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The belief that mathematics ability is a fixed trait is particularly common and may be a key reason for many students' disinterest and underperformance in mathematics. This study investigates how mathematics teaching practices might contribute to students' beliefs about mathematics ability being a fixed or malleable trait (mindset). Through a synthesis of existing literature and an analysis of data from classroom observations, this article presents a framework of teaching practices and identifies how varying implementations of these practices can be classified along a continuum from conveying fixed-mindset messages to conveying growth-mindset messages related to mathematics ability.

This paper presents a major reformulation of the standard theory of Fowler–Nordheim (FN) tunnelling and cold field electron emission (CFE). Mathematical analysis and physical interpretation become easier if the principal field emission elliptic function

This article introduces an interview-based instrument that was created for the purposes of characterizing the visions of high-quality mathematics instruction of teachers, principals, mathematics coaches, and district leaders and tracking changes in those visions over time. The instrument models trajectories of perceptions of high-quality instruction along what have been identified in the literature as critical dimensions of mathematics classroom practice.

Given a grid presentation of a knot (or link) K in the three-sphere, we describe a Heegaard diagram for the knot complement in which the Heegaard surface is a torus and all elementary domains are squares. Using this diagram, we obtain a purely combinatorial description of the knot Floer homology of K.

For any quantum algorithm operating on pure states, we prove that the presence of multi-partite entanglement, with a number of parties that increases unboundedly with input size, is necessary if the quantum algorithm is to offer an exponential speed-up over classical computation. Furthermore, we prove that the algorithm can be efficiently simulated classically to within a prescribed tolerance η even if a suitably small amount of global entanglement is present. We explicitly identify the occurrence of increasing multi-partite entanglement in Shor's algorithm. Our results do not apply to quantum algorithms operating on mixed states in general and we discuss the suggestion that an exponential computational speed-up might be possible with mixed states in the total absence of entanglement. Finally, despite the essential role of entanglement for pure-state algorithms, we argue that it is nevertheless misleading to view entanglement as a key resource for quantum-computational power.

We tested whether informing women about stereotype threat is a useful intervention to improve their performance in a threatening testing situation. Men and women completed difficult math problems described either as a problem-solving task or as a math test. In a third (teaching-intervention) condition, the test was also described as a math test, but participants were additionally informed that stereotype threat could interfere with women's math performance. Results showed that women performed worse than men when the problems were described as a math test (and stereotype threat was not discussed), but did not differ from men in the problemsolving condition or in the condition in which they learned about stereotype threat. For women, attributing anxiety to gender stereotypes was associated with lower performance in the math-test condition but improved performance in the teaching-intervention condition. The results suggest that teaching about stereotype threat might offer a practical means of reducing its detrimental effects.

Recent studies have revealed the complex structure of nerve signals in axons. There is experimental evidence that the propagation of an electrical signal (action potential) is accompanied by mechanical and thermal effects. In this paper, first, an overview is presented on experimental results and possible mechanisms of electromechanophysiological couplings which govern the signal formation in axons. This forms a basis for building up a mathematical model describing an ensemble of waves. Three basic physical mechanisms responsible for coupling are (i) electric-lipid bi-layer interaction resulting in the mechanical wave in biomembrane; (ii) electric-fluid interaction resulting in the mechanical wave in the axoplasm; (iii) electric-fluid interaction resulting in the temperature change in axoplasm. The influence of possible changes in variables which could have a role for interactions are analysed and the concept of internal variables introduced for describing the endothermic processes. The previously proposed mathematical model is modified reflecting the possible physical explanation of these interactions.

Educational research literature on mathematical modelling is extensive. However, not much attention has been paid to empirical investigations of its scholarly knowledge from the perspective of didactic transposition processes. This paper reports from an interview study of mathematical modelling activities involving nine professional model constructors. The research question was: How can mathematical modelling by professional mathematical model constructors be characterised? The analysis of our interview data inspired by the coding procedure of grounded theory led us to the description of three main types of modelling activities as a characterisation of mathematical modelling as a professional task. In data-generated modelling the models are developed principally from quantitative data drawing on no or only some assumed knowledge of the system being modelled, while in theory-generated modelling the models are developed based on established theory. In the third activity, model-generated modelling, the development of new models is based on already established models. For all types, the use of computer support and communication between clients, constructors and other experts are central aspects. Finally, the three types of modelling activities are related to existing theoretical descriptions of mathematical modelling and the relevance of the study for mathematical modelling in education is discussed.

The purpose of this work is to obtain an array of experimental data, kinetic dependences of the protein extracting from the particles of dispersed muscle tissue of hydrobionts (Alaska pollack), its mathematical description, and to find the molecular diffusion coefficients of proteins in the system of equations using solving the inverse problem of the modeling. This work’s novelty is to study swelling and protein extraction from animal tissue and its mathematical description. This study has established that the diffusion of proteins from the muscle tissue of hydrobionts during stirring suspension (raw material-extractant) occurs at a significantly higher rate than diffusion from raw materials of plant origin. It indicates the contribution of a convective component to the swelling process and dissolution of particles due to intensive blending and easy differentiability of muscular tissue particles of hydrobionts. Based on the solution of the mathematical model, the molecular diffusion coefficient of proteins from raw materials of animal origin (muscle tissue of hydrobionts) (3÷8)×10−9 m2/s, which exceeds the diffusion coefficients of carbohydrates from raw materials of plant origin (e.g., sugar 0.3×10−9 m2/s) by more than an order of magnitude. Researchers can use the obtained molecular diffusion coefficient to calculate the parameters of the diffusion process of proteins during its extraction from dispersed particles of hydrobionts.