Meanings of fractions as demonstrated by future primary teachers in the initial phase of teacher education

Fractions are a fundamental content of primary-level education and must therefore be included in the training courses for primary school teachers. Experts argue that deep understanding is required to improve primary school teachers’ knowledge of this mathematical concept (Ball, 1990; Cramer, Post & del Mas, 2002; Newton, 2008). Our study focuses on the part-whole relationship as a crucial foundation in working with fractions. This paper characterizes some of the meanings of this relationship for a group of future primary school teachers

Meaning of the part-whole relation and the concept of fraction for primary teachers

The part-whole relation is complex and raises questions that affect different disciplines. Researchers have proposed different interpretations of the notions of fraction and rational number (e.g., Behr, Lesh, Post & Silver, 1983; Kieren, 1976). We highlight three kinds of relations in the study of rational numbers—the part whole-relation, the part-part relation, and the functional relation—through which we organize the different subconstructs of rational number. We claim that the meaning of fractions should be understood through three components: their mathematical structure, their representations and their senses

Dirac points merging and wandering in a model Chern insulator

We present a model for a Chern insulator on the square lattice with complex first and second neighbor hoppings and a sublattice potential which displays an unexpectedly rich physics. Similarly to the celebrated Haldane model, the proposed Chern insulator has two topologically non-trivial phases with Chern numbers $\pm1$. As a distinctive feature of the present model, phase transitions are associated to Dirac points that can move, merge and split in momentum space, at odds with Haldane's Chern insulator where Dirac points are bound to the corners of the hexagonal Brillouin zone. Additionally, the obtained phase diagram reveals a peculiar phase transition line between two distinct topological phases, in contrast to the Haldane model where such transition is reduced to a point with zero sublattice potential. The model is amenable to be simulated in optical lattices, facilitating the study of phase transitions between two distinct topological phases and the experimental analysis of Dirac points merging and wandering

Mirror therapy and self-care autonomy after stroke: an intervention program

Background: In patients with middle cerebral artery (MCA) stroke, changes in upper limb function lead to dependence on others for self-care. In the process of recovering autonomy/independence, there is evidence on the effectiveness of sensory stimulation techniques in the motor recovery after stroke. Objective: To assess the effect of mirror therapy on the self-care autonomy of patients with hemiplegia/hemiparesis due to MCA stroke. Methodology: Cross-sectional and quasi-experimental study with a quantitative approach, a before-and-after design, and a non-equivalent control group. A nonprobability sample of 30 participants was selected. Results: Gains in grip strength, joint range of motion, and manual dexterity of the upper limb were more significant in the experimental group but without statistically significant differences between groups. Conclusion: Despite the more significant evolution of the experimental group, mirror therapy was not effective in the motor recovery of the upper limb. Further studies are needed in this area using randomized designs, larger samples, and focused on self-care