Using Records of Practice to Bridge from Teachers’ Mathematical Problem Solving to Classroom Practice

It is often the case that high quality mathematics education professional development involves enhancing both teachers’ content knowledge and their pedagogical skills, specifically for teaching mathematics. However, when teachers are immersed in their own learning of mathematics, they are often unaware of the facilitators’ instructional decisions and moves that influence their own learning outcomes, as well as how these might apply to their future teaching. Thus, while the teachers can appreciate their new understanding of content, they may not have added significantly to their understanding of the instructor’s pedagogical moves that facilitated their growth. As a result, teachers may leave even high quality professional development without assurance that they will be able to adjust their teaching in ways that support their own students’ meaningful learning of mathematics. In this work we describe one way in which professional development can both enhance teachers’ subject matter knowledge and help to transform these new understandings into pedagogical content knowledge; the mathematics content sessions provide the platform for reflection on pedagogy. To facilitate this reflection, a “record of practice” is created by facilitators, and thereafter utilized for participants and facilitators to identify and analyze critical moments in the mathematics content session. This paper offers two specific examples of records of practice and how they were used, as well as teachers’ reactions and insights. It also discusses various formats of records of practice, the logistics of developing them, and ends with the potential benefits of using records of practice in professional development for teachers

On Solving a Generalized Chinese Remainder Theorem in the Presence of Remainder Errors

In estimating frequencies given that the signal waveforms are undersampled multiple times, Xia et. al. proposed to use a generalized version of Chinese remainder Theorem (CRT), where the moduli are $M_1, M_2, \cdots, M_k$ which are not necessarily pairwise coprime. If the errors of the corrupted remainders are within \tau=\sds \max_{1\le i\le k} \min_{\stackrel{1\le j\le k}{j\neq i}} \frac{\gcd(M_i,M_j)}4, their schemes can be used to construct an approximation of the solution to the generalized CRT with an error smaller than $\tau$. Accurately finding the quotients is a critical ingredient in their approach. In this paper, we shall start with a faithful historical account of the generalized CRT. We then present two treatments of the problem of solving generalized CRT with erroneous remainders. The first treatment follows the route of Wang and Xia to find the quotients, but with a simplified process. The second treatment considers a simplified model of generalized CRT and takes a different approach by working on the corrupted remainders directly. This approach also reveals some useful information about the remainders by inspecting extreme values of the erroneous remainders modulo $4\tau$. Both of our treatments produce efficient algorithms with essentially optimal performance. Finally, this paper constructs a counterexample to prove the sharpness of the error bound $\tau$

An introduction to the spectrum, symmetries, and dynamics of spin-1/2 Heisenberg chains

Quantum spin chains are prototype quantum many-body systems. They are employed in the description of various complex physical phenomena. The goal of this paper is to provide an introduction to the subject by focusing on the time evolution of a Heisenberg spin-1/2 chain and interpreting the results based on the analysis of the eigenvalues, eigenstates, and symmetries of the system. We make available online all computer codes used to obtain our data.Comment: 8 pages, 3 figure

Phage inducible islands in the gram-positive cocci

The SaPIs are a cohesive subfamily of extremely common phage-inducible chromosomal islands (PICIs) that reside quiescently at specific att sites in the staphylococcal chromosome and are induced by helper phages to excise and replicate. They are usually packaged in small capsids composed of phage virion proteins, giving rise to very high transfer frequencies, which they enhance by interfering with helper phage reproduction. As the SaPIs represent a highly successful biological strategy, with many natural Staphylococcus aureus strains containing two or more, we assumed that similar elements would be widespread in the Gram-positive cocci. On the basis of resemblance to the paradigmatic SaPI genome, we have readily identified large cohesive families of similar elements in the lactococci and pneumococci/streptococci plus a few such elements in Enterococcus faecalis. Based on extensive ortholog analyses, we found that the PICI elements in the four different genera all represent distinct but parallel lineages, suggesting that they represent convergent evolution towards a highly successful lifestyle. We have characterized in depth the enterococcal element, EfCIV583, and have shown that it very closely resembles the SaPIs in functionality as well as in genome organization, setting the stage for expansion of the study of elements of this type. In summary, our findings greatly broaden the PICI family to include elements from at least three genera of cocci

Nine-year comparison of presentation and management of acute coronary syndromes in Ireland: a national cross-sectional survey

BACKGROUND: Shorter time to treatment is associated with lower mortality in acute coronary syndromes (ACS). A previous (1994) survey showed substantial delays for acute myocardial infarction (AMI) in Ireland. The present study compared current practice with 1994 and surveyed acute coronary syndromes as a more complete contemporary evaluation of critical cardiac care than assessing AMI alone. METHODS: Following ethics committee approval, all centres (N = 39) admitting acute cardiac patients to intensive/coronary care unit provided information on 1365 episodes. A cross-sectional survey design was employed. RESULTS: Since 1994, median hospital arrival to thrombolysis time was reduced by 41% (76 to 45 minutes). Thrombolysis was delivered more often in the emergency department in 2003 (48% vs 2%). Thrombolysis when delivered in the emergency department was achieved faster than thrombolysis delivered in intensive/coronary care (35 mins v 60 mins; z = 5.62, p < .0001). Suspected AMI patients who did not subsequently receive thrombolysis took longer to present to hospital (5 h vs 2 h 34 mins; z = 7.33, p < .0001) and had longer transfer times to the intensive/coronary care unit following arrival (2 h 17 mins vs 1 h 10 mins; z = 8.92, p < .0001). Fewer confirmed AMI cases received thrombolysis in 2003 (43% vs 58%). There was an increase in confirmed cases of AMI from 1994 (70% to 87%). CONCLUSIONS: Substantial improvements in time to thrombolysis have occurred since 1994, probably relating to treatment provision in emergency departments. Patient delay pre-hospital is still the principal impediment to effective treatment of ACS. A recent change of definition of AMI may have precluded an exact comparison between 1994 and 2003 data

Population Structure of a Hybrid Clonal Group of Methicillin-Resistant Staphylococcus aureus, ST239-MRSA-III

The methicillin-resistant Staphylococcus aureus (MRSA) clonal group known as ST239-MRSA-III is notable for its hybrid origin and for causing sustained hospital epidemics worldwide since the late 1970s. We studied the population structure of this MRSA clonal group using a sample of 111 isolates that were collected over 34 years from 29 countries. Genetic variation was assessed using typing methods and novel ascertainment methods, resulting in approximately 15 kb of sequence from 32 loci for all isolates. A single most parsimonious tree, free of homoplasy, partitioned 28 haplotypes into geographically-associated clades, including prominent European, Asian, and South American clades. The rate of evolution was estimated to be approximately 100× faster than standard estimates for bacteria, and dated the most recent common ancestor of these isolates to the mid-20th century. Associations were discovered between the ST239 phylogeny and the ccrB and dru loci of the methicillin resistance genetic element, SCCmec type III, but not with the accessory components of the element that are targeted by multiplex PCR subtyping tools. In summary, the evolutionary history of ST239 can be characterized by rapid clonal diversification that has left strong evidence of geographic and temporal population structure. SCCmec type III has remained linked to the ST239 chromosome during clonal diversification, but it has undergone homoplasious losses of accessory components. These results provide a population genetics framework for the precise identification of emerging ST239 variants, and invite a re-evaluation of the markers used for subtyping SCCmec

Tracking cryptic SARS-CoV-2 Lineages Detected in NYC Wastewater

Tracking SARS-CoV-2 genetic diversity is strongly indicated because diversifying selection may lead to the emergence of novel variants resistant to naturally acquired or vaccine-induced immunity. To monitor New York City (NYC) for the presence of novel variants, we deep sequence most of the receptor binding domain coding sequence of the S protein of SARS-CoV-2 isolated from the New York City wastewater. Here we report detecting increasing frequencies of novel cryptic SARS-CoV-2 lineages not recognized in GISAID’s EpiCoV database. These lineages contain mutations that had been rarely observed in clinical samples, including Q493K, Q498Y, E484A, and T572N and share many mutations with the Omicron variant of concern. Some of these mutations expand the tropism of SARS-CoV-2 pseudoviruses by allowing infection of cells expressing the human, mouse, or rat ACE2 receptor. Finally, pseudoviruses containing the spike amino acid sequence of these lineages were resistant to different classes of receptor binding domain neutralizing monoclonal antibodies. We offer several hypotheses for the anomalous presence of these lineages, including the possibility that these lineages are derived from unsampled human COVID-19 infections or that they indicate the presence of a non-human animal reservoir