‘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

'Don't think in your head, think aloud': ICT and exploratory talk in the primary school mathematics classroom

This paper arises out of research into classroom activities conducted with Year 5 and Year 6 primary school students (9-10 year-olds). The study applied the ‘Thinking Together’ approach developed by Mercer and colleagues at the Open University in mathematics lessons involving the use of ICT. The study describes the use of mathematics software to promote collaborative thinking and exploratory talk in the mathematics classroom. Teachers were given training in the Thinking Together approach. They then conducted a series of lessons with students and explicitly taught them how to work and talk collaboratively to solve mathematical problems at the computer. These lessons were video-recorded and the transcripts analysed for evidence of ‘exploratory talk’. This paper reports on the role of the teacher, the students and the computer in developing exploratory talk

Practising Mathematics Teacher Education: Expanding The Realm of Possibilities

It is often said that student teachers’ underlying beliefs of what mathematics consists of and how it should be taught are restricted in two ways. On the one hand, future elementary teachers in general use only weak mathematical conceptions, which often do not help them to realise their educational ambitions. On a general educational level, many of these students advocate discovery learning and collective problem solving, but when it comes down to the mathematical activities that have to be prepared, their experience of “traditional” school mathematics is of little help. On the other hand, future (higher) secondary teachers mostly are very well prepared with respect to formal academic mathematics when entering mathematics education programmes, either because they have already passed a mathematical formation at university or because their teacher education programmes emphasise the study of academic mathematics and not of educational or didactical modules

Measurement of Two-Qubit States by a Two-Island Single Electron Transistor

We solve the master equations of two charged qubits measured by a single-electron transistor (SET) consisted of two islands. We show that in the sequential tunneling regime the SET current can be used for reading out results of quantum calculations and providing evidences of two-qubit entanglement, especially when the interaction between the two qubits is weak

Spin-based all-optical quantum computation with quantum dots: understanding and suppressing decoherence

We present an all-optical implementation of quantum computation using semiconductor quantum dots. Quantum memory is represented by the spin of an excess electron stored in each dot. Two-qubit gates are realized by switching on trion-trion interactions between different dots. State selectivity is achieved via conditional laser excitation exploiting Pauli exclusion principle. Read-out is performed via a quantum-jump technique. We analyze the effect on our scheme's performance of the main imperfections present in real quantum dots: exciton decay, hole mixing and phonon decoherence. We introduce an adiabatic gate procedure that allows one to circumvent these effects, and evaluate quantitatively its fidelity

Forming conjectures within a spreadsheet environment

This paper is concerned with the use of spreadsheets within mathematical investigational tasks. Considering the learning of both children and pre-service teaching students, it examines how mathematical phenomena can be seen as a function of the pedagogical media through which they are encountered. In particular, it shows how pedagogical apparatus influence patterns of social interaction, and how this interaction shapes the mathematical ideas that are engaged with. Notions of conjecture, along with the particular faculty of the spreadsheet setting, are considered with regard to the facilitation of mathematical thinking. Employing an interpretive perspective, a key focus is on how alternative pedagogical media and associated discursive networks influence the way that students form and test informal conjectures