Supporting the development of algebraic thinking in middle school: a closer look at students’ informal strategies

This study investigated how 31 sixth-, seventh-, and eighth-grade middle school students who had not previously, nor were currently taking a formal Algebra course, approached word problems of an algebraic nature. Specifically, these algebraic word problems were of the form x + (x + a) + (x + b) = c or ax + bx + cx = d. An examination of students’ understanding of the relationships expressed in the problems and how they used this information to solve problems was conducted. Data included the students’ written responses to problems, field notes of researcher-student interactions while working on the problems, and follow-up interviews. Results showed that students had many informal strategies for solving the problems with systematic guess and check being the most common approach. Analysis of researcher-student interactions while working on the problems revealed ways in which students struggled to engage in the problems. Support mechanisms for students who struggle with these problems are suggested. Finally, implications are provided for drawing upon students’ informal and intuitive knowledge to support the development of algebraic thinking.     

Fritz Reiche and the Emergency Committee in Aid of Displaced Foreign Scholars

I discuss the family background and early life of the German theoretical physicist Fritz Reiche (1883–1969) in Berlin; his higher education at the University of Berlin under Max Planck (1858–1947); his subsequent work at the University of Breslau with Otto Lummer (1860–1925); his return to Berlin in 1911, where he completed his Habilitation thesis in 1913, married Bertha Ochs the following year, became a friend of Albert Einstein (1879–1955), and worked during and immediately after the Great War. In 1921 he was appointed as ordentlicher Professor of Theoretical Physics at the University of Breslau and worked there until he was dismissed in 1933. He spent the academic year 1934–1935 as a visiting professor at the German University in Prague and then returned to Berlin, where he remained until, with the crucial help of his friend Rudolf Ladenburg (1882–1952) and vital assistance of the Emergency Committee in Aid of Displaced Foreign Scholars, he, his wife Bertha, and their daughter Eve were able to emigrate to the United States in 1941 (their son Hans had already emigrated to England in 1939).From 1941–1946 he held appointments at the New School for Social Research in New York, the City College of New York, and Union College in Schenectady, New York, and then was appointed as an Adjunct Professor of Physics at New York University, where his contract was renewed year-by-year until his retirement in 1958.     

置换群关于变换后基的强生成集的算法

本文给出置换群关于变换后的基的强生成集的一种算法及其理论证明,并讨论了既约强生成集和最小强生成集的问题。     

On σ-commutativity in Menger algebras of n-place functions

We present an abstract characterization of various types of Menger algebras of n-place functions allowing certain permutations of variables.     

Spectral properties of matrix polynomials in the max algebra

We consider the spectral properties of matrix polynomials over the max algebra. In particular, we show how the Perron-Frobenius theorem for the max algebra extends to such polynomials and illustrate the relevance of this for multistep difference equations in the max algebra. We also present a number of inequalities for the largest max eigenvalue of a matrix polynomial.     

有限格蕴涵代数的零化子

研究有限格蕴涵代数的零化子,找出有限格蕴涵代数所有理想的零化子,并证明对有限格蕴涵代数的理想做零化子运算(记为0*)是一个逆序对合算子,因此在由有限格蕴涵代数L的所有理想所组成的集合∑(L)上定义一个蕴涵算子,则(∑(L),O,L,0*,)构成一个格蕴涵代数。     

玻恩和沃尔夫合著的《光学原理》一书写作过程

《光学原理》(Principles of Optics)一书是玻恩与沃尔夫合著的、倍受好评的光学经典名著。回顾、梳理与品味这一名著的撰写过程,会得到诸多感悟。教书育人、著书立说的物理教育家以及物理学家,都可以从中得到一把尺子,用以衡量自己的工作。这有利于中国物理学的发展。     

KamiWaAI – interactive 3D sketching with Java based on Cl(4, 1) conformal model of Euclidean space

This paper introduces the new interactive Java sketching software KamiWaAi, recently developed at the University of Fukui. Its graphical user interface enables the user without any knowledge of both mathematics or computer science, to do full three dimensional “drawings” on the screen. The resulting constructions can be reshaped interactively by dragging its points over the screen. The programming approach is new. KamiWaAi implements geometric objects like points, lines, circles, spheres, etc. directly as software objects (Java classes) of the same name. These software objects are geometric entities mathematically defined and manipulated in a conformal geometric algebra, combining the five dimensions of origin, three space and infinity. Simple geometric products in this algebra represent geometric unions, intersections, arbitrary rotations and translations, projections, distance, etc. To ease the coordinate free and matrix free implementation of this fundamental geometric product, a new algebraic three level approach is presented. Finally details about the Java classes of the new GeometricAlgebra software package and their associated methods are given. KamiWaAi is available for free internet download.     

The Politics of Memory: Otto Hahn and the Third Reich

As President of the Kaiser Wilhelm Society and its successor, the Max Planck Society, from 1946 until 1960, Otto Hahn (1879–1968) sought to portray science under the Third Reich as a purely intellectual endeavor untainted by National Socialism. I outline Hahn’s activities from 1933 into the postwar years, focusing on the contrast between his personal stance during the National Socialist period, when he distinguished himself as an upright non-Nazi, and his postwar attitude, which was characterized by suppression and denial of Germany’s recent past. Particular examples include Hahn’s efforts to help Jewish friends; his testimony for colleagues involved in denazification and on trial in Nuremberg; his postwar relationships with émigré colleagues, including Lise Meitner; and his misrepresentation of his wartime work in the Kaiser Wilhelm Institute for Chemistry.     

Attention to meaning by algebra teachers

Non-attendance to meaning by students is a prevalent phenomenon in school mathematics. Our goal is to investigate features of instruction that might account for this phenomenon. Drawing on a case study of two high school algebra teachers, we cite episodes from the classroom to illustrate particular teaching actions that de-emphasize meaning. We categorize these actions as pertaining to (a) purpose of new concepts, (b) distinctions in mathematics, (c) mathematical terminology, and (d) mathematical symbols. The specificity of the actions that we identify allows us to suggest several conjectures as to the impact of the teaching practices observed on student learning: that students will develop the belief that mathematics involves executing standard procedures much more than meaning and reasoning, that students will come to see mathematical definitions and results as coincidental or arbitrary, and that students’ treatment of symbols will be largely non-referential.