The role of the researcher’s epistemology in mathematics education: an essay on the case of proof

Is there a shared meaning of “mathematical proof” among researchers in mathematics education? Almost all researchers may agree on a formal definition of mathematical proof. But beyond this minimal agreement, what is the state of our field? After three decades of activity in this area, being familiar with the most influential pieces of work, I realize that the sharing of keywords hides important differences in the understanding. These differences could be obstacles to scientific progress in this area, if they are not made explicit and addressed as such. In this essay I take a sample of research projects which have impacted the teaching and learning of mathematical proof, in order to describe where the gaps are. Then I suggest a possible scientific programme which aspires to strengthen the research practice in this domain. Eventually, I make the additional claim that this programme could hold for other areas of research in mathematics education.     

Cation mobility upon adsorption of methanol in NaY faujasite type zeolite: A molecular dynamics study compared to dielectric relaxation experiments

Molecular Dynamics simulations have been carried out to address the question of cation migration upon adsorption of methanol in NaY Faujasite system as a function of the loading. It has been shown that at low and intermediate loadings, SII cations can migrate toward the center of the supercage due strong interaction with the adsorbates, followed by hopping of SI' cations from the sodalite cage into the supercage to fill vacant SII sites. SI cations mainly remain trapped in their initial sites whatever the loading. At higher loading, only limited motion is observed for SII cations due to steric effects induced by the adsorbates within the supercage. These simulated results are in good agreement with those extracted by Complex Impedance Spectroscopy measurements, which provided the evolution of the number of extraframework cations in the different crystallographic sites as a function of the treatment temperature.     

Internal derivatization of oligonucleotides with selenium for X-ray crystallography using MAD.

We have developed a route for the synthesis of 2'-selenium uridine analogues and oligonucleotides containing selenium labels, and have demonstrated for the first time a new strategy to covalently derivatize nucleotides with selenium for phase and structure determination in X-ray crystallography.     

Study of a nanocomposite based on a conducting polymer: polyaniline

A simple way to obtain a conducting nanocomposite is described, and the conducting particles are characterized. Core-shell particles [polystyrene-polyaniline (PANI)] have been obtained by the dispersion process from three types of polystyrene latexes: a no-cross-linked core stabilized by a nonylphenolethoxylate (NP40) and two cross-linked cores stabilized by NP40 and a mixture NP40/Surfamid (a surfactant bearing an amide group). The surface of these particles has been extensively characterized by X-ray photoelectron spectroscopy (XPS), atomic force microscopy, and scanning electron microscopy. A maximum coverage of 94% was obtained for the high PANI content as revealed by XPS analysis. A better coverage was obtained for the cross-linked polystyrene latex stabilized by the Surfamid. The amide group of this surfactant allows the H-bonding formation with the PANI backbone and, thus, improves the conductivity. It was shown that a uniform coverage of the core particles was not required to ensure a good conductivity.     

NO2 disproportionation for the IR characterisation of basic zeolites

NO2 disproportionation on alkaline zeolites is used to generate nitrosonium (NO+) and nitrate ions on the surface, and the infrared vibrations observed are very sensitive to the cation chemical hardness and to the basicity of zeolitic oxygen atoms.     

Complex gold nanostructures derived by templating from diatom frustules

Diatom frustules have been used for the first time as templates for the fabrication of gold nanostructures; high-precision replicas featuring complex three-dimensional gold nanostructures from the nano- to the microscale were achieved.     

Quantum cryptography protocols robust against photon number splitting attacks for weak laser pulse implementations

We introduce a new class of quantum key distribution protocols, tailored to be robust against photon number splitting (PNS) attacks. We study one of these protocols, which differs from the original protocol by Bennett and Brassard (BB84) only in the classical sifting procedure. This protocol is provably better than BB84 against PNS attacks at zero error.     

Direct femtosecond laser writing of waveguides in As2S3 thin films

Single-channel waveguides and Y couplers were fabricated in chalcogenide thin films by use of femtosecond laser pulses from a 25-MHz repetition rate Ti:sapphire laser. Refractive-index differentials (delta n > 10(-2)) were measured through interferometric microscopy and are higher than the typical values reported for oxide glasses. The dependence of the index differential on the peak intensity reveals the nonlinear nature of the photosensitivity in arsenic trisulfide below its bandgap energy, and the refractive-index change is correlated to the photoinduced structural changes inferred by Raman spectroscopy data. A free-electron model to predict the parametric dependence of delta n is proposed.     

Molecular dynamics simulations of a silver atom in water: evidence for a dipolar excitonic state

The properties of a silver atom in bulk water were studied for the first time by molecular dynamics simulations using two complementary mixed quantum-classical approaches. The first one consists of treating by quantum mechanics one electron only, which interacts with a classical silver cation and solvent through one-electron pseudopotentials. The second one is Car-Parrinello molecular dynamics that treats all the valence electrons quantum-mechanically. Very good agreement is obtained between these two methods, and the calculated absorption spectrum of the solvated silver atom agrees very well with experimental data. Both simulations reveal that the silver atom is in the critical region for the appearance of a dipolar excitonic state and exhibits a dipole moment of approximately 2 D with large fluctuations of +/-1 D. The structure of the solvation shell is also analyzed.