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Recently, there has been a growing emphasis on training teachers to integrate computational thinking (CT) practices into disciplinary instruction. However, many current approaches involve a "top-down" method, where CT concepts and teacher training are dictated by external CT “experts,” often in an abstract and generalized manner, rather than being developed collaboratively or contextually with the teachers. These approaches typically treat CT as a set of abstract concepts, which can fail to promote a holistic understanding of the purposes and disciplinary value of CT. Consequently, teachers may feel less inclined to integrate CT into their regular teaching practice beyond the confines of professional development sessions. Furthermore, teachers are frequently positioned as novices awaiting the transmission of relevant CT knowledge rather than as agentive knowledge-builders with valuable expertise. This can undermine their autonomy, ownership, adaptability, and long-term commitment to implementing CT effectively in their teaching practice. We propose an alternative, “bottom-up” approach to supporting teachers in CT integration through a collaborative partnership between researchers and practitioners. We share evidence that this partnership led to understanding CT as inherently contextualized and productive for disciplinary problem-solving. 

Traditional professional development aimed at integrating computational thinking (CT) into K-12 classrooms frequently fails to link abstract technical terminology with teachers' personal experiences or real-world situations, which can impede overall teacher understanding and effective classroom implementation. This paper investigates an alternative method that employs personal storytelling to introduce CT to educators. We discovered that storytelling helped build emotional connections and prompted deeper reflections on CT concepts. Participants shared how CT appears in various settings, including teaching, parenting, and outdoor activities, which transformed their understanding and forged significant connections between theory and practice. 

Paraproducts are a special subclass of the multilinear Calderón-Zygmund operators, and their Lebesgue space estimates in the full multilinear range are characterized by the norm of the symbol. In this note, we characterize the Sobolev space boundedness properties of multilinear paraproducts in terms of a suitable family of Triebel-Lizorkin type norms of the symbol. Coupled with a suitable wavelet representation theorem, this characterization leads to a new family of Sobolev space T(1)-type theorems for multilinear Calderón-Zygmund operators. 

We quantify the Sobolev space norm of the Beltrami resolvent \((I- \mu S)^{-1}\), where \(S\) is the Beurling–Ahlfors transform, in terms of the corresponding Sobolev space norm of the dilatation \(\mu\) in the critical and supercritical ranges. Our estimate entails as a consequence quantitative self-improvement inequalities of Caccioppoli type for quasiregular distributions with dilatations in \(W^{1,p}\), \(p \ge 2\). Our proof strategy is then adapted to yield quantitative estimates for the resolvent \((I-\mu S_\Omega)^{-1}\) of the Beltrami equation on a sufficiently regular domain \(\Omega\), with \(\mu\in W^{1,p}(\Omega)\). Here, \(S_\Omega\) is the compression of \(S\) to a domain \(\Omega\). Our proofs do not rely on the compactness or commutator arguments previously employed in related literature. Instead, they leverage the weighted Sobolev estimates for compressions of Calderón–Zygmund operators to domains, recently obtained by the authors, to extend the Astala–Iwaniec–Saksman technique to higher regularities. 

Abstract As we approach the era of quantum advantage, when quantum computers (QCs) can outperform any classical computer on particular tasks, there remains the difficult challenge of how to validate their performance. While algorithmic success can be easily verified in some instances such as number factoring or oracular algorithms, these approaches only provide pass/fail information of executing specific tasks for a single QC. On the other hand, a comparison between different QCs preparing nominally the same arbitrary circuit provides an insight for generic validation: a quantum computation is only as valid as the agreement between the results produced on different QCs. Such an approach is also at the heart of evaluating metrological standards such as disparate atomic clocks. In this paper, we report a cross-platform QC comparison using randomized and correlated measurements that results in a wealth of information on the QC systems. We execute several quantum circuits on widely different physical QC platforms and analyze the cross-platform state fidelities.