Characteristics of Second Graders' Mathematical Writing

This study compared the characteristics of second graders' mathematical writing between an intervention and comparison group. Two six‐week Project M2 units were implemented with students in the intervention group. The units position students to communicate in ways similar to mathematicians, including engaging in verbal discourse where they themselves make sense of the mathematics through discussion and debate, writing about their reasoning on an ongoing basis, and utilizing mathematical vocabulary while communicating in any medium. Students in the comparison group learned from the regular school curriculum. Students in both the intervention and comparison groups conveyed high and low levels of content knowledge as indicated in archived data from an open‐response end‐of‐the‐year assessment. A multivariate analysis of variance indicated several differences favoring the intervention group. Both the high‐ and low‐level intervention subgroups outperformed the comparison group in their ability to (a) provide reasoning, (b) attempt to use formal mathematical vocabulary, and (c) correctly use formal mathematical vocabulary in their writing. The low‐level intervention subgroup also outperformed the respective comparison subgroup in their use of (a) complete sentences and (b) linking words. There were no differences between groups in their attempt at writing and attempts at and usage of informal mathematical vocabulary.     

Evolution processes of coupled thermal qubits

We investigate the quantum speed limit time (QSLT) of quantum evolution before thermal equilibrium of two coupled qubits each of which is coupled to a separate thermal bath at the same temperature within the Born-Markov approximation. The evolution process in one particular initial state can change between speed-up and speed-down two times before reaching equilibrium. We call this double cusp behaviour. This behaviour is an anomalous phenomenon in evolution processes in the weak-coupling Markovian regime. We study QSLT corresponding to all pure initial energy eigenstates and categorise them. In addition, we also display the conditions for double cusp behaviour in terms of temperature, qubit interaction and frequency.     

Enhancing Formative Assessment Practice and Encouraging Middle School Mathematics Engagement and Persistence

In the transition to middle school, and during the middle school years, students' motivation for mathematics tends to decline from what it was during elementary school. Formative assessment strategies in mathematics can help support motivation by building confidence for challenging tasks. In this study, the authors developed and piloted a professional development program, Learning to Use Formative Assessment in Mathematics with the Assessment Work Sample Method (AWSM) to build middle school math teachers' understanding of the characteristics of high‐quality formative assessment processes and increases their ability to use them in their classrooms. AWSM proved to be feasible to implement in the middle school setting. It improved teachers' practice of formative assessment, especially in their feedback practices, regardless of their pedagogical content knowledge at entry. Results from focus groups suggested that teachers were better able to implement ungraded practice and student self‐ and peer‐assessment after AWSM, and that students were more willing to engage in complex problem solving.     

MEXIT: Maximal un-coupling times for stochastic processes

Classical coupling constructions arrange for copies of the same Markov process started at two different initial states to become equal as soon as possible. In this paper, we consider an alternative coupling framework in which one seeks to arrange for two different Markov (or other stochastic) processes to remain equal for as long as possible, when started in the same state. We refer to this “un-coupling” or “maximal agreement” construction as MEXIT, standing for “maximal exit”. After highlighting the importance of un-coupling arguments in a few key statistical and probabilistic settings, we develop an explicit MEXIT construction for stochastic processes in discrete time with countable state-space. This construction is generalized to random processes on general state-space running in continuous time, and then exemplified by discussion of MEXIT for Brownian motions with two different constant drifts.     

Preservice teachers’ mathematics task modification for emergent bilinguals

Implementing mathematically challenging tasks is difficult for teachers when working with emergent bilinguals because cognitively demanding tasks in mathematics commonly have high language demand. Currently, inadequate teacher preparation for teaching emergent bilinguals is becoming a significant concern in the United States as this population of students is rapidly growing. This study investigated how two mathematics preservice teachers (PSTs) support middle school emergent bilinguals to understand cognitively demanding mathematical problems through task modification. Fieldwork with a concurrent intervention was designed for the PSTs to work with emergent bilinguals in a one‐on‐one setting. The PSTs modified cognitively demanding mathematics tasks and designed a lesson for the emergent bilinguals based on the modified tasks. The results revealed that the task modification made by the PSTs tended to shift from reducing cognitive demands in mathematics and language to maintaining the demands through learning strategies of contextual support.     

On the domain of fractional Laplacians and related generators of Feller processes

In this paper we study the domain of the generator of stable processes, stable-like processes and more general pseudo- and integro-differential operators which naturally arise both in analysis and as infinitesimal generators of Lévy- and Lévy-type (Feller) processes. In particular we obtain conditions on the symbol of the operator ensuring that certain (variable order) Hölder and Hölder–Zygmund spaces are in the domain. We use tools from probability theory to investigate the small-time asymptotics of the generalized moments of a Lévy or Lévy-type process ${(X_t)}_{t\ge 0}$,

$\underset{t\to 0}{\lim }?\frac{1}{t}({\mathbb{E}}^{x}f(X_t)?f(x)),x\in {\mathbb{R}}^d,$

for functions f which are not necessarily bounded or differentiable. The pointwise limit exists for fixed $x\in {\mathbb{R}}^d$ if f satisfies a Hölder condition at x. Moreover, we give sufficient conditions which ensure that the limit exists uniformly in the space of continuous functions vanishing at infinity. As an application we prove that the domain of the generator of ${(X_t)}_{t\ge 0}$ contains certain Hölder spaces of variable order. Our results apply, in particular, to stable-like processes, relativistic stable-like processes, solutions of Lévy-driven SDEs and Lévy processes.     

Bounds and inequalities for single server loss systems

We consider a single server loss system in which arrivals occur according to a doubly stochastic Poisson process with a stationary ergodic intensity functionλ _t . The service times are independent, exponentially distributed r.v.'s with meanμ ⁻¹, and are independent of arrivals. We obtain monotonicity results for loss probabilities under time scaling as well as under amplitude scaling ofλ _t . Moreover, using these results we obtain both lower and upper bounds for the loss probability.     

强相关平稳Gauss过程高水平穿过和最大值的弱收敛

设｛ξ（ｔ），ｔ≥０｝是强相关不可微Ｇａｕｓ过程，即其相关函数Ｒ（ｔ）满足：Ｒ（ｔ）ｌｏｇｔ→γ＞０（ｔ→∞）和Ｒ（ｔ）＝１－Ｃ｜ｔ｜^α＋ｏ（｜ｔ｜^α）（ｔ→０），且０＜α≤２，Ｃ＞０．本文建立了｛ξ（ｔ），ｔ≥０｝对一个和多个高水平的ε-上穿点过程的极限定理，并给出了最大值的极限分布和局部ε-最大值的联合渐近分布．     

Stochastic resonance in an ion channel following the non-Arrhenius gating rate

Stochastic resonance (SR) is a novel cooperative phenomenon occurring in nonlinear systems due to coupling of an ambient noise and an external signal. Biological systems may use SR mechanism to detect the signal efficiently from an external environment. A number of studies have addressed the SR in artificial ion channels considering external voltages as noises. More important than these external noises is the internal, thermal noise which changes the channel conformations essential for biological functions. In this work, we consider that the channel gating rates follow a non-Arrhenius temperature dependence derived from experimental data of a real biological channel. Using the Monte-Carlo simulations, we find that in this channel SR occurs near a physiological temperature in a very distinctive manner compared with that for the Arrhenius gating model.