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Mathematical Knowledge for Teaching Proof (MKT-P) has been recognized as an important component of fostering student engagement with mathematical reasoning and proof. This study is one component of a larger study aimed at exploring the nature of MKT-P. The present study examines qualitative differences in feedback given by STEM majors, in-service and pre-service secondary mathematics teachers on hypothetical students’ arguments. The results explicate key distinctions in the feedback provided by these groups, indicating that this is a learnable skill. Feedback is cast as a component of MKT-P, making the results of this study significant empirical support for the construct of MKT-P as a type of knowledge that is unique to teachers. 

Mathematical Knowledge for Teaching Proof (MKT-P) has been recognized as an important component of fostering student engagement with mathematical reasoning and proof. This study is one component of a larger study aimed at exploring the nature of MKT-P. The present study examines qualitative differences in feedback given by STEM majors, in-service and pre-service secondary mathematics teachers on hypothetical students’ arguments. The results explicate key distinctions in the feedback provided by these groups, indicating that this is a learnable skill. Feedback is cast as a component of MKT-P, making the results of this study significant empirical support for the construct of MKT-P as a type of knowledge that is unique to teachers. 

It has been suggested that integrating reasoning and proof in mathematics teaching requires a special type of teacher knowledge - Mathematical Knowledge for Teaching Proof (MKT-P). Yet, several important questions about the nature of MKT-P remain open, specifically, whether MKT-P is a type of knowledge specific to teachers, and whether MKT-P can be improved through intervention. We explored these questions by comparing performance on an MKT-P questionnaire of in-service secondary mathematics teachers, undergraduate STEM majors, and pre-service secondary mathematics teachers. The latter group completed the questionnaire twice- before and after participating in a capstone course, Mathematical Reasoning and Proving for Secondary Teachers. Our data suggest that MKT-P is indeed a special kind of knowledge specific to teachers and it can be improved through interventions. 

The benefits of using video in teacher education as a tool for reflection and for developing professional expertise have long been recognized. Recent introduction of 360 video technology holds promise to extend these benefits as it allows prospective teachers to reflect on their own performance by considering the classroom from multiple perspectives. This study examined nine prospective secondary teachers’ (PSTs) noticing and self-reflection on the 360 recordings of their own teaching. The PSTs, enrolled in a capstone course Mathematical Reasoning and Proving for Secondary Teachers, taught a proof-oriented lesson to small groups of students in local schools while capturing their teaching with 360 video cameras. We analyzed the PSTs’ written comments on their video and reflection reports to identify the categories of noticing afforded by the 360 technology as well as the instances of PSTs’ learning. The results point to the powerful potential of 360 videos for supporting PSTs’ self-reflection and professional growth. 

We analyze 5108 AFGKM stars with at least five high-precision radial velocity points, as well as Gaia and Hipparcos astrometric data, utilizing a novel pipeline developed in previous work. We find 914 radial velocity signals with periods longer than 1000 days. Around these signals, 167 cold giants and 68 other types of companions are identified, through combined analyses of radial velocity, astrometry, and imaging data. Without correcting for detection bias, we estimate the minimum occurrence rate of the wide-orbit brown dwarfs to be 1.3%, and find a significant brown-dwarf valley around 40M_Jup. We also find a power-law distribution in the host binary fraction beyond 3 au, similar to that found for single stars, indicating no preference of multiplicity for brown dwarfs. Our work also reveals nine substellar systems (GJ 234 B, GJ 494 B, HD 13724 b, HD 182488 b, HD 39060 b and c, HD 4113 C, HD 42581 d, HD 7449 B, and HD 984 b) that have previously been directly imaged, and many others that are observable at existing facilities. Depending on their ages, we estimate that an additional 10–57 substellar objects within our sample can be detected with current imaging facilities, extending the imaged cold (or old) giants by an order of magnitude.

 

Abstract Asteroseismology of bright stars has become increasingly important as a method to determine the fundamental properties (in particular ages) of stars. The Kepler Space Telescope initiated a revolution by detecting oscillations in more than 500 main-sequence and subgiant stars. However, most Kepler stars are faint and therefore have limited constraints from independent methods such as long-baseline interferometry. Here we present the discovery of solar-like oscillations in α Men A, a naked-eye ( V = 5.1) G7 dwarf in TESS’s southern continuous viewing zone. Using a combination of astrometry, spectroscopy, and asteroseismology, we precisely characterize the solar analog α Men A ( T eff = 5569 ± 62 K, R ⋆ = 0.960 ± 0.016 R ⊙ , M ⋆ = 0.964 ± 0.045 M ⊙ ). To characterize the fully convective M dwarf companion, we derive empirical relations to estimate mass, radius, and temperature given the absolute Gaia magnitude and metallicity, yielding M ⋆ = 0.169 ± 0.006 M ⊙ , R ⋆ = 0.19 ± 0.01 R ⊙ , and T eff = 3054 ± 44 K. Our asteroseismic age of 6.2 ± 1.4 (stat) ± 0.6 (sys) Gyr for the primary places α Men B within a small population of M dwarfs with precisely measured ages. We combined multiple ground-based spectroscopy surveys to reveal an activity cycle of P = 13.1 ± 1.1 yr for α Men A, a period similar to that observed in the Sun. We used different gyrochronology models with the asteroseismic age to estimate a rotation period of ∼30 days for the primary. Alpha Men A is now the closest ( d = 10 pc) solar analog with a precise asteroseismic age from space-based photometry, making it a prime target for next-generation direct-imaging missions searching for true Earth analogs.