Expanding conceptions of utility: middle school students’ perspectives on the usefulness of mathematics

This research explores how adolescents conceptualize the usefulness of mathematics. Integrating sociocultural theory with the study of utility value, this study uses open-ended survey items and interview tasks to examine conceptions of usefulness among a group of predominantly Latinx middle school students. Findings reveal that students primarily conceptualized the usefulness of mathematics in two ways. First, students considered the applicability of mathematics content, focusing on applications of mathematics in everyday life and future jobs/careers. Second, students considered the usefulness of features of the learning experience, such as the form of interaction and structure of the activity. Both conceptions are compared to existing conceptions of usefulness in the literature, and implications for classroom practice and future research are discussed.     

Using the Knowledge Quartet to quantify mathematical knowledge in teaching: the development of a protocol for Initial Teacher Education

This study examined trainee teachers' mathematical knowledge in teaching (MKiT) over their final year in a US Initial Teacher Education (ITE) programme. This paper reports on an exploratory methodological approach taken to use the Knowledge Quartet to quantify MKiT through the development of a new protocol to code trainees' teaching of mathematics lessons. This approach extends Rowland's et al. work on the Knowledge Quartet (KQ). Justification for using the KQ to quantify MKiT, and the potential benefits such an attempt might provide those involved with ITE, are discussed. It is suggested that quantified MKiT data based on the Knowledge Quartet can be used to consider MKiT development in novice teachers in order to inform ITE programmes and form new theoretical loops between theory and practice in teacher education.     

Moments of wishart distribution

Moments of central and noncentral Wishart distributions are obtained by differentiating their characteristic functions applying matrix derivative techniques, using a special operator which takes into account the symmetry of the matrices. As a special case, higher moments of the multivariate normal distribution are obtained, arranged automatically in a square matrix form     

The inverse inertia problem for graphs: Cut vertices, trees, and a counterexample

Let G be an undirected graph on n vertices and let S(G) be the set of all real symmetric n×n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. The inverse inertia problem for G asks which inertias can be attained by a matrix in S(G). We give a complete answer to this question for trees in terms of a new family of graph parameters, the maximal disconnection numbers of a graph. We also give a formula for the inertia set of a graph with a cut vertex in terms of inertia sets of proper subgraphs. Finally, we give an example of a graph that is not inertia-balanced, which settles an open problem from the October 2006 AIM Workshop on Spectra of Families of Matrices described by Graphs, Digraphs and Sign Patterns. We also determine some restrictions on the inertia set of any graph.     

Computing inertia sets using atoms

We consider the problem of computing inertia sets for graphs. By using tools for combining the inertia sets of smaller graphs we can reduce this problem to understanding the inertia sets for three-connected graphs that are not joins. We term such graphs atoms and give the inertia sets for all atoms on at most seven vertices. This can be used to compute the inertia sets for all graphs on at most seven vertices.     

Learning to See: Making Sense of the Mathematics of Change in Middle School

In this article, we will describe the results of a study of 6th grade students learning about the mathematics of change. The students in this study worked with software environments for the computer and the graphing calculator that included a simulation of a moving elevator, linked to a graph of its velocity vs. time. We will describe how the students and their teacher negotiated the mathematical meanings of these representations, in interaction with the software and other representational tools available in the classroom. The class developed ways of selectively attending to specific features of stacks of centimeter cubes, hand-drawn graphs, and graphs (labeled velocity vs. time) on the computer screen. In addition, the class became adept at imagining the motions that corresponded to various velocity vs. time graphs. In this article, we describe this development as a process of learning to see mathematical representations of motion. The main question this article addresses is: How do students learn to see mathematical representations in ways that are consistent with the discipline of mathematics? This revised version was published online in July 2006 with corrections to the Cover Date.     

An alternative to ratio method in sample surveys

Summary This paper examines a simple transformation which enables the use of product method in place of ratio method. The convenience with the former, proposed by Murthy [3], is that expressions for bias and mean square error (mse) can be exactly evaluated. The optimum situation in the minimum mse sense and allowable departures from this optimum are indicated. The procedure requires a good guess of a certain parameter, which does not seem very restrictive for practice. Two methods for dealing with the bias of the estimator are mentioned. An extension to use multiauxiliary information is outlined.     

素理想完全分裂的一个判别法则

本文给出了数域K内一个素理想在K的有限次正规扩域L内完全分裂的一个判别法则。