Numeracy equipment and Year 3 children : "bright, shiny stuff" or supporting the development of part-whole thinking?

New Zealand teachers' use of equipment has increased as a result of their participation in the Numeracy Development Project. The purpose of this study was to discover how closely the children's reasons for their equipment choices matched their teachers' reasons for including the same pieces of equipment in their numeracy programmes. In the teachers' reasons for equipment choices, the surface features of equipment seemed equally important as the conceptual development the equipment can support. In contrast, the reasons given for equipment choices by the 34 Year 3 children who were interviewed were almost exclusively concerned with how the equipment might help them to solve the given problem. The children's success rates at solving the problem declined as the equipment became more structured; this paralleled the teachers' equipment choices. The equipment choices of the four teachers interviewed in this study were not strongly consistent with the equipment use recommended in the NDP materials.

Numeracy equipment and Year 3 children : "bright, shiny stuff" or supporting the development of part-whole thinking?

New Zealand teachers' use of equipment has increased as a result of their participation in the Numeracy Development Project. The purpose of this study was to discover how closely the children's reasons for their equipment choices matched their teachers' reasons for including the same pieces of equipment in their numeracy programmes. In the teachers' reasons for equipment choices, the surface features of equipment seemed equally important as the conceptual development the equipment can support. In contrast, the reasons given for equipment choices by the 34 Year 3 children who were interviewed were almost exclusively concerned with how the equipment might help them to solve the given problem. The children's success rates at solving the problem declined as the equipment became more structured; this paralleled the teachers' equipment choices. The equipment choices of the four teachers interviewed in this study were not strongly consistent with the equipment use recommended in the NDP materials.

Much ADO about failures: a fault-aware model for compositional verification of strongly consistent distributed systems

Despite recent advances, guaranteeing the correctness of large-scale distributed applications without compromising performance remains a challenging problem. Network and node failures are inevitable and, for some applications, careful control over how they are handled is essential. Unfortunately, existing approaches either completely hide these failures behind an atomic state machine replication (SMR) interface, or expose all of the network-level details, sacrificing atomicity. We propose a novel, compositional, atomic distributed object (ADO) model for strongly consistent distributed systems that combines the best of both options. The object-oriented API abstracts over protocol-specific details and decouples high-level correctness reasoning from implementation choices. At the same time, it intentionally exposes an abstract view of certain key distributed failure cases, thus allowing for more fine-grained control over them than SMR-like models. We demonstrate that proving properties even of composite distributed systems can be straightforward with our Coq verification framework, Advert, thanks to the ADO model. We also show that a variety of common protocols including multi-Paxos and Chain Replication refine the ADO semantics, which allows one to freely choose among them for an application's implementation without modifying ADO-level correctness proofs.

Strong and Efficient Consistency with Consistency-aware Durability

We introduce consistency-aware durability or C ad , a new approach to durability in distributed storage that enables strong consistency while delivering high performance. We demonstrate the efficacy of this approach by designing cross-client monotonic reads , a novel and strong consistency property that provides monotonic reads across failures and sessions in leader-based systems; such a property can be particularly beneficial in geo-distributed and edge-computing scenarios. We build O rca , a modified version of ZooKeeper that implements C ad and cross-client monotonic reads. We experimentally show that O rca provides strong consistency while closely matching the performance of weakly consistent ZooKeeper. Compared to strongly consistent ZooKeeper, O rca provides significantly higher throughput (1.8--3.3×) and notably reduces latency, sometimes by an order of magnitude in geo-distributed settings. We also implement C ad in Redis and show that the performance benefits are similar to that of C ad ’s implementation in ZooKeeper.

Least squares type estimations for discretely observed nonergodic Gaussian Ornstein-Uhlenbeck processes of the second kind

<abstract><p>We consider the nonergodic Gaussian Ornstein-Uhlenbeck processes of the second kind defined by $ dX_t = \theta X_tdt+dY_t^{(1)}, t\geq 0, X_0 = 0 $ with an unknown parameter $ \theta &gt; 0, $ where $ dY_t^{(1)} = e^{-t}dG_{a_{t}} $ and $ \{G_t, t\geq 0\} $ is a mean zero Gaussian process with the self-similar index $ \gamma\in (\frac{1}{2}, 1) $ and $ a_t = \gamma e^{\frac{t}{\gamma}} $. Based on the discrete observations $ \{X_{t_i}:t_i = i\Delta_n, i = 0, 1, \cdots, n\} $, two least squares type estimators $ \hat{\theta}_n $ and $ \tilde{\theta}_n $ of $ \theta $ are constructed and proved to be strongly consistent and rate consistent. We apply our results to the cases such as fractional Brownian motion, sub-fractional Brownian motion, bifractional Brownian motion and sub-bifractional Brownian motion. Moreover, the numerical simulations confirm the theoretical results.</p></abstract>

On quantile based co-risk measures and their estimation

Conditional Value-at-Risk (CoVaR) is defined as the Value-at-Risk of a certain risk given that the related risk equals a given threshold (CoVaR=) or is smaller/larger than a given threshold (CoVaR</CoVaR≥). We extend the notion of Conditional Value-at-Risk to quantile based co-risk measures that are weighted mixtures of CoVaR at different levels and hence involve the stochastic dependence that occurs among the risks and that is captured by copulas. We show that every quantile based co-risk measure is a quantile based risk measure and hence fulfills all related properties. We further discuss continuity results of quantile based co-risk measures from which consistent estimators for CoVaR< and CoVaR≥ based risk measures immediately follow when plugging in empirical copulas. Although estimating co-risk measures based on CoVaR= is a nontrivial endeavour since conditioning on events with zero probability is necessary we show that working with so-called empirical checkerboard copulas allows to construct strongly consistent estimators for CoVaR= and related co-risk measures under very mild regularity conditions. A small simulation study illustrates the performance of the obtained estimators for special classes of copulas.