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\"Principles to Actions: Ensuring Mathematical Success for All\" (NCTM 2014, p. 10) contains eight research-informed teaching practices that have been shown to support students' mathematical thinking and learning. Two teaching practices highlighted herein are \"to elicit and use evidence of students' thinking\" and \"support students' productive struggle in learning mathematics.\" Through enacting these two teaching practices, teachers can support students in ways that do not take over the thinking of the students. Such communication involves teachers determining \"how\" students are thinking mathematically and, in the moment, being able to respond in a way that supports students building on their thinking. In this article, the authors share a set of communication moves that teachers can use to support efforts to \"elicit and use evidence of students' thinking\" and \"support students' productive struggle in learning mathematics\" and then illustrate how a teacher can implement such moves while working with students.

Preservice mathematics teachers are entrusted with developing their future students' interest in and ability to do mathematics effectively. Various policy documents place importance on being able to reason about and prove mathematical claims. However, it is not enough for these preservice teachers, and their future students, to have a narrow focus on only one type of proof (demonstration proof), as opposed to other forms of proof, such as generic example proofs, pictorial proofs, and so on. This article examines the effectiveness of a course on reasoning-and-proving on preservice teachers' awareness of and abilities to recognize and construct generic example proofs. The findings support assertions that such a course can and does change preservice teachers' capability with generic example proofs.

Teacher responses to student mathematical thinking (SMT) matter because the way in which teachers respond affects student learning. Although studies have provided important insights into the nature of teacher responses, little is known about the extent to which these responses take into account the potential of the instance of SMT to support learning. This study investigated teachers’ responses to a common set of instances of SMT with varied potential to support students’ mathematical learning, as well as the productivity of such responses. To examine variations in responses in relation to the mathematical potential of the SMT to which they are responding, we coded teacher responses to instances of SMT in a scenario-based interview. We did so using a scheme that analyzes who interacts with the thinking (Actor), what they are given the opportunity to do in those interactions (Action), and how the teacher response relates to the actions and ideas in the contributed SMT (Recognition). The study found that teachers tended to direct responses to the student who had shared the thinking, use a small subset of actions, and explicitly incorporate students’ actions and ideas. To assess the productivity of teacher responses, we first theorized the alignment of different aspects of teacher responses with our vision of responsive teaching. We then used the data to analyze the extent to which specific aspects of teacher responses were more or less productive in particular circumstances. We discuss these circumstances and the implications of the findings for teachers, professional developers, and researchers.

In the last decade, mathematics teacher educators have begun to design learning opportunities for preservice mathematics teachers using a pedagogies-of-practice perspective. In particular, learning cycles provide a structure for engaging PSTs in learning to teach through the use of representations, approximations, and decompositions of practice (Grossman et al., 2009). In this article, we provide details of one learning cycle designed to support secondary mathematics preservice teachers' learning to elicit and use evidence of student thinking and pose purposeful questions (National Council of Teachers of Mathematics, 2014). Through qualitative analyses conducted on learning reflections, we provide evidence of the impact on engagement of this cycle through the lens of the Framework for Learning to Teach (Hammerness et al., 2005).

The field of teacher education is currently experiencing a shift towards curricula focused on practices for teaching and professional coursework designed around different pedagogies of practice (e.g., decomposition of, representations of, and approximations of practice). Researchers and research-informed documents have identified a variety of mathematics teaching practices, one of which is facilitating classroom mathematics discourse. Further, researchers have identified some key practices that constitute facilitating classroom mathematics discourse such as assessing questions and advancing questions (Smith, Bill, & Hughes, 2008) and judicious telling (Lobato, Clarke, & Ellis, 2005). In a methods course designed to engage preservice secondary mathematics teachers (PSMTs) in course activities designed around pedagogies of practice and focused on Types of Teacher Talk (TTT) (i.e., assessing questions, advancing questions, and judicious telling), this study sought to examine what PSMTs learn about TTT and in what ways the course activities were connected to their learning. The design of the study was informed by the interpretive research genre and the study employed qualitative data collection and analysis techniques to explore four case studies of the PSMTs’ learning of TTT. The results of the study indicate that the PSMTs, over the course of the semester, constructed conceptions of TTT that were oriented towards focusing on and promoting students’ mathematical thinking. In addition, while there were similarities among the PSMTs’ conceptions of TTT, they constructed these conceptions in individually distinct ways. Study implications include the design of preservice teacher education professional coursework that engages preservice teachers in multiple and extended experiences with a small set of teaching practices.

Tidal disruption events (TDEs) are bursts of electromagnetic energy that are released when supermassive black holes at the centres of galaxies violently disrupt a star that passes too close 1 . TDEs provide a window through which to study accretion onto supermassive black holes; in some rare cases, this accretion leads to launching of a relativistic jet 2 – 9 , but the necessary conditions are not fully understood. The best-studied jetted TDE so far is Swift J1644+57, which was discovered in γ-rays, but was too obscured by dust to be seen at optical wavelengths. Here we report the optical detection of AT2022cmc, a rapidly fading source at cosmological distance (redshift z  = 1.19325) the unique light curve of which transitioned into a luminous plateau within days. Observations of a bright counterpart at other wavelengths, including X-ray, submillimetre and radio, supports the interpretation of AT2022cmc as a jetted TDE containing a synchrotron ‘afterglow’, probably launched by a supermassive black hole with spin greater than approximately 0.3. Using four years of Zwicky Transient Facility 10 survey data, we calculate a rate of 0.0 2 − 0.01 + 0.04 Gpc −3 yr −1 for on-axis jetted TDEs on the basis of the luminous, fast-fading red component, thus providing a measurement complementary to the rates derived from X-ray and radio observations 11 . Correcting for the beaming angle effects, this rate confirms that approximately 1 per cent of TDEs have relativistic jets. Optical surveys can use AT2022cmc as a prototype to unveil a population of jetted TDEs. A series of early-time, multiwavelength observations of an optical transient, AT2022cmc, indicate that it is a relativistic jet from a tidal disruption event originating from a supermassive black hole.

In recent years, certain luminous extragalactic optical transients have been observed to last only a few days 1 . Their short observed duration implies a different powering mechanism from the most common luminous extragalactic transients (supernovae), whose timescale is weeks 2 . Some short-duration transients, most notably AT2018cow (ref.  3 ), show blue optical colours and bright radio and X-ray emission 4 . Several AT2018cow-like transients have shown hints of a long-lived embedded energy source 5 , such as X-ray variability 6 , 7 , prolonged ultraviolet emission 8 , a tentative X-ray quasiperiodic oscillation 9 , 10 and large energies coupled to fast (but subrelativistic) radio-emitting ejecta 11 , 12 . Here we report observations of minutes-duration optical flares in the aftermath of an AT2018cow-like transient, AT2022tsd (the ‘Tasmanian Devil’). The flares occur over a period of months, are highly energetic and are probably nonthermal, implying that they arise from a near-relativistic outflow or jet. Our observations confirm that, in some AT2018cow-like transients, the embedded energy source is a compact object, either a magnetar or an accreting black hole. Observations of optical flares from AT2022tsd (the ‘Tasmanian Devil’) show that they have durations on the timescale of minutes, occur over a period of months, are highly energetic, are probably nonthermal and have supernova luminosities.

In the version of this article initially published, there was in an error in the third-to-last sentence of the abstract, now reading, in part, “we calculate a rate of 0.02–0.01 +0.04 Gpc–3 yr–1”, where Gpc was spelled out as gigapascals, not gigaparsecs. Also, the scale label (2″) was missing in the lower-left corner of Fig. 1b. The errors have been corrected in the HTML and PDF versions of the article.

In recent years, certain luminous extragalactic optical transients have been observed to last only a few days. Their short observed duration implies a different powering mechanism from the most common luminous extragalactic transients (supernovae) whose timescale is weeks. Some short-duration transients, most notably AT2018cow, display blue optical colours and bright radio and X-ray emission. Several AT2018cow-like transients have shown hints of a long-lived embedded energy source, such as X-ray variability, prolonged ultraviolet emission, a tentative X-ray quasiperiodic oscillation, and large energies coupled to fast (but subrelativistic) radio-emitting ejecta. Here we report observations of minutes-duration optical flares in the aftermath of an AT2018cow-like transient, AT2022tsd (the \"Tasmanian Devil\"). The flares occur over a period of months, are highly energetic, and are likely nonthermal, implying that they arise from a near-relativistic outflow or jet. Our observations confirm that in some AT2018cow-like transients the embedded energy source is a compact object, either a magnetar or an accreting black hole.