‘Warrant’ revisited: Integrating mathematics teachers’ pedagogical and epistemological considerations into Toulmin’s model for argumentation

In this paper, we propose an approach to analysing teacher arguments that takes into account field dependence—namely, in Toulmin’s sense, the dependence of warrants deployed in an argument on the field of activity to which the argument relates. Freeman, to circumvent issues that emerge when we attempt to determine the field(s) that an argument relates to, proposed a classification of warrants (a priori, empirical, institutional and evaluative). Our approach to analysing teacher arguments proposes an adaptation of Freeman’s classification that distinguishes between: epistemological and pedagogical a priori warrants, professional and personal empirical warrants, epistemological and curricular institutional warrants, and evaluative warrants. Our proposition emerged from analyses conducted in the course of a written response and interview study that engages secondary mathematics teachers with classroom scenarios from the mathematical areas of analysis and algebra. The scenarios are hypothetical, grounded on seminal learning and teaching issues, and likely to occur in actual practice. To illustrate our proposed approach to analysing teacher arguments here, we draw on the data we collected through the use of one such scenario, the Tangent Task. We demonstrate how teacher arguments, not analysed for their mathematical accuracy only, can be reconsidered, arguably more productively, in the light of other teacher considerations and priorities: pedagogical, curricular, professional and personal

Simulated Atmospheric N Deposition Alters Fungal Community Composition and Suppresses Ligninolytic Gene Expression in a Northern Hardwood Forest

High levels of atmospheric nitrogen (N) deposition may result in greater terrestrial carbon (C) storage. In a northern hardwood ecosystem, exposure to over a decade of simulated N deposition increased C storage in soil by slowing litter decay rates, rather than increasing detrital inputs. To understand the mechanisms underlying this response, we focused on the saprotrophic fungal community residing in the forest floor and employed molecular genetic approaches to determine if the slower decomposition rates resulted from down-regulation of the transcription of key lignocellulolytic genes, by a change in fungal community composition, or by a combination of the two mechanisms. Our results indicate that across four Acer-dominated forest stands spanning a 500-km transect, community-scale expression of the cellulolytic gene cbhI under elevated N deposition did not differ significantly from that under ambient levels of N deposition. In contrast, expression of the ligninolytic gene lcc was significantly down-regulated by a factor of 2–4 fold relative to its expression under ambient N deposition. Fungal community composition was examined at the most southerly of the four sites, in which consistently lower levels of cbhI and lcc gene expression were observed over a two-year period. We recovered 19 basidiomycete and 28 ascomycete rDNA 28S operational taxonomic units; Athelia, Sistotrema, Ceratobasidium and Ceratosebacina taxa dominated the basidiomycete assemblage, and Leotiomycetes dominated the ascomycetes. Simulated N deposition increased the proportion of basidiomycete sequences recovered from forest floor, whereas the proportion of ascomycetes in the community was significantly lower under elevated N deposition. Our results suggest that chronic atmospheric N deposition may lower decomposition rates through a combination of reduced expression of ligninolytic genes such as lcc, and compositional changes in the fungal community

Mathematics teacher change in a collaborative environment: to what extent and how

This article reports on a study into how collaborative contexts influence the professional development of an early-career primary teacher, Julia. We describe the process of change by which Julia manages to make her planning to teach mathematics more flexible so as to adapt to student difficulties, and we analyse the role that joint reflection plays in promoting this change. In order to understand the how of this influence, we carried out an analysis of the interactions within the group from Julia’s point of view, following a dialogical approach to discourse. We believe that it is in and through the interactions that Julia constructs her interpretation of the opinions, critiques and suggestions expressed. This interpretation conditions the extent of her involvement and moulds the influence of the context on her professional development. The presence of skilled collaborators (Day, 1993) proved decisive in promoting this development

Competences of Mathematics Teachers in Diagnosing Teaching Situations and Offering Feedback to Students:Specificity, Consistency and Reification of Pedagogical and Mathematical Discourses

In the study we report in this chapter, we investigate the competences of mathematics pre- and in-service teachers in diagnosing situations pertaining to mathematics teaching and in offering feedback to the students at the heart of said situations. To this aim we deploy a research design that involves engaging teachers with situation-specific tasks in which we invite participants to: solve a mathematical problem; examine a (fictional yet research-informed) solution proposed by a student in class and a (fictional yet research-informed) teacher response to the student; and, describe the approach they themselves would adopt in this classroom situation. Participants were 23 mathematics graduates enrolled in a post-graduate mathematics education programme, many already in-service teachers. They responded to a task that involved debating the identification of a tangent line at an inflection point of a cubic function through resorting to the formal definition of tangency or the function graph. Analysis of their written responses to the task revealed a great variation in the participants’ diagnosing and addressing of teaching issues – in this case involving the role of visualisation in mathematical reasoning. We describe this variation in terms of a typology of four interrelated characteristics that emerged from the data analysis: consistency between stated beliefs/knowledge and intended practice, specificity of the response to the given classroom situation, reification of pedagogical discourses, and reification of mathematical discourses. We propose that deploying the theoretical construct of these characteristics in tandem with our situation-specific task design can contribute towards the identification – as well as reflection upon and development – of mathematics teachers’ diagnostic competences in teacher education and professional development programmes