Regulation of Teacher Elicitations in the Mathematics Classroom

Using the perspective of instructional conversation, we investigated how one teacher regulated student participation and conceptual reasoning in the middle-school mathematics classroom. We examined the elicitations—questions and provocative statements—made by the teacher over a four-day algebra lesson. Analyses showed how the teacher systematically regulated the level of cognitive complexity of his elicitations in reaction to students' responses. When students gave inaccurate or incomplete answers, the teacher tended to reduce the level of cognitive complexity needed to respond to a subsequent elicitation, with the apparent impact being that he scaffolded participation and reasoning. When students provided responses that were mathematically accurate, the teacher usually increased the elicitation level, which subsequently engaged students in more sophisticated forms of reasoning.     

The Process of Constructing Main Ideas From Text

A discourse-analytical approach to children's knowledge is offered, focusing on extracts of classroom talk between a teacher and a kindergarten class. After rejecting the possibility of defining concepts and memories as cognitive states prior to or underlying discourse, the talk is examined for how participants define and deal with such notions as a feature of the sequential and rhetorical organization of discourse. An argument is advanced for the analysis of knowledge, reality, and education as public, interactionally managed participants' concerns that can be studied as discursive practices. It is argued that the study of discourse does not ignore nondiscursive realms of mind and reality but permits analysis of how those things are defined, so that any explanatory appeal to what is beyond or behind the talk is unwarranted.     

An Oral Proof in a Geometry Class: How Linguistic Tools Can Help Map the Content of a Proof

Recommendations that teachers promote argument and discourse in their mathematics classrooms anticipate researchers' needs for methods for examining and analyzing such talk. One form of discourse is oral arguments, including proofs. We ask: How can we track the development of an oral argument by a teacher and her/his students? We illustrate a method that combines Systemic Functional Linguistics (SFL) and Toulmin's argumentation scheme to examine how speakers logically connect different parts of an argument. We suggest that conjunction analysis can aid a researcher to map the content of a proof that has been constructed in class discussion. Using data from a discussion of a geometry proof, we show that different types of conjunctions enabled the teacher and the students to connect various components of an argument and, also, different arguments. The article illustrates how conjunction analysis can support and deepen what Toulmin's scheme for arguments can reveal about oral discussions.     

Studying the organization in action: Membership categorization and interaction analysis

A current set of concerns in ethnomethodology and conversation analysis includes the question of how conversation analysis (CA) can deal with studies of social structure or studies of talk in institutional settings.In this paper a focus is placed on how the accomplishment of "work" and "categorization" are interrelated. Two particular instances are examined: a ski school and a package delivery service. Membership categorization is shown to be a complex, on-going, interactive accomplishment. The parties act in ways that are "predicatively-bound" (i.e. predicates of action, rights, obligations, etc.) which allow inferences to be made by each of the parties about the other based on these actions; these enable each to accept/confirm/validate the other's self-categorization and to produce, via their own actions, activities that are congruent with the other's self-categorization. Activities of the parties are category-relevant and category-generative. Thus, "work" or "the work of the organization", (e.g. for Choice, a package delivery service), is being accomplished in and through the talk and interaction of the parties.     

Power in the Classroom: How the Classroom Environment Shapes Students' Relationships With Each Other and With Concepts

Changes in participant structures in classroom environments are often examined in terms of their effects on student learning. In this study, we proposed a way of examining participant structures in terms of power. According to Wertsch (1998) "the emergence of new cultural tools transforms power and authority" (p. 65). When researchers or teachers introduce new tools into classrooms including new participant structures, they create the potential for transforming many relationships of power: between students and teachers, among students, and between students and the material being studied. Using data from a 6th-grade classroom involved in a science unit, we consider how these transformations of power play out. In considering the role of participant structures, we look at how a match between the participant structures and the structure of the discipline can positively affect these relationships of power.     

Dramatic conversation: An exploratory application of Burton's discourse model

The British sociolinguists Sinclair and Coulthard have generated a systematic approach to the analysis of natural conversation using classroom discourse. Burton has articulated an innovative development of this model that she claims is applicable to a broad range of spoken discourse. She focuses on the language of dramatic plays. Her model is explained, and in an attempt to assess its viability, an application to a brief exchange in a Tennessee Williams play and its Italian translation is proposed. Interesting differences between the interactive structures of the two texts are revealed, indicating that Burton's model may be fruitful for the analysis of fictional dialogue. However, theatre talk may differ in as yet unknown ways from natural conversation.     

Sacred and secret stories in professional knowledge landscapes: learner agency in teacher professional learning

Set in a schooling practitioner research context, this article critiques instrumental forms of teacher professional development. An account of teacher professional learning and development is provided, as an inter-relational construct, where teachers reflect on practice to inquire into their students’ learning and foster their own and others specific interests and talents in creative ways. In an Aotearoa/New Zealand context, teachers are mandated to inquire into what their students need to know and understand, the strategies required for teachers to attain these learning goals, and how effective the approach implemented is in enhancing student learning. Emerging from qualitative case study research, a teacher’s narrative account of a student-initiated critical thinking episode in a secondary science classroom is analysed to demonstrate sacred, secret and cover stories of professional knowledge landscapes. Findings highlight the importance of agentic teacher-enacted forms of practitioner inquiry that support critical and collaborative reflective practice.     

Intertextuality in Read-Alouds of Integrated Science-Literacy Units in Urban Primary Classrooms: Opportunities for the Development of Thought and Language

The nature and evolution of intertextuality was studied in 2 urban primary-grade classrooms, focusing on read-alouds of an integrated science-literacy unit. The study provides evidence that both debunks deficit theories for urban children by highlighting funds of knowledge that these children bring to the classroom and the sense they make of them and demonstrates how intertextuality in classroom discourse creates learning opportunities as teachers and students interact with texts broadly defined. Changes over time of the various forms of intertextuality emerged in the 2 classes, thereby showing how intertextuality is both an act of discourse and an act of mind. A particular type of intertextuality-event intertextuality-dominated throughout the read-alouds and revealed how hybrid classroom discourse, incorporating both narrative and scientific genre features, allowed the emergence of classroom discourse that shared characteristics with scientific language.     

“Can We Say the N-word?”: Exploring Psychological Safety During Race Talk

Classroom interactions that engender psychologically safe climates have rich developmental potential; these interactions often facilitate engagement in academic risk-taking and exploration (Cooper, 2013). On the contrary, classroom interactions that are threatening to adolescents’ identities send messages of devaluation and can negatively impact feelings of belonging and academic achievement, particularly for youth of color (Sherman et al., 2013). In efforts to help explicate the interrelations between racial-identity linked risks and threats and psychological safety in classroom learning contexts, this exploratory case study examined two 9th grade English Language Arts teachers’ negotiation of attunement, authenticity, and power sharing during race talk. Our in-depth analysis of these classrooms suggested the following teacher moves may be particularly important for psychological safety during race talk: teacher moves that (a) support or inhibit power sharing, (b) indicate high or low levels of teacher attunement, and (c) demonstrate or mask the teacher’s authenticity. These results have implications for understanding the co-construction of psychological safety in classrooms during identity-salient and racialized interactions.     

Observation of children with attention‐deficit hyperactivity (ADHD) problems in three natural classroom contexts

Background. Observation studies of students with attention‐deficit hyperactivity disorder (ADHD) problems in natural classroom situations are costly and relatively rare. Aims. The study enquired how teacher ratings are anchored in actual student classroom behaviours, and how the behaviour of children with ADHD problems differs from their classmates. The authors attempted to broaden the usual focus on disruptive and inattentive behaviours to elucidate the role of various on‐task behaviours, as well as considering differences between classroom contexts. Sample. DSM‐III‐R criteria were used in conjunction with a teacher rating scale to select a sample of 55 students with ADHD problems, and 55 matched controls from a population of 569 primary school students. Method. Students were observed in their natural classrooms using the Munich Observation of Attention Inventory (MAI; Helmke, 1988 ). Correlations between teacher reports and observation codes were computed, and systematic differences between students with ADHD problems and controls in different classroom contexts were examined using a generalized linear mixed model (GLMM). Results. Global teacher reports showed moderate to strong correlations with observed student behaviours. Expected on‐task behaviour demonstrated the strongest relationship (r >?.70) with teacher reports. As hypothesized, the children with ADHD were more disruptive and inattentive than their matched peers. They were also less often inconspicuous on‐task as expected by their teachers. However, their behaviour was assigned to two other on‐task categories more often than their peers, and this raised their total on‐task behaviour to over 66%. Situational differences were found for all codes as well, which mostly affected all students in a similar way, not just students with ADHD. Conclusions. ADHD related behaviours are pervasive across the classroom situations coded. Teachers appear to distinguish between desirable and undesirable on‐task behaviours. Nevertheless, assisting students with ADHD problems requires shaping both. Future studies need to include more differentiated codes for various types of on‐task behaviours and also need to code the lesson context concurrently.     

Physical education and fostering relationships

Rather than presenting an academic paper, I wanted to simply examine my own perspective as a physical educator and classroom teacher and the importance of creating relationships with children. As a relatively new physical educator and recent Masters of Education graduate of the University of Toronto at OISE, but experienced classroom teacher working in a Toronto public school, spirituality at first appeared to be the farthest thing that affected both my life and the life of my classes. During the last two years, I became increasingly aware of the connection between physical education and feelings of enthusiasm and perseverance that have helped my students to see themselves in positive ways. The relationship developed between teacher and student had been apparent to me but I had not realised how important until I began graduate school and reflecting on both my classroom practice and how it had extended into the gymnasium. I often thought that physical education teachers tended to focus on the physical aspects and skills, but instead I found that it was indeed making a shared connection with my students through the various physical and everyday activities as of the highest importance. The shared connections encourage feelings of perseverance, and fit together between mind, body and spirit, which also encouraged active participation and success. What I had suspected to be true for the classroom really was true anywhere in the school community. My experiences as a physical educator and classroom teacher helped me create positive learning environments for children as they struggled with academic and physical activities. With this paper, I am asking academics to help teachers such as myself to understand the correlation to something we as teachers take for granted as part of our everyday teacher–student relationship.     

Mathematical Reasoning: Adults' Ability to Make the Inductive-Deductive Distinction

In this study, I examined adults' ability to distinguish necessary deductive and indeterminate inductive forms of argument in mathematics. Only 30% of a sample of college students distinguished deductive and inductive forms of argument and experienced deductively derived conclusions as necessary and inductively derived conclusions as uncertain. Forty percent failed to distinguish deductive and inductive forms and experienced inductively derived and deductively derived conclusions as necessary. Thirty percent distinguished deductive and inductive arguments but experienced deductively derived and inductively derived conclusions as uncertain. As observed in other reasoning domains, the introduction of personal beliefs or knowledge about the argument content appeared to affect adult reasoners' application of knowledge about forms of argument and judgments of necessity. The results suggest the following conclusions. Adults' experience of the conclusions from mathematical inductive and deductive arguments as provisional conclusions or necessary conclusions depends on a complex coordination involving ability to attend to abstract premise-conclusion relations and beliefs about the nature of mathematical objects and regularities. Thus, two major achievements are involved, explaining the low numbers able to judge necessity in mathematics.     

Indispensability and Holism

It is claimed that the indispensability argument for the existence of mathematical entities (IA) works in a way that allows a proponent of mathematical realism to remain agnostic with regard to how we establish that mathematical entities exist. This is supposed to be possible by virtue of the appeal to confirmational holism that enters into the formulation of IA. Holism about confirmation is supposed to be motivated in analogy with holism about falsification. I present an account of how holism about falsification is supposed to be motivated. I argue that the argument for holism about falsification is in tension with how we think about confirmation and with two principles suggested by Quine for construing a plausible variety of holism. Finally, I show that one of Quine’s principles does not allow a proponent of mathematical realism to remain agnostic with regard to how we establish that mathematical entities exist.     

The role of the degree of acquaintance with teachers on students’ interpersonal perceptions of their teacher

Some researchers suggest that the way a teacher is perceived during the first encounter with a classroom group resembles, and therefore is predictive of, later classroom processes as perceived by students. This study investigated the degree to which interpersonal ratings (i.e., control and affiliation), given by secondary school students to teachers unknown to them, are comparable to ratings given by students who are well acquainted with that teacher, by means of a quasi-experimental design. Ratings concerning a teacher students had not met before, based on a 5 min video vignette, were compared to ratings given by students who had been taught by the same teacher for at least a year (N $\,{=}\,$ 489). Students at no acquaintance perceived a teacher’s interpersonal traits in a similar way as students who were familiar with that teacher. It is concluded that students’ early interpersonal perceptions of a teacher largely resemble perceptions formed by students that know this teacher well.     

Tell Us More: Reading Comprehension,Engagement, and Conceptual Press Discourse

This study examines interactions between teachers and students during reading comprehension instruction to determine how certain patterns of teacher–student talk support student comprehension achievement and reading engagement. The central focus of the study is conceptual press discourse, a pattern of teacher response that includes requests for evidence, examples, clarification, and elaboration. Hierarchical Linear Modeling analysis of data from 21 fourth- and fifth-grade classrooms (495 students) indicated that in classrooms where teachers more frequently used discourse patterns that reduced conceptual press, students demonstrated weaker comprehension and engagement outcomes.     

Equal Sharing and Shared Meaning: The Development of Fraction Concepts in a First-Grade Classroom

Work on learning from a situative point of view suggests that to understand how children's thinking about a domain such as fractions develops, research needs to take into account the socially organized ways of thinking in which children participate during instruction. The aim of the study reported in this article was to explore children's fraction learning in a 1st-grade classroom in which the teacher elicited and built on children's informal knowledge of fractions. Instruction revolved around children's solutions and discussions of equal-sharing tasks. The study provides an account of children's learning that examines the relation between classroom talk and children's evolving fraction concepts, with a focus on the analysis of several key classroom interactions that resulted in cognitive change. Pretests and posttests indicated that children's understanding of fractions changed in important ways. The results suggest that how children think about fractions is influenced not only by how their own knowledge is structured but, perhaps more profoundly, by how the context for thinking about and discussing fractions is structured.