Math discourse is a technique that works as well virtually as it does on paper or in face-to-face classrooms, according to experts.

“Detracking” math teachers is tough because many educators resist upending their routines or challenging informal hierarchies, and PD initiatives to make it happen are limited.

A sizable body of research shows that intensive, one-on-one coaching can improve instructional practice and student achievement more than other professional development offerings for teachers.

Experts recommend emphasizing language skills, avoiding assumptions about ability based on broad student labels, and focusing on students’ strengths rather than their weaknesses.

A program in Howard County, Md., is built on the insight that children can have strong emotions around academics, and those emotions can sabotage learning.

Educators are changing instructional priorities, altering lessons, and working on ways to help teachers grow professionally, all in an effort to raise math achievement.

  Photos by Whitney Curtis

The children exposed to high levels of lead-laced drinking water from Michigan's Flint River are entering schools now and the school system is straining to meet their special education needs.

Monte Carlo simulation is useful to compute or estimate expected functionals of random elements if those random samples are possible to be generated from the true distribution. However, when the distribution has some unknown parameters, the samples must be generated from an estimated distribution with the parameters replaced by some estimators, which causes a statistical error in Monte Carlo estimation. This paper considers such a statistical error and investigates the asymptotic distributions of Monte Carlo-based estimators when the random elements are not only the real valued, but also functional valued random variables. We also investigate expected functionals for semimartingales in details. The consideration indicates that the Monte Carlo estimation can get worse when a semimartingale has a jump part with unremovable unknown parameters.

We show, by an explicit construction, that a mixture of univariate Gaussians with variance 1 and means in $[-A,A]$ can have $Omega(A^2)$ modes. This disproves a recent conjecture of Dytso, Yagli, Poor and Shamai [IEEE Trans. Inform. Theory, Apr. 2020], who showed that such a mixture can have at most $O(A^2)$ modes and surmised that the upper bound could be improved to $O(A)$. Our result holds even if an additional variance constraint is imposed on the mixing distribution. Extending the result to higher dimensions, we exhibit a mixture of Gaussians in $mathbb{R}^d$, with identity covariances and means inside $[-A,A]^d$, that has $Omega(A^{2d})$ modes.

In the Gaussian sequence model $Y= heta_0 + varepsilon$ in $mathbb{R}^n$, we study the fundamental limit of approximating the signal $ heta_0$ by a class $Theta(d,d_0,k)$ of (generalized) splines with free knots. Here $d$ is the degree of the spline, $d_0$ is the order of differentiability at each inner knot, and $k$ is the maximal number of pieces. We show that, given any integer $dgeq 0$ and $d_0in{-1,0,ldots,d-1}$, the minimax rate of estimation over $Theta(d,d_0,k)$ exhibits the following phase transition: egin{equation*} egin{aligned} inf_{widetilde{ heta}}sup_{ hetainTheta(d,d_0, k)}mathbb{E}_ heta|widetilde{ heta} - heta|^2 asymp_d egin{cases} kloglog(16n/k), & 2leq kleq k_0,\ klog(en/k), & k geq k_0+1. end{cases} end{aligned} end{equation*} The transition boundary $k_0$, which takes the form $lfloor{(d+1)/(d-d_0) floor} + 1$, demonstrates the critical role of the regularity parameter $d_0$ in the separation between a faster $log log(16n)$ and a slower $log(en)$ rate. We further show that, once encouraging an additional '$d$-monotonicity' shape constraint (including monotonicity for $d = 0$ and convexity for $d=1$), the above phase transition is eliminated and the faster $kloglog(16n/k)$ rate can be achieved for all $k$. These results provide theoretical support for developing $ell_0$-penalized (shape-constrained) spline regression procedures as useful alternatives to $ell_1$- and $ell_2$-penalized ones.

Area under ROC curve (AUC) is a widely used performance measure for classification models. We propose two new distributionally robust AUC maximization models (DR-AUC) that rely on the Kantorovich metric and approximate the AUC with the hinge loss function. We consider the two cases with respectively fixed and variable support for the worst-case distribution. We use duality theory to reformulate the DR-AUC models and derive tractable convex optimization problems. The numerical experiments show that the proposed DR-AUC models -- benchmarked with the standard deterministic AUC and the support vector machine models - perform better in general and in particular improve the worst-case out-of-sample performance over the majority of the considered datasets, thereby showing their robustness. The results are particularly encouraging since our numerical experiments are conducted with training sets of small size which have been known to be conducive to low out-of-sample performance.

A graph homomorphism is a map between two graphs that preserves adjacency relations. We consider the problem of sampling a random graph homomorphism from a graph $F$ into a large network $mathcal{G}$. We propose two complementary MCMC algorithms for sampling a random graph homomorphisms and establish bounds on their mixing times and concentration of their time averages. Based on our sampling algorithms, we propose a novel framework for network data analysis that circumvents some of the drawbacks in methods based on independent and neigborhood sampling. Various time averages of the MCMC trajectory give us various computable observables, including well-known ones such as homomorphism density and average clustering coefficient and their generalizations. Furthermore, we show that these network observables are stable with respect to a suitably renormalized cut distance between networks. We provide various examples and simulations demonstrating our framework through synthetic networks. We also apply our framework for network clustering and classification problems using the Facebook100 dataset and Word Adjacency Networks of a set of classic novels.

We study the problem of parameter estimation for a non-ergodic Gaussian Vasicek-type model defined as $dX_t=(mu+ heta X_t)dt+dG_t, tgeq0$ with unknown parameters $ heta>0$ and $muinR$, where $G$ is a Gaussian process. We provide least square-type estimators $widetilde{ heta}_T$ and $widetilde{mu}_T$ respectively for the drift parameters $ heta$ and $mu$ based on continuous-time observations ${X_t, tin[0,T]}$ as $T ightarrowinfty$.

Our aim is to derive some sufficient conditions on the driving Gaussian process $G$ in order to ensure that $widetilde{ heta}_T$ and $widetilde{mu}_T$ are strongly consistent, the limit distribution of $widetilde{ heta}_T$ is a Cauchy-type distribution and $widetilde{mu}_T$ is asymptotically normal. We apply our result to fractional Vasicek, subfractional Vasicek and bifractional Vasicek processes. In addition, this work extends the result of cite{EEO} studied in the case where $mu=0$.

The $p$-tensor Curie-Weiss model is a two-parameter discrete exponential family for modeling dependent binary data, where the sufficient statistic has a linear term and a term with degree $p geq 2$. This is a special case of the tensor Ising model and the natural generalization of the matrix Curie-Weiss model, which provides a convenient mathematical abstraction for capturing, not just pairwise, but higher-order dependencies. In this paper we provide a complete description of the limiting properties of the maximum likelihood (ML) estimates of the natural parameters, given a single sample from the $p$-tensor Curie-Weiss model, for $p geq 3$, complementing the well-known results in the matrix ($p=2$) case (Comets and Gidas (1991)). Our results unearth various new phase transitions and surprising limit theorems, such as the existence of a 'critical' curve in the parameter space, where the limiting distribution of the ML estimates is a mixture with both continuous and discrete components. The number of mixture components is either two or three, depending on, among other things, the sign of one of the parameters and the parity of $p$. Another interesting revelation is the existence of certain 'special' points in the parameter space where the ML estimates exhibit a superefficiency phenomenon, converging to a non-Gaussian limiting distribution at rate $N^{frac{3}{4}}$. We discuss how these results can be used to construct confidence intervals for the model parameters and, as a byproduct of our analysis, obtain limit theorems for the sample mean, which provide key insights into the statistical properties of the model.

This paper studies a service system in which arriving customers are provided with information about the delay they will experience. Based on this information they decide to wait for service or to leave the system. The main objective is to estimate the customers' patience-level distribution and the corresponding potential arrival rate, using knowledge of the actual workload process only. We cast the system as a queueing model, so as to evaluate the corresponding likelihood function. Estimating the unknown parameters relying on a maximum likelihood procedure, we prove strong consistency and derive the asymptotic distribution of the estimation error. Several applications and extensions of the method are discussed. In particular, we indicate how our method generalizes to a multi-server setting. The performance of our approach is assessed through a series of numerical experiments. By fitting parameters of hyperexponential and generalized-hyperexponential distributions our method provides a robust estimation framework for any continuous patience-level distribution.

Hyperbolic balance laws with uncertain (random) parameters and inputs are ubiquitous in science and engineering. Quantification of uncertainty in predictions derived from such laws, and reduction of predictive uncertainty via data assimilation, remain an open challenge. That is due to nonlinearity of governing equations, whose solutions are highly non-Gaussian and often discontinuous. To ameliorate these issues in a computationally efficient way, we use the method of distributions, which here takes the form of a deterministic equation for spatiotemporal evolution of the cumulative distribution function (CDF) of the random system state, as a means of forward uncertainty propagation. Uncertainty reduction is achieved by recasting the standard loss function, i.e., discrepancy between observations and model predictions, in distributional terms. This step exploits the equivalence between minimization of the square error discrepancy and the Kullback-Leibler divergence. The loss function is regularized by adding a Lagrangian constraint enforcing fulfillment of the CDF equation. Minimization is performed sequentially, progressively updating the parameters of the CDF equation as more measurements are assimilated.

In this paper we obtain the limit distribution for partial sums with a random number of terms following a class of mixed Poisson distributions. The resulting weak limit is a mixing between a normal distribution and an exponential family, which we call by normal exponential family (NEF) laws. A new stability concept is introduced and a relationship between {alpha}-stable distributions and NEF laws is established. We propose estimation of the parameters of the NEF models through the method of moments and also by the maximum likelihood method, which is performed via an Expectation-Maximization algorithm. Monte Carlo simulation studies are addressed to check the performance of the proposed estimators and an empirical illustration on financial market is presented.

We develop a general framework for data-driven approximation of input-output maps between infinite-dimensional spaces. The proposed approach is motivated by the recent successes of neural networks and deep learning, in combination with ideas from model reduction. This combination results in a neural network approximation which, in principle, is defined on infinite-dimensional spaces and, in practice, is robust to the dimension of finite-dimensional approximations of these spaces required for computation. For a class of input-output maps, and suitably chosen probability measures on the inputs, we prove convergence of the proposed approximation methodology. Numerically we demonstrate the effectiveness of the method on a class of parametric elliptic PDE problems, showing convergence and robustness of the approximation scheme with respect to the size of the discretization, and compare our method with existing algorithms from the literature.

Nina Gantert, Markus Heydenreich, Timo Hirscher.

Source: Bernoulli, Volume 26, Number 2, 1269--1293.

Abstract:
We investigate a model for opinion dynamics, where individuals (modeled by vertices of a graph) hold certain abstract opinions. As time progresses, neighboring individuals interact with each other, and this interaction results in a realignment of opinions closer towards each other. This mechanism triggers formation of consensus among the individuals. Our main focus is on strong consensus (i.e., global agreement of all individuals) versus weak consensus (i.e., local agreement among neighbors). By extending a known model to a more general opinion space, which lacks a “central” opinion acting as a contraction point, we provide an example of an opinion formation process on the one-dimensional lattice $mathbb{Z}$ with weak consensus but no strong consensus.

YOU could call it state-sponsored sanctimony. In times of crisis or national emergency we’re all urged to pull in the same direction and put partisan politics behind us. How dare you talk about inequality and the plight of the disadvantaged at a time like this? Those who tend to be loudest in rebuking these social pariahs are often those who stand to benefit most from any suspension of scrutiny.

The Long Beach, Calif., school district is deploying a multifaceted strategy to put more students of color in high-level math courses and help them succeed.

The largest experiment to date comparing commercial math curricula gives a slight edge to two popular programs.

Marlo Warburton, a 7th and 8th grade math teacher at Longfellow Arts and Technology Middle School in Berkeley, Calif., shares how greeting her students in the morning and expressing appreciation during dismissal are valuable opportunities for character building and for fostering teacher-student rela

In 2007, Joel Rose conceived an idea for an innovative, blended way to teach middle school math. Today, it has spread to over 40 schools reaching 13,000 students. Here's how.

It's part of the foundation's $425 million research and development push, announced last fall.

Students with dyslexia often struggle with math fluency as well, and scholars at a recent conference put a spotlight on some of the possible connections.

Makeda Brome, Pia Hansen, Linda Gojak, Marian Small, Kenneth Baum and David Krulwich share their thoughts on the biggest challenges facing math teachers.

Teachers must reject the idea that math is like eating vegetables, says former offensive lineman and current mathematician John Urschel.

An overemphasis on calculus in high school may be harming students, writes Dickinson College professor Jeffrey Forrester.

A student-centered approach to teaching mathematics enables students to develop conceptual understanding and to grow as confident mathematicians.

Timed tests and "drill-and-kill" approaches to math education can leave students with long-lasting anxiety, writes researcher Jennifer Ruef. There's a better way to teach the subject.

Better mathematics screening procedures may help schools choose students for 8th grade Algebra 1 classes who will be able to successfully complete the course, according to a study by the Regional Educational Laboratory West.

Though boys typically perform better in mathematics, a new study shows that girls' superior verbal skills tend to make them better at arithmetic.

High school math classes should be broadened to focus on goals beyond college and careers, including teaching the math students will need to be literate participants in civic life.

Helping students to categorize different types of word problems can help elementary-age students tackle a common challenge in math class, according to a new analysis of 21 studies in the journal Review of Educational Research.

A handful of educators across the country are quietly making the case that math may be the missing piece in civics education.

Comic books and graphic novels, popular in many language arts and social studies classes, are just now tiptoeing into the world of K-12 math.

Boys and girls start out on the same biological footing when it comes to math, finds the first neuroimaging study of math gender differences in children, published this month in the journal Science of Learning.

Political numeracy is as important as it is overlooked, argues Wellesley mathematics professor Ismar Volić.

Math discourse is a technique that works as well virtually as it does on paper or in face-to-face classrooms, according to experts.

Educators are changing instructional priorities, altering lessons, and working on ways to help teachers grow professionally, all in an effort to raise math achievement.

One in 4 teachers feel anxious doing math. This is having a big impact on what happens in the classroom.

A program in Howard County, Md., is built on the insight that children can have strong emotions around academics, and those emotions can sabotage learning.

Boys and girls start out on the same biological footing when it comes to math, according to the first neuroimaging study of math gender differences in children.

Boys and girls start out on the same biological footing when it comes to math, finds the first neuroimaging study of math gender differences in children, published this month in the journal Science of Learning.

For a moment, Wall Street seemed to be inching away from Saudi Arabia. Now, it’s already inching back.

The Culture, Tourism, Europe and External Affairs Committee has today issued a call for views from the culture and tourism industry on the current Covid-19 crisis.