)
 Linguistic approaches such as analysis of process–participant configurations can identify the ways students are positioned by the discourses they engage in, while analysis of modality and mood/ speech function reveals how students’ actions are regulated by the teacher as they learn mathematics.
Findings:
 One linguistic approach to exploring these issues is analysis of thematic pat terns. Thematic patterns are the relationships between the constructs being learned that are built up by teachers and students in interaction or are presented in curricu lum materials as particular topics are developed. Analysis of thematic patterns offers a method for identifying key concepts and seeing how they are presented to students and how students take them up, providing a powerful tool for investigating the knowledge that is being made available to students and the learning that is taking place. It can answer questions such as: Is the mathematics that students are being offered accurate, appropriate and challenging? Does it connect with their prior knowledge and help them move toward more complex understandings? (111)
 Chapman tracks the evolution of the key mathematical concepts by identifying the grammatical participants (nouns and noun phrases italicized in the excerpts above) and the relationships between them that are constructed in the discourse through the grammatical processes they are involved in (verbs and verb phrases underlined in the excerpts above). Identifying grammatical processes and participants and creating schematic representations of thematic patterns offer a means of exploring how mathematical meaning is built up over time in classroom discourse. It has the potential to illuminate aspects of the mathematics that students are presented with, as well as providing a means to chart their taking up of mathematics concepts and to trace where their understanding is different from that of the teacher or the textbook (113)

Thematic pattern analysis provides evidence of the integrity of the mathemat ics that is construed in the classroom and thus can be used to illuminate equity issues related to access and achievement. Through analysis of the language of the textbook materials, for example, the ways mathematics concepts are presented in the official curriculum can be explored and thematic patterns developed in different kinds of classrooms and different contexts of learning can be compared, in order to look closely at the nature of the mathematics that is being offered to students (for an example of this, see HerbelEisenmann and Otten 2011). This helps us recognize whether or not students in different contexts have equal access to under standing mathematical meanings. (113)
In addition, the thematic patterns in the mathematics students bring from their home contexts can be compared with the thematic patterns of the ‘official’ mathematics in order to recognize how mathemat ics itself can be construed in various ways. The analysis of thematic patterns can assist in evaluating equity in access and achievement by revealing the mathematics represented in the discourse of teachers, students and materials.
 From a functional linguistics perspective, processes can be categorized into different types, based on their meanings and the grammatical configurations they present.2 Both instructional materials and classroom discourse can be analyzed to identify the kinds of processes that are prominent and what they are accomplishing for the speaker/ writer, in order to recognize how pedagogical materials represent mathematics and how those “official” representations are taken up in the classroom (e.g. HerbelEisenmann 2007; HerbelEisenmann and Wagner 2005; Morgan 2005). For example, researchers can explore who is acting and thinking at different points in the discourse or who is being directed to act or think in some way, by identifying and comparing the use of material and mental processes and the participants in them. Material processes are processes that are ‘external’; doing processes that describe actions or happenings in the world. Mental processes are ‘internal’; processes of thinking, perceiving, feeling and experiencing in human consciousness. Researchers can examine relational processes, processes of defining, describing, having and being, to explore what is defined and described. Each of these different process types has its own set of participant roles(114115)

The student text, on the other hand, obscures the agency involved in defining through use of the passive voice; we are not told who gives the ratio a special name. The analysis of the rela tionships between processes and participants also yields other interesting findings here. Morgan observes that, after presenting the definition, the authors of the math ematics research paper use the nominalization this viewpoint (a nominalization is the linguistic construal of a process as a thing; here, the giving of the somewhat non standard definition becomes this viewpoint). The viewpoint is then an agent in the process makes the actions transparent; in other words, this abstraction is presented as an actor, performing the process of making the action more or less transparent. (116)
 Morgan suggests that when textbooks use procedural discourse as an organizational strategy, constructing the students as actors who are to perform particular actions, they “may make students more likely to perceive mathematics as consisting of a set of procedures and hence, perhaps, to find it more difficult to engage with relational or logical aspects of the subject” (Morgan 2006, p. 228). Texts that obscure human agency, on the other hand, “may contribute to difficulties for some students in seeing themselves as potential math ematicians” (p. 228).

when the teacher asks What question would I be asking myself in my head as I start that problem? (p. 29), the teacher is develop ing in students the notion that the solving of the problem is also engaging them in a thinking process and that this linking of reflection and action is part of what math ematics is about. This, along with other key moves that the teacher makes, positions the students as mathematical thinkers and positions mathematics as something that is about meaning, has reasons and requires engagement in reasoning and explain ing. (See also González 2011, for examples of this kind of analysis related to what geometry teachers expect students to be responsible for.) (117)
 A linguistic construct that is often used for this purpose is modality. Modality is the linguistic system used to mark degrees of possibility, usuality, obligation, inclination and ability. Analysis of the modality of possibility (what may or might be) helps us explore the strength of claims being made and whether knowl edge is being presented as contingent or absolute. Analysis of the modality of obli gation (what must, should, or ought to be done) helps us explore what is being construed as necessary and, consequently, how learners are being regulated. Analysis of the modality of ability (what can or could be) helps us look at how learners are being positioned as competent to engage in particular activities. Modality can be realized in modal verbs (e.g. can, could, may, might, must, ought, should, etc.), as well as in other language structures such as adverbs (e.g. maybe, usually). (118)

Morgan’s analysis of modality across a range of texts enables her to argue that in academic mathematics and higherlevel textbooks, definitions are presented as dynamic and evolving, open to decisionmaking by the mathematician and incorporating ambiguity, while lowerlevel textbooks typically present defi nitions as static. She shows how professional mathematics texts are very different from school texts in the kind of reader they construct and that pedagogical texts aimed at different levels of students also vary in the positioning of the reader as more or less invited into or involved in the decisionmaking activity that the text constructs (119)
 Linguistic constructs that are useful for analysis of classroom interaction include mood and speech function. Mood is a general linguistics construct, referring to the three possibilities for structuring a clause in terms of interaction: declarative, inter rogative and imperative. Speech function refers to the meaning made by the mood choice, which is not always congruent with the grammatical form, as different moods can construe the same speech function, creating variation in discourse that is related to power and authority, as well as cultural conventions and practices. For example, to request that a student do something, a teacher may use any of the three grammatical moods, saying Please take out your books (imperative); Would you take out your books? (interrogative); I’m looking around for people who have their books out (declarative) (120)

Analysis of thematic patterns, pro cess–participant configurations and modality enables researchers to explore the integrity of mathematics itself and also how concepts are developed over time in pedagogical discourse, as well as to examine how the mathematics is presented; for example, whether as contingent or absolute. Analysis of process–participant con figurations, modality and mood/speech function enables researchers to explore the processes through which knowledge is developed, focusing on the agency of stu dents and the authoritativeness of the teacher, as well as the role of the teacher as mediator of learning, and how students are positioned as learners through classroom and pedagogical discourses. (122)
 More research is needed to explore patterns of discourse in different settings and with diverse groups of students, analyzing the language used in mathematics class rooms and in mathematics pedagogical materials to illuminate the equity issues dis cussed in this chapter, as well as other issues of importance. We need more research that compares texts of different types, reveals problems with different wording and analyzes representations of mathematics at different levels and in different topics. We also need better understanding of how mathematics knowledge evolves over a unit of study of different topics at different levels to help teachers move between everyday and technical ways of making mathematical meaning. Analyses of the texts used for instruction and the ways teachers and students use and respond to those texts can identify challenges in the language and suggest ways to enable more effective presen tation of content and how best to engage students in diverse contexts.

A key issue in the study of equity is to identify forms of discourse that bring together the mathematics that enables students to achieve with ways of engaging with this mathematics that empower students to question critically – forms of dis course that afford students agency in their own learning and engage them with a mathematics that is open to exploration and discussion. (123)
 Related to the learner, linguistic analysis can reveal differences in positioning of students and show how different contexts for learning mathematics may be more effective for particular groups of learners than others. The language analysis tools described here offer mathematics classroom researchers evidence that illuminates issues of access, achievement, power and identity, revealing the nature of the knowledge that teachers make available in mathematics classrooms and how stu dents take up new knowledge in the context of actually doing mathematics, as well as the ways students are positioned and engaged as they learn.
Posted on September 21, 2012 by B.J.D.Armas
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