Using tasks to explore teacher knowledge in situation-specific contexts

Research often reports an overt discrepancy between theoretically/out-of context expressed teacher beliefs about mathematics and pedagogy and actual practice. In order to explore teacher knowledge in situation-specific contexts we have engaged mathematics teachers with classroom scenarios (Tasks) which: are hypothetical but grounded on learning and teaching issues that previous research and experience have highlighted as seminal; are likely to occur in actual practice; have purpose and utility; and, can be used both in (pre- and in-service) teacher education and research through generating access to teachers’ views and intended practices. The Tasks have the following structure: reflecting upon the learning objectives within a mathematical problem (and solving it); examining a flawed (fictional) student solution; and, describing, in writing, feedback to the student. Here we draw on the written responses to one Task (which involved reflecting on solutions of x+x−1=0 of 53 Greek in-service mathematics teachers in order to demonstrate the range of teacher knowledge (mathematical, didactical and pedagogical) that engagement with these tasks allows us to explore

Teacher beliefs and the didactic contract on visualisation.

This paper explores secondary teachers’ views on the role of visualisation in the justification of a claim in the mathematics classroom and how these views could influence instruction. We engaged 91 teachers with tasks that invited them to: reflect on/solve a mathematical problem; examine flawed (fictional) student solutions; and, describe, in writing, feedback to students. Eleven teachers were also interviewed. Here we draw on the interviews and the responses to one Task (which involved recognising a line as a tangent to a curve at an inflection point) in order to explore the influence on the teachers’ feedback to students of: persistent images of the tangent line; beliefs about the sufficiency of a visual argument; and, beliefs about the role of visual arguments in student learning. We focus particularly on the influence on the didactical contract regarding mathematical reasoning that teachers with a variation of beliefs about the role of visualisation are likely to offer their students. We conclude with a concise description of a didactical contract which maintains a role for proof in the mathematics classroom that is not disjoint from the creative parts of visually-based classroom activity and that reflects an essential intellectual need. We also conclude with crediting the combination ‘task engagement-followed by-interview’ for the identification of subtle issues regarding the teachers’ pedagogical and epistemological beliefs and for the raising of their awareness of these issues

On-Surface Reaction between Tetracarbonitrile-Functionalized Molecules and Copper Atoms

Self-assembly at surfaces has proven to be very efficient in creating ordered, atomically controlled organic nanostructures. On-surface synthesis has recently emerged as a promising strategy to create stable structures bound by strong and irreversible covalent bonds. Here we present on-surface reaction between pyrazino phenanthroquinoxaline-tetracarbonitrile (PPCN) molecules and copper atoms on a Au(111) substrate. The reaction is monitored in ultrahigh vacuum conditions by scanning tunneling microscopy and X-ray photoelectron spectroscopy (XPS). After a 475 K annealing, phthalocyanine cyclization occurs around a copper atom; the increase of annealing temperature to 540 and 675 K leads to the formation of 1D and 2D phthalocyanine polymers, respectively. This reaction is confirmed by the modification of the Cu 2p_3/2 and C 1s XPS spectra upon annealing