‘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

Teaching and Learning of Calculus

This survey focuses on the main trends in the field of calculus education. Despite their variety, the findings reveal a cornerstone issue that is strongly linked to the formalism of calculus concepts and to the difficulties it generates in the learning and teaching process. As a complement to the main text, an extended bibliography with some of the most important references on this topic is included. Since the diversity of the research in the field makes it difficult to produce an exhaustive state-of-the-art summary, the authors discuss recent developments that go beyond this survey and put forward new research questions

Structural Transformations of Ni<inf>1 –</inf><inf>x</inf>Cu<inf> x</inf>Fe<inf>2</inf>O<inf>4</inf> Nanoparticles Depending on the Number of Cu Ions

© 2020, Pleiades Publishing, Ltd. Abstract: Changes in the structure and properties of the Ni1 – xCuxFe2O4 ferrites-spinel magnetic nanoparticles depending on the Cu ion concentration (0 ≤ x ≤ 1.0) have been studied using Mossbauer spectroscopy. It has been found that, with the increasing of the Cu ion concentration, the structure of nanoparticles changes from the structure of reverse spinel (NiFe2O4) to that of mixed spinel (CuFe2O4). It has been shown that the hydrothermal synthesis method makes it possible to obtain single-phase nanoscale particles with a very narrow size distribution and ideal magnetic ordering, which are promising for biomedical applications. The relationship between the distribution of cations on sublattices, the value of the inversion parameter, and the Cu ion concentration has been established