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  1. Although the paradigm wars between quantitative and qualitative research methods and the associated epistemologies may have settled down in recent years within the mathematics education research community, the high value placed on quantitative methods and randomized control trials remain as the gold standard at the policy-making level (USDOE, 2008). Although diverse methods are valued in the mathematics education community, if mathematics educators hope to influence policy to cultivate more equitable education systems, then we must engage in rigorous quantitative research. However, quantitative research is limited in what it can measure by the quantitative tools that exist. In mathematics education, it seems as though the development of quantitative tools and studying their associated validity and reliability evidence has lagged behind the important constructs that rich qualitative research has uncovered. The purpose of this study is to describe quantitative instruments related to mathematics teacher behavior and affect in order to better understand what currently exists in the field, what validity and reliability evidence has been published for such instruments, and what constructs each measure. 1. How many and what types of instruments of mathematics teacher behavior and affect exist? 2. What types of validity and reliability evidence are published for these instruments?more »3. What constructs do these instruments measure? 4. To what extent have issues of equity been the focus of the instruments found?« less
    Free, publicly-accessible full text available April 1, 2023
  2. Context.   Tycho ’s supernova remnant (SNR) is associated with the historical supernova (SN) event SN 1572 of Type Ia. The explosion occurred in a relatively clean environment, and was visually observed, providing an age estimate. This SNR therefore represents an ideal astrophysical test-bed for the study of cosmic-ray acceleration and related phenomena. A number of studies suggest that shock acceleration with particle feedback and very efficient magnetic-field amplification combined with Alfvénic drift are needed to explain the rather soft radio spectrum and the narrow rims observed in X-rays. Aims. We show that the broadband spectrum of Tycho ’s SNR can alternatively be well explained when accounting for stochastic acceleration as a secondary process. The re-acceleration of particles in the turbulent region immediately downstream of the shock should be efficient enough to impact particle spectra over several decades in energy. The so-called Alfvénic drift and particle feedback on the shock structure are not required in this scenario. Additionally, we investigate whether synchrotron losses or magnetic-field damping play a more profound role in the formation of the non-thermal filaments. Methods. We solved the full particle transport equation in test-particle mode using hydrodynamic simulations of the SNR plasma flow. The background magneticmore »field was either computed from the induction equation or follows analytic profiles, depending on the model considered. Fast-mode waves in the downstream region provide the diffusion of particles in momentum space. Results. We show that the broadband spectrum of Tycho can be well explained if magnetic-field damping and stochastic re-acceleration of particles are taken into account. Although not as efficient as standard diffusive shock acceleration, stochastic acceleration leaves its imprint on the particle spectra, which is especially notable in the emission at radio wavelengths. We find a lower limit for the post-shock magnetic-field strength ∼330  μ G, implying efficient amplification even for the magnetic-field damping scenario. Magnetic-field damping is necessary for the formation of the filaments in the radio range, while the X-ray filaments are shaped by both the synchrotron losses and magnetic-field damping.« less
  3. The angular size of a star is a critical factor in determining its basic properties. Direct measurement of stellar angular diameters is difficult: at interstellar distances stars are generally too small to resolve by any individual imaging telescope. This fundamental limitation can be overcome by studying the diffraction pattern in the shadow cast when an asteroid occults a star, but only when the photometric uncertainty is smaller than the noise added by atmospheric scintillation. Atmospheric Cherenkov telescopes used for particle astrophysics observations have not generally been exploited for optical astronomy due to the modest optical quality of the mirror surface. However, their large mirror area makes them well suited for such high-time-resolution precision photometry measurements. Here we report two occultations of stars observed by the Very Energetic Radiation Imaging Telescope Array System (VERITAS) Cherenkov telescopes with millisecond sampling, from which we are able to provide a direct measurement of the occulted stars’ angular diameter at the ≤0.1 mas scale. This is a resolution never achieved before with optical measurements and represents an order of magnitude improvement over the equivalent lunar occultation method. We compare the resulting stellar radius with empirically derived estimates from temperature and brightness measurements, confirming the latter canmore »be biased for stars with ambiguous stellar classifications.« less