Proof in School Mathematics: Insights from Psychological Research into Students' Ability for Deductive Reasoning

There are currently increased efforts to make proof central to school mathematics throughout the grades. Yet, realizing this goal is challenging because it requires that students master several abilities. In this article we focus on one such ability, namely, the ability for deductive reasoning, and we review psychological research to enhance what is currently known in mathematics education research about this ability in the context of proof and to identify important directions for future research. We first offer a conceptualization of proof, which we use to delineate our focus on deductive reasoning. We then review psychological research on the development of students' ability for deductive reasoning to see what can be said about the ages at which students become able to engage in certain forms of deductive reasoning. Finally, we review two psychological theories of deductive reasoning to offer insights into cognitively guided ways to enhance students' ability for deductive reasoning in the context of proof.     

Nonlinear function approximation: Computing smooth solutions with an adaptive greedy algorithm

In contrast to linear schemes, nonlinear approximation techniques allow for dimension independent rates of convergence. Unfortunately, typical algorithms (such as, e.g., backpropagation) are not only computationally demanding, but also unstable in the presence of data noise. While we can show stability for a weak relaxed greedy algorithm, the resulting method has the drawback that it requires in practise unavailable smoothness information about the data.In this work we propose an adaptive greedy algorithm which does not need this information but rather recovers it iteratively from the available data. We show that the generated approximations are always at least as smooth as the original function and that the algorithm also remains stable, when it is applied to noisy data. Finally, the applicability of this algorithm is demonstrated by numerical experiments.     

Subtotally positive and Monge matrices

A real matrix is called k-subtotally positive if the determinants of all its submatrices of order at most k are positive. We show that for an m × n matrix, only mn inequalities determine such class for every k, 1 ? k ? min(m,n). Spectral properties of square k-subtotally positive matrices are studied. Finally, completion problems for 2-subtotally positive matrices and their additive counterpart, the anti-Monge matrices, are investigated. Since totally positive matrices are 2-subtotally positive as well, the presented necessary conditions for this completion problem are also necessary conditions for totally positive matrices.     

The Method of Fundamental Solutions for Stokes flows with a free surface

We investigate the use of the Method of Fundamental Solutions (MFS) for solving Stokes flow problems with a free surface. We apply the method to the creeping planar Newtonian extrudate-swell problem and study the effect of the surface tension on the free surface. The results are in good agreement with existing finite element and boundary element solutions. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 667–678, 1998