The beginnings of the R.L. Moore school of topology

The roots of the R.L. Moore school of point set topology were formed around 1900, with Moore's most direct influences coming from his University of Texas teacher, G.B. Halsted, and from his graduate school teachers at the University of Chicago, mainly O. Veblen and E.H. Moore. It was recognized as a school by the 1930s as Moore and his students achieved recognition for their work and as this group interacted with the Polish school of topology. In addition to the mathematical subject, the other main factor that helped to identify this group as a school was the distinctive method of teaching Moore used to introduce the subject to students.     

On the use of history of mathematics: an introduction to Galileo's study of free fall motion

In this paper, we report on an experimental activity for discussing the concepts of speed, instantaneous speed and acceleration, generally introduced in first year university courses of calculus or physics. Rather than developing the ideas of calculus and using them to explain these basic concepts for the study of motion, we led 82 first year university students through Galileo's experiments designed to investigate the motion of falling bodies, and his geometrical explanation of his results, via simple dynamic geometric applets designed with GeoGebra. Our goal was to enhance the students’ development of mathematical thinking. Through a scholarship of teaching and learning study design, we captured data from students before, during and after the activity. Findings suggest that the historical development presented to the students helped to show the growth and evolution of the ideas and made visible authentic ways of thinking mathematically. Importantly, the activity prompted students to question and rethink what they knew about speed and acceleration, and also to appreciate the novel concepts of instantaneous speed and acceleration at which Galileo arrived.     

数学建模课内容和教学方法的探讨

拟就以下内容进行了探讨 .(i)该课程究竟应该讲什么内容、怎样讲 ,才能使学生在较短的时间内 ,掌握数学建模的基本知识和基本方法 ;(ii)该课程怎样与数学实验更好地结合起来 ,以培养学生的动手能力 ;(iii)该课程应采用什么样的教学手段和教学方法 ,才能加大课堂信息量 ,加强直观性和趣味性等 .我们的解决方法是 :(i)以介绍建立数学模型为主 ,按数学知识内容的不同来选取数学模型的典型案例 ,通过案例介绍 ,使学生学会怎样建立模型 .(ii)适当介绍数学软件包 ,让学生掌握运用软件包来求解模型能力 .(iii)做大作业 ,教员给出题目 ,学生自己收集资料、讨论、上机求解 ,最后写出报告 .(iv)开展多媒体教学 ,对主要的教学内容进行模块化教学 ,将建模分成 1 4个专题 ,做成 1 4个多媒体课件     

Teachers' ways of listening and responding to students' emerging mathematical models

In this paper, I present the results of a case study of the practices of four experienced secondary teachers as they engaged their students in the initial development of mathematical models for exponential growth. The study focuses on two related aspects of their practices: (a) when, how and to what extent they saw and interpreted students' ways of thinking about exponential functions and (b) how they responded to the students' thinking in their classroom practice. Through an analysis of the teachers' actions in the classroom, I describe the teachers' developing knowledge when using modeling tasks with secondary students. The analysis suggests that there is considerable variation in the approaches that teachers take in listening to and responding to students' emerging mathematical models. Having a well-developed schema for how students might approach the task enabled one teacher to press students to express, evaluate, and revise their emerging models of exponential growth. Implications for the knowledge needed to teach mathematics through modeling are discussed.     

Student Difficulties with Accessing and Using Mathematical Knowledge

This article examines the issue of why students fail to activate and use mathematical knowledge during problem solving when it is known that they possess the required knowledge. This issue is explored by analyzing problem-solving attempts of a high-achieving student and a low-achieving student in the domain of plane geometry. On the basis of these data and other literature, three major sources of mathematical knowledge-access difficulties are identified that might be considered by classroom teachers, including a student's (1) dispositional state, (2) management of the problem-solving process, and (3) state of organization of his or her mathematical knowledge. It is argued that teaching practices that place emphasis on careful management of problem-solving activity could help students activate and extend the use of mathematical knowledge acquired in lesson activities.     

Alice Bache Gould: mathematician in search of war work, 1918

Alliances between American mathematics and the military were not well defined when the United States entered World War I in April of 1917. Although academic mathematicians – like other American faculty members and students – were among the strongest supporters of the cause, it took time to identify how and where to utilize their technical training in the war effort. This paper investigates the case of Alice Bach Gould, one of E H Moore's mathematics graduate students at the University of Chicago, and her efforts to support US military endeavors with her scientific expertise. Although Gould's substantial research accomplishments in Spanish archives eventually surpassed her mathematical achievements, her quest for war work in 1917 nonetheless illustrates the difficulty of contributing mathematical training to patriotic service in the United States during World War I.     

The missing link in teachers’ knowledge about common fractions division

The current study explored the difficulties teachers encounter when teaching common fractions division, focusing on teachers’ knowledge concerning this issue. Nine teachers who study towards a M.Ed. degree in mathematics education demonstrated the algorithms they apply in order to solve fractions division problems, described how they teach the subject, and attempted to explain a student's mistake, in understanding a word problem involving dividing by fraction. The findings indicate there is a missing link in the teachers’ pedagogical capability, stemming from insufficient content knowledge. They presented different solution algorithms and reported using constructivist teaching methods, yet the methods they described couldn't lead a student to understand the logic behind the algorithm they teach (invert-and-multiply – multiplication by an inverse number, in accordance with the requirements of the curriculum). Furthermore, the participating teachers did not possess specialized mathematics content knowledge (SCK) and knowledge of content and students (KCS), enabling them to identify the source of a student's misconception.     

对于工学硕士的《数理统计》教学的研究

工学硕士的《数理统计》教学 ,要根据工学硕士的培养目标和数理统计课程的特点编写教材、确定讲授内容和授课方式 .力求使学员掌握处理随机数据的思考方法 ,培养解决问题、分析问题和善于应用知识的能力 .改革《数理统计》教学 ,实现和普及多媒体教学是时代发展的趋势 .     

Using Sociocultural Theory to Teach Mathematics: A Vygotskian Perspective

This study describes an elementary teacher's implementation of sociocultural theory in practice. Communication is central to teaching with a sociocultural approach and to the understanding of students; teachers who use this theory involve students in explaining and justifying their thinking. In this study ethnographic research methods were used to collect data for 4 1/2 months in order to understand the mathematical culture of this fourth‐grade class and to portray how the teacher used a sociocultural approach to teach mathematics. To portray this teaching approach, teaching episodes from the teacher's mathematics lessons are described, and these episodes are analyzed to demonstrate how students created taken‐as‐shared meanings of mathematics. Excerpts from interviews with the teacher are also used to describe this teacher's thinking about her teaching.     

Expanding in-service mathematics teachers’ horizons in creative work using technology

This article describes the experiences gained from a seminar in the teaching of mathematical reasoning and problem solving designed to prepare in-service high school mathematics teachers to teach genuine mathematical activity in a computer-based environment. Presented with a set of unfamiliar tasks and activities, the participants were encouraged to investigate each of them, using the Geometer's Sketchpad software, and then to justify their assertions accordingly. In the exploratory process the student teachers make the major mathematical contributions while the teacher plays the role of facilitator. The mathematics teachers began to realize the power of technology in teaching mathematics and were pleased to return to their classrooms with a great number of experiences and ideas for immediate use.     

Accommodation in the formal world of mathematical thinking

ABSTRACT

In this study, we examined a mathematician and one of his students’ teaching journals and thought processes concurrently as the class was moving towards the proof of the Fundamental Theorem of Galois Theory. We employed Tall's framework of three worlds of mathematical thinking as well as Piaget's notion of accommodation to theoretically study the narratives. This paper reveals the pedagogical challenges of proving an elegant theory as the events unfolded. Although the mathematician was conscious of the students’ abilities as he carefully made the path accessible, the disparity between the mind of the mathematician and the student became apparent.     

Special Issue Article: A Look at Technology's Role in Professional Development of Mathematics Teachers at the Middle School Level

In last month's issue of School Science and Mathematics, Glenda Lappan addressed the dilemma of supporting teachers in continuing to grow professionally in four domains: Learning more mathematics content; improving their pedagogical skills; improving assessment skills; and addapting the curriculum to the needs of their students. Though it is clear that this kind of learning is best done in the context of what is going on in the classroom, teachers' schedules and other problems make it difficult to implement a coherent classroom based, professional development program. In this article, two more knowledge domains are added to this mix: the ability to (a) use technology and (b) teach with technology effectively. Since the process of learning and teaching is a dynamic one, the author describes his vision of the classroom as a laboratory where teachers get to practice and improve in these six areas and get feedback from an audience of their peers. Reflections are based on a current project in Paterson, New Jersey where the author helps middle school teachers use computer software to improve their mathematics teaching and learning.     

A Collaborative Effort to Improve University Engineering Instruction

This study focused on the instructional development of an assistant professor of environmental engineering in collaboration with science education and higher education faculty members. One semester of data was collected in the assistant professor's environmental engineering laboratory class as he endeavored to address his teaching goals. Data collection included pre and post interviews with the assistant professor, students, and program coordinator, and collection of course documents, such as the course syllabus and assignments. In addition, all of the classroom sessions were observed and videotaped, and a midsemester video stimulated‐recall interview was conducted. Results show the assistant professor made growth in the areas of questioning strategies, “think time” for students, increased class participation, and the implementation of a student‐designed field research project. Implications include that new professors can benefit from peer faculty support, and they and their students can benefit when the new professors recognize the complementary nature of research and teaching.     

Preparation for Practice: Elementary Preservice Teachers Learning and Using Scientific Classroom Discourse Community Instructional Strategies

Despite historical national efforts to improve elementary science education, science instruction continues to be marginalized, varying by state. This study was designed to address the ongoing challenge of educating elementary preservice teachers (PSTs) to teach science. Elementary PSTs are one of the science education community's major links to schools and science education reform. However, they often lack a strong background in science, knowledge of effective science teaching strategies, and consequently have low confidence and self‐efficacy. This investigation explored the initial learning of elementary PSTs using an interdisciplinary model of a scientific classroom discourse community during a science methods course. Findings post‐methods course suggested that the PSTs gained confidence in how to teach inquiry‐based elementary science and recognized inquiry‐based science as an effective means for engaging student learning. Additionally, PSTs embraced the interdisciplinary model as one that benefits students' learning and effectively uses limited time in a school day.     

Beginner’s mind and the middle years mathematics student

ABSTRACT

The Zen concept of beginner's mind describes how one's level of awareness can open one's mind to growth and possibilities, an attitude that would be beneficial for many mathematics students. In this naturalistic case study, two small groups of middle years students engage in the same mathematical task, one group demonstrating the characteristics of beginner's mind and the other not. One group focuses on making sure that all its members can follow the steps of one member's proposed solution. The other group, which displays characteristics of beginner's mind, explores the task thoroughly, and in its openness to ideas it notices mathematically salient details in the task that the first group overlooks completely. This paper suggests that through beginner's mind, students may develop habits of mind that enable them to attend to and think about mathematical possibilities deeply.     

The Influences of Three Interventions on Prospective Elementary Teachers' Beliefs About the Knowledge Base Needed for Teaching Mathematics

To meet the challenge to reform mathematics education, effective opportunities to learn are needed to promote prospective elementary school teachers' development of the knowledge base that supports teaching for mathematical proficiency. This article describes three professional development interventions and their influence on prospective teachers' beliefs about mathematics, how children learn mathematics, and mathematics teaching. The three interventions consisted of problem‐solving journals, structured interviews, and peer teaching that were integrated in a PreK‐6 mathematics methods course. Results of precourse and postcourse survey data are included that measured 24 prospective teachers' beliefs about the knowledge base needed to teach elementary school mathematics. Data indicated that using these interventions and other course experiences facilitated change in the prospective teachers' beliefs, with a shift toward reform‐oriented mathematics education perspectives.     

Ky Fan (1914�C2010), he spent every waking moment thinking about mathematics

This survey article on Dr. Ky Fan summarizes his versatile achievements and fundamental contributions in the fields of topological groups, nonlinear and convex analysis, operator theory, linear algebra and matrix theory, mathematical programming, and approximation theory, etc., and as well reveals Fan’s exemplary mathematical formation opening up the beauty of pure mathematics, with natural conditions, concise statements and elegant proofs. This article contains a brief biography of Dr. Fan and epitomizes his life. He was not only a great mathematician, but also a very serious teacher known to be extremely strict to his students. He loved his motherland and made generous donations for promoting mathematical development in China. He devoted his life to mathematics, continued his research and published papers till 85 years old.     

Calculators in Elementary School Mathematics Instruction

Due to the increased availability of hand-held calculators, teachers at all grade levels must now face the prospect of having to change both how they teach mathematics as well as what mathematics they teach. Since most teachers did not learn mathematics with the help of technology, they need time to adjust to both a new learning environment and a new teaching one. Through federal funds, the Texas Education Agency has created mathematics staff development modules which help teachers learn about calculators, mathematics, and the integration of calculators in mathematics instruction. This article presents games based upon those included in the staff development modules. Each game was designed to promote exploration of mathematical relationships via a calculator, specifically, Texas Instrument's Math Explorer.     

A case study of pedagogy of mathematics support tutors without a background in mathematics education

This study investigates the pedagogical skills and knowledge of three tertiary-level mathematics support tutors in a large group classroom setting. This is achieved through the use of video analysis and a theoretical framework comprising Rowland's Knowledge Quartet and general pedagogical knowledge. The study reports on the findings in relation to these tutors’ provision of mathematics support to first and second year undergraduate engineering students and second year undergraduate science students. It was found that tutors are lacking in various pedagogical skills which are needed for high-quality learning amongst service mathematics students (e.g. engineering/science/technology students), a demographic which have low levels of mathematics upon entering university. Tutors teach their support classes in a very fast didactic way with minimal opportunities for students to ask questions or to attempt problems. It was also found that this teaching method is even more so exaggerated in mandatory departmental mathematics tutorials that students take as part of their mathematics studies at tertiary level. The implications of the findings on mathematics tutor training at tertiary level are also discussed.     

Students' Retention of Mathematical Knowledge and Skills in Differential Equations

This study investigates students' retention of mathematical knowledge and skills in two differential equations classes. Posttests and delayed posttests after 1 year were administered to students in inquiry‐oriented and traditional classes. The results show that students in the inquiry‐oriented class retained conceptual knowledge, as seen by their performance on modeling problems, and retained equal proficiency in procedural problems, when compared with students in the traditionally taught classes. The results of this study add additional support to the claim that teaching for conceptual understanding can lead to longer retention of mathematical knowledge.