Stages in the History of Algebra with Implications for Teaching

In this article, we take a rapid journey through the history of algebra, noting the important developments and reflecting on the importance of this history in the teaching of algebra in secondary school or university. Frequently, algebra is considered to have three stages in its historical development: the rhetorical stage, the syncopated stage, and the symbolic stage. But besides these three stages of expressing algebraic ideas, there are four more conceptual stages which have happened along side of these changes in expressions. These stages are the geometric stage, where most of the concepts of algebra are geometric ones; the static equation-solving stage, where the goal is to find numbers satisfying certain relationships; the dynamic function stage, where motion seems to be an underlying idea, and finally, the abstract stage, where mathematical structure plays the central role. The stages of algebra are, of course not entirely disjoint from one another; there is always some overlap. We discuss here high points of the development of these stages and reflect on the use of these historical stages in the teaching of algebra. Commentary from a Mathematics Educator Bill Barton. See also the last page. Commentary from a Mathematics Educator     

Recording the Creative Process: An Empirical Basis for Practice‐Integrated Research in the Arts

A case is made for a form of narrative reporting (the Creative Process Journal) as a methodology for practice‐integrated research in the arts. It is argued that this stage of research creativity, which applies in all domains of academic study but is often not reported, is fundamental to the kind of arts research which allocates practice a central role. The practical and technological character of making a CPJ, and its consequent benefits to the maker‐researcher are outlined.     

Examining the Exam: A Critical Look at The California Critical Thinking Skills Test

This paper examines the content of The California Critical Thinking Skills Test (1990). This report is not a statistical review. Instead it brings under scrutiny the content of the exam. This content will be of interest to the general reader, because the issues range from logic to ethics to pedagogy, and to questions of evidential and epistemological support. Anyone interested in clear thought and expression will find these issues of significance. Although the exam has a number of strengths and has the clearest instructions of all the presently available Critical Thinking exams, the content of 9 of the exams 34 questions is defective, namely the content of questions 6, 7, 8, 19, 21, 23, 24, 29, and 33. These questions make errors in critical thinking. Hence, no statistical results pertaining to the administration of these questions to students can be acceptable. The remaining questions are acceptable as to content. But until the problems are corrected, those who may use the exam should remove the defective questions from test administration or from data collection and reporting.The scope of the exam also is quite limited, but this may be unavoidable for any instrument designed to be completed in about an hour. Further, the scores resulting from any such testing can be understood only as a measure of minimal competency (below which remediation likely is needed) for the skills tested, but not as an adequate measure of critical thinking.Disclosure: Three of the authors are engaged in producing and marketing a critical thinking test. Though this paper was written before any of us considered developing such a test, the reader should be informed. Each of the writers has exercised considerable care to avoid any bias, and we thank our independent reviewers for helping us in this regard as well.     

我的网上教学三原则

比尔.佩尔茨博士是纽约州立大学赫基默社区学院（Herkimer County Community College）（HC-CC）大学心理学教授。比尔.佩尔茨教授可以算作是HCCC大学的元老级教授,他是在1968年8月进入该校任教,第二年这所大学才开始正式运营。在38年的教学生涯中,他曾经担任过人文社科部主席,远程学习部主任等职,但是他一直没有中断他最热爱的事业-教学。1994年,他获得了“纽约州立大学系统校长优秀教学奖”;2003年,由于他在网络教学方面的尝试和出色的成果,获得斯隆基金会“Sloan-C2003年度优秀网上教学奖”-这些大奖是对他在教学方面研究和钻研的肯定。佩尔茨发表了多篇论文,近年来,他的研究兴趣集中在学生和教师对网络异步教学的满意度方面的研究和实践,目前他正在研究在虚拟学习环境中有哪些教学因素影响学生的学业成就。除了担任全职的网络教师以外,佩尔茨还是HCCC网络学院的负责人,HCCC教学设计专家。此外,他还担任SLN的主要培训师,在过去的五年里,他已经把1000多名来自纽约州立大学系统各个大学的普通教师培养成了网络教师。他还代表纽约州立大学系统担任Merlot项目的心理学科代表,Merlot是美国高校开展的一项精品网络教学资源建设和共享计划,目的是建立美国高等教育精品资源库。     

A Canadian Perspective on the Uncertain Future of Distance Education

Distance education as a field of educational endeavor is at a crucial juncture in its historical development. The notion of learning at a distance has gained wide acceptance across the developed world. Instructors, physically and temporally separated from learners using newly emerging information and communication technologies, are widespread. Most postsecondary institutions have employed, or plan to develop, forms of “distance‐like” educational activities for both campus‐based and remote students. Yet, as acceptance has increased for such learning activities, many distance educators feel that they are no longer recognized as leaders in the emerging field of technology‐enhanced, hybrid, flexible learning environments. This article asks a series of questions which will help to illuminate the challenges facing distance educators at the beginning of the 21st century.     

Temperamental Origins of Child and Adolescent Behavior Problems: From Age Three to Age Fifteen

We assessed relations between early temperament and behavior problems across 12 years in an unselected sample of over 800 children. Temperament measures were drawn from behavior ratings made by examiners who observed children at ages 3, 5, 7, and 9. Factor analyses revealed 3 dimensions at each age: Lack of Control, Approach, and Sluggishness . Temperament dimensions at ages 3 and 5 were correlated in theoretically coherent ways with behavior problems that were independently evaluated by parents and teachers at ages 9 and 11, and by parents at ages 13 and 15. Lack of Control was more strongly associated with later externalizing behavior problems than with internalizing problems; Approach was associated with fewer internalizing problems among boys; and Sluggishness was weakly associated with both anxiety and inattention, especially among girls. Lack of Control and Sluggishness were also associated with fewer adolescent competencies. These results suggest that early temperament may have predictive specificity for the development of later psychopathology.