미국의 “전국 수학 교사 협의회”(National Council of Teachers of Mathematics, NCTM)는 1989년부터 〈학교 수학의 교육과정과 평가 규준〉(1989), 〈수학 가르침(교수)의 전문성 규준〉(1991), 〈학교 수학의 평가(시험) 규준〉(NCTM, 1995), 〈학교 수학의 원리와 규준〉(2000)을 출판하여 미국의 수학 교육 의 전망(목표, 나아갈 길)과 규준(실행 지침)을 제시하였다. 수학 교사들로 구성된 미국의 NCTM은 학생, 학부모, 학교 행정가 등 많은 사람들과 힘을 합하여 모든 학생들에게 수준 높은 수학 교육을 받을 수 있는 여건(환경, 기회)을 조성하는 데 구심점의 역할을 하였다. 한편 많은 관련 단체들은 여러 배경과 능력을 가진 학생들이 전문성을 지닌 교사(특수 교사를 일컫는 밀이 아니다. 수학 교과를 이해하고 수학의 전문성과 특수성을 가르칠 수 있는 일반 교사를 일컫는 말이다.)로부터 미래를 대비해 평등하고, 진취적이며, 지원이 잘 이루어지고, 공학 도구(IT)가 잘 갖춰진 환경에서 중요한 수학적 아이디어를 이해하면서 학습할 수 있는 수학 교실(미국에서는 우리나라처럼 수학 교사가 수학 시간에 학생의 방(교실: Homeroom)에 찾아가지 않고 학생들이 선생의 방(수학 교실: Classroom)을 찾아온다. 전형적인 수학 교실의 사진은 2쪽에 나와 있다.)을 만들기 위해 함께 힘썼다. NCTM에서 출간한 여러 규준들은 우리나라의 제6차와 제7차 교육과정에도 큰 영향을 미쳤다. 이 글에서는 NCTM(2000)에서 제시한 학습 원리를 간단히 살펴본 다음 이를 중심으로 현재 미국 수학 교육의 교수ㆍ학습 이론의 동향을 살펴본다.

미국 NCTM의 Principles and Standards for School Mathematics에 나타난 수학과 교수,학습의 이론

The principles and most-recent results of high-power THz generation through optical rectification using a tilted optical pulse front are described. Single-cycle THz pulses of multimicrojoule energies are generated at kHz repetition rates, and average THz power levels exceeding 1 mW can be generated at kHz-MHz repetition rates. Applications in nonlinear THz spectroscopy and THz coherent control are discussed.

Generation of high-power terahertz pulses by tilted-pulse-front excitation and their application possibilities

Topological quantum materials exhibit fascinating properties1–3, with important applications for dissipationless electronics and fault-tolerant quantum computers4,5. Manipulating the topological invariants in these materials would allow the development of topological switching applications analogous to switching of transistors6. Lattice strain provides the most natural means of tuning these topological invariants because it directly modifies the electron–ion interactions and potentially alters the underlying crystalline symmetry on which the topological properties depend7–9. However, conventional means of applying strain through heteroepitaxial lattice mismatch10 and dislocations11 are not extendable to controllable time-varying protocols, which are required in transistors. Integration into a functional device requires the ability to go beyond the robust, topologically protected properties of materials and to manipulate the topology at high speeds. Here we use crystallographic measurements by relativistic electron diffraction to demonstrate that terahertz light pulses can be used to induce terahertz-frequency interlayer shear strain with large strain amplitude in the Weyl semimetal WTe2, leading to a topologically distinct metastable phase. Separate nonlinear optical measurements indicate that this transition is associated with a symmetry change to a centrosymmetric, topologically trivial phase. We further show that such shear strain provides an ultrafast, energy-efficient way of inducing robust, well separated Weyl points or of annihilating all Weyl points of opposite chirality. This work demonstrates possibilities for ultrafast manipulation of the topological properties of solids and for the development of a topological switch operating at terahertz frequencies. Terahertz light pulses induce transitions between a topological and a trivial phase in the Weyl semimetal WTe2 through an interlayer shear strain.

An ultrafast symmetry switch in a Weyl semimetal.

In the last two decades non-equilibrium spectroscopies have evolved from avant-garde studies to crucial tools for expanding our understanding of the physics of strongly correlated materials. The possibility of obtaining simultaneously spectroscopic and temporal information has led to insights that are complementary to (and in several cases beyond) those attainable by studying the matter at equilibrium. From this perspective, multiple phase transitions and new orders arising from competing interactions are benchmark examples where the interplay among electrons, lattice and spin dynamics can be disentangled because of the different timescales that characterize the recovery of the initial ground state. For example, the nature of the broken-symmetry phases and of the bosonic excitations that mediate the electronic interactions, eventually leading to superconductivity or other exotic states, can be revealed by observing the sub-picosecond dynamics of impulsively excited states. Furthermore, recent experimental...

Ultrafast optical spectroscopy of strongly correlated materials and high-temperature superconductors: a non-equilibrium approach

Terahertz time-domain spectroscopy (THz-TDS) is a powerful technique for material’s characterization and process control. It has been used for contact-free conductivity measurements of metals, semiconductors, 2D materials, and superconductors. Furthermore, THz-TDS has been used to identify chemical components such as amino acids, peptides, pharmaceuticals, and explosives, which makes it particularly valuable for fundamental science, security, and medical applications. This tutorial is intended for a reader completely new to the field of THz-TDS and presents a basic understanding of THz-TDS. Hundreds of articles and many books can be consulted after reading this tutorial. We explore the basic concepts of TDS and discuss the relationship between temporal and frequency domain information. We illustrate how THz radiation can be generated and detected, and we discuss common noise sources and limitations for THz-TDS. This tutorial concludes by discussing some common experimental scenarios and explains how THz-TDS measurements can be used to identify materials, determine complex refractive indices (phase delay and absorption), and extract conductivity.

Tutorial: An introduction to terahertz time domain spectroscopy (THz-TDS)

We present a combined theoretical and experimental investigation of the generation of few-cycle terahertz (THz) pulses via the nonlinear effect of optical rectification and of their coherent detection via electro-optic sampling. The effects of dispersive velocity matching, absorption of the optical and the THz waves, crystal thickness, pulse diameter, pump pulse duration, and two-photon absorption are discussed. The theoretical calculations are compared with the measured spectra of THz pulses that have been generated and detected in crystals of the highly nonlinear organic salt 4-N,N-dimethylamino-4′-N′-methyl stilbazolium tosylate (DAST). The results are found to be in agreement with the theory. By the selection of the optical pump wavelength between 700 and 1600 nm, we achieved several maxima of the overall generation and detection efficiency in the spectral range between 0.4 and 6.7 THz, with an optimum at 2 THz generated with 1500 nm laser pulses.

Generation of terahertz pulses through optical rectification in organic DAST crystals: theory and experiment

using low-temperature solution growth. We also demonstrate the growth of single crystalline thin films of DSTMS with an area of up to 6! 5 mm 2 and a thickness between 5‐30 lm. Nonlinear optical measurements reveal that DSTMS possesses large nonlinear optical susceptibilities with v111 (2) =(430±40) pmV ‐1 at 1.9 lm. Highly efficient generation of broadband THz waves with THz electric field strengths of more than 4 kVcm ‐1 using 160 fs laser pump pulses at a wavelength k=1.45 lm and DSTMS crystals has been demonstrated.

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Large-Size Bulk and Thin-Film Stilbazolium-Salt Single Crystals for Nonlinear Optics and THz Generation†

A series of nonlinear optical chromophores based on configurationally locked polyene are synthesized, and the single crystal growth from the melt are investigated. The chromophores consist of a π-conjugated hexatriene bridge between the dialkylamino or methoxy electron donors and the dicyanomethylidene electron acceptor. The physical and nonlinear optical properties and the single crystal X-ray structures of these chromophores are characterized. Specifically, the chromophore 2-{3-[2-(4-dimethylaminophenyl)vinyl]-5,5-dimethylcyclohex-2-enylidene}malononitrile (DAT2) exhibits a strong powder second-harmonic generation signal of about 2 orders of magnitude greater than that of urea and a large temperature difference of 80 °C between the melting temperature and the recrystallization temperature. Finally, we demonstrate the growth of single crystalline thin films of DAT2 by a Bridgman-type melt growth technique, which gives a new direction for organic nonlinear optical crystal engineering using melt−growth tec...

Organic Nonlinear Optical Crystals Based on Configurationally Locked Polyene for Melt Growth

Three series of ionic stilbazolium salts with different counteranions have been synthesized to investigate the effect of counteranion size to crystal packing. Single crystals of five new salts were successfully grown from solution by the slow cooling and slow evaporation techniques. X-ray studies revealed that three of them show the triclinic space group P1 and two crystallize in the monoclinic space group Cc and P21/c, respectively. Kurtz powder tests revealed that out of the eight new stilbazolium salts, six possess noncentrosymmetric structures and three show large powder second-harmonic generation efficiencies higher or similar to that of 4-N,N-dimethylamino-4‘-N‘-methyl-stilbazolium tosylate (DAST). The optimal crystalline packing for electro-optics with perfectly aligned chromophores was achieved in three of the investigated compounds. The reasons for considerably different macroscopic nonlinearities of these compounds have been explored in terms of intermolecular interactions.

Molecular Engineering of Stilbazolium Derivatives for Second-Order Nonlinear Optics

In this paper, we analyse the graphs produced by 93 prospective primary school teachers in a task comparing two statistical variables. We define a level of semiotic complexity that serves to classify the graphs produced. We relate this level to the level in reading the graphs and the conclusion obtained. Results show that only some participants produce a graph with enough complexity to get an adequate conclusion and part of them either do not reach an adequate level of reading or do not get a complete conclusion in relation to the research question.

Análisis de la complejidad semiótica de los gráficos producidos por futuros profesores de educación primaria en una tarea de comparación de dos variables estadísticas

Blanca Ruiz

Papers

Generation of terahertz pulses through optical rectification in organic DAST crystals: theory and experiment

Large-Size Bulk and Thin-Film Stilbazolium-Salt Single Crystals for Nonlinear Optics and THz Generation†

Organic Nonlinear Optical Crystals Based on Configurationally Locked Polyene for Melt Growth

Molecular Engineering of Stilbazolium Derivatives for Second-Order Nonlinear Optics

Análisis de la complejidad semiótica de los gráficos producidos por futuros profesores de educación primaria en una tarea de comparación de dos variables estadísticas