Introducing Young Children to the Role of Assumptions in Proving

The notion of assumptions permeates school mathematics, but instruction tends to highlight this notion only in the advanced grades. In this article, I argue that it is important for even young children to develop a sense of the role of assumptions in proving, and I investigate what it might mean and look like for instruction to promote this goal. Toward this end, I study an episode from third grade that describes the first time that the students in the class were introduced in a deliberate and explicit way to the role of assumptions in proving. The central role of the mathematical task in the episode is identified, and features of mathematical tasks that can generate rich mathematical activity in the intersection of assumptions and proving are discussed. In addition, issues of the role of teachers in fostering productive interactions between students and mathematical tasks that have those features are considered.     

Young Children Do Not Hold the Classic Earth's Shadow Misconception to Explain Lunar Phases

This research explored young children's early thoughts about natural phenomena and investigated sources of influence toward their knowledge construction. Two Piagetian interviews were conducted with four children. Each child was questioned about two phenomena in particular: (a) the moon and its changing appearance (moon phases) and (b) the formation of shadows. The first Piagetian interview involved a lunar protocol, and the second (given a week later) was a shadow protocol. The external interest of this research study concerns when and why do children develop the commonly held Earth's shadow misconception as the cause of the moon's phases. Findings showed that none of the young children believed that various lunar phases were due to an Earthly shadow cast upon the moon. The shadow interview results revealed that young children had primitive understandings of the shadow concept. In fact, two of the four children indicated that no light source was needed to produce a shadow.     

Slutsky定理在样本分位数的极限分布证明上的应用

为了更好地理解和应用样本分位数的极限分布,利用Slutsky定理,推导了样本分位数的极限分布.     

辅助函数在不等式证明中的应用

将不等式问题转化为函数问题,利用函数性质来研究、解决不等式问题.掌握不等式证明的一种函数思想方法,从而提高分析问题与解决问题的能力.     

Justifying and Proving in the Cabri Environment

This paper describes a long term teaching experiment carried out with students from the 9th–10th grades. Geometrical constructions in the Cabri environment were selected as a specific field of experience, within which the sense of theory may emerge. The idea of construction constitutes the key to accessing the idea of theorem, moving from a generic idea of justification towards the idea of validating within a geometrical system. The study aims at clarifying the role of the Cabri environment in this teaching-learning processes: analysis of protocols shows the possible evolution of a justification into a proof but at the same time indicates that this evolution is not expected to be simple and spontaneous.This revised version was published online in September 2005 with corrections to the Cover Date.     

On the Unavoidable Uncertainty of Truth in Dynamic Geometry Proving

The aim of this note is to discuss some issues posed by the emergency of universal interfaces able to decide on the truth of geometric statements. More specifically, we consider a recent GeoGebra module allowing general users to verify standard geometric theorems. Working with this module in the context of Varignon’s theorem, we were driven—by the characteristics of the GeoGebra interface—to perform a quite detailed study of the very diverse fate of attempting to automatically prove this statement, when using two different construction procedures. We highlight the relevance—for the theorem proving output—of expression power of the dynamic geometry interface, and we show that the algorithm deciding about the truth of some—even quite simple—statements can fall into a not true and not false situation, providing a source of confusion for a standard user and an interesting benchmark for geometers interested in discovering new geometric facts.     

Assumptions regarding right censoring in the presence of left truncation

Clinical studies using complex sampling often involve both truncation and censoring, where there are options for the assumptions of independence of censoring and event and for the relationship between censoring and truncation. In this paper, we clarify these choices, show certain equivalences, and provide examples.     

Relaxing the Assumptions of the Multilevel Single Linkage Algorithm

In this paper we relax the assumptions of a well known algorithm for continuous global optimization, Multilevel Single Linkage (MLSL). It is shown that the good theoretical properties of MLSL are shared by a slightly different algorithm, Non-monotonic MLSL (NM MLSL), but under weaker assumptions. The main difference with MLSL is the fact that in NM MLSL some non-monotonic sequences of sampled points are also considered in order to decide whether to start or not a local search, while MLSL only considers monotonic decreasing sequences. The modification is inspired by non-monotonic methods for local searches. This revised version was published online in July 2006 with corrections to the Cover Date.     

Encouraging Young Women to Stay in the Mathematics Pipeline: Mathematics Camps for Young Women

For two summers, week‐long residential mathematics programs were held for high school women, with the primary goal of encouraging them to continue their study of mathematics. The activities were designed to rekindle their excitement about mathematics and to support the idea that women should learn advanced mathematics. This paper reports the findings of the data collected to assess the programs. Statistically significant changes were found in student attitudes, confidence level, willingness to continue working on a problem, and perception of the value of group work. Journal entries confirmed the statistical analysis. Follow‐up surveys also indicated that the summer programs changed the way the participants perceived mathematics and mathematics courses.     

惯性定律的新证法

本文从实对称矩阵的角度 ,给出了惯性定律的一种新证法 .这种证法可使我们从已给的一个实二次型 f( x1 ,x2 ,… ,xn) =∑ni,j=1aijxixj 的矩阵 A=( aij) n× n的诸元素 ,来直接研究这个二次型的性质的这一体系更加完整 .同时 ,本文还大大地改进了雅可比方法 ,使雅可比的方法更加完美 ,应用更加广泛 .     

论线性代数证明题教学

本文强调了线性代数证明题教学在培养学生能力和提高素质方面的重要性 ,并通过例子分析在证明题教学过程中怎样培养学生的各方面能力 .