A climbing girl's reflections about angles

The main research question in this paper is whether a climbing discourse can be a resource for a school-geometry discourse. The text is based on a 12-year old girl's story from an exciting climbing trip during her summer holiday. The girl uncovers some of her knowledge that had been invisible to her; she is guided to see some relations between her climbing and her understanding of angles. In the beginning, this girl believes her story does not concern angles at all. The tools for uncovering angles in her story are based on different levels of visibility and objects of the climbing discourse combined with different conceptions of space. The girl develops her consciousness about angles as natural elements in her climbing activity and she is guided to see the angle as an object of her climbing discourse.     

Learning about teaching for understanding through the study of tutoring

In this study, we conducted a fine-grained analysis of an expert tutor's (Nancy Mack) tutorial actions as she attempted, successfully, to help students learn fractions with understanding. Our analysis revealed that, as Mack tutored students in two different research studies, she took two types of tutorial actions previously unrecorded in the literature. By analyzing her actions using a methodology involving production rules, we suggest how her content knowledge, pedagogical content knowledge, and her knowledge of her students were interrelated and how they impacted on her instructional decisions and teaching actions. We also provide an example of how using production rules can be useful to discern some of the complexities involved in teaching and tutoring.     

The Role of the Visual System in Emotion Perception

Looking at a person’s expression is a good way of telling what she feels—what emotions she has. Why is that? Is it because we see her emotion, or is it because we infer her mental state from her expression? My claim is that there is a sense in which we do see the person’s emotion. I first argue that expressions are physical events that carry information about the emotions that produce them. I then examine evidence suggesting that specific brain areas and structures are involved in the process that extracts such information and makes it available in the content of visual experience. I consider only what happens in early stages of visual processing and make no claim about the role of simulation and empathy.     

Investigating teachers’ appraisal of unexpected moments and underlying values: an exploratory case in the context of changing mathematics classroom discourse

This article provides an exploratory case study that examines what one teacher indicated as unexpected as she worked to become more purposeful about her classroom discourse practices. We found that she highlighted three areas as being unexpected: (1) aspects of lesson enactment; (2) characteristics of student learning and (3) her own intentionality or purposefulness. We interpret these instances through the lens of the systemic functional linguistics (SFL) appraisal framework in order to understand how she evaluates the instances she highlights and connect these instances to the literature on values in the teaching and learning of mathematics.     

A Teacher's Journey with a New Generation Handheld: Decisions,Struggles, and Accomplishments

In this technology‐oriented age, teachers face daily decisions regarding the use of advanced digital technologies—graphing calculators, dynamic geometry software, blogs, wikis, podcasts and the like—to enhance student mathematical understanding in their classrooms. In this case study, the authors use the Technological, Pedagogical, and Content Knowledge (TPACK) model in conjunction with a five‐stage developmental model, which can be used to describe growth in TPACK to describe the initial attempts of a teacher, Jane, to develop TPACK as she learns and attempts to integrate an advanced teaching technology into her classroom, namely the TI‐Nspire graphing calculator. The study tracks her struggles to reconcile some traditional beliefs about how students learn with her desire to be responsive to what she perceives as affordances of advanced digital technologies. Main data collection methods were journal writing, observations, document analysis, and interviews. Using the five‐stage developmental model, we saw that this experience helped Jane to move among different stages. This study showed that the TPACK model with the five‐stage developmental model can be a beneficial tool for researchers to study teachers' professional growth and is also a valuable tool for teachers to reflect on their own growth.     

Making sense of qualitative geometry: The case of Amanda

This article presents a case study of a seven-year-old girl named Amanda who participated in an eighteen-week teaching experiment I conducted in order to model the development of her intuitive and informal topological ideas. I designed a new dynamic geometry environment that I used in each of the episodes of the teaching experiment to elicit these conceptions and further support their development. As the study progressed, I found that Amanda developed significant and authentic forms of geometric reasoning. It is these newly identified forms of reasoning, which I refer to as “qualitative geometry,” that have implications for the teaching and learning of geometry and for research into students’ mathematical reasoning.     

Beliefs,Practical Knowledge,and Context: A Longitudinal Study of a Beginning Biology Teacher's 5E Unit

The purpose of this three‐year case study was to understand how a beginning biology teacher (Alice) designed and taught a 5E unit on natural selection, how the unit changed when she took a position in a different school district, and why the changes occurred. We examined Alice's developing beliefs about science teaching and learning, practical knowledge, and perceptions of school context in relation to the 5E unit. Data sources consisted of interviews, classroom observations, and lesson materials. We found that Alice placed more emphasis on the explore phase, less emphasis on the engage and explain phases, and removed the elaborate phase over time. Alice's beliefs about science teaching and learning acted as a filter for making sense of practical knowledge and perceptions of context. Although her beliefs were student centered, they aligned with discovery learning in which little intervention from the teacher is required. We discuss how her beliefs, practical knowledge, and perceptions of context explained the changes in her practice. This study sheds insight into the nature of beliefs and how they relate to the 5E lesson phases, as well as the different lenses for viewing the 5E instructional model. Implications for science teacher preparation and induction programs are discussed.     

The powers and pitfalls of algorithmic knowledge: a case study

This study examines one child's use of computational procedures over a period of 3 years in an urban elementary school where teachers were using a standards-based curriculum. From a sociocultural perspective, the use of standard algorithms to solve mathematical problems is viewed as a cultural tool that both enables and constrains particular practices. As this student appropriated and mastered procedures for addition, subtraction, multiplication and division, she could solve problems that involved fairly straightforward computations or where she could easily model the action to determine an appropriate computation. At the same time, her use of these algorithms, along with other readily available tools, such as her fingers or multiplication tables, constrained her ability to reflect on the tens-structure of the number system, an effect that had serious consequences for her overall mathematical achievement. The results of this study suggest that even when not directly introduced, algorithms have such strong currency that they can mediate more reform-oriented instruction.     

Das Kardinalzahlkonzept

This case-study investigates different aspects of the concept of cardinality of an eighteen-year-old student with mental retardation. At the age of six she could not relate number words, finger and objects in counting. These errors still persist in the classroom situation. This investigation shows that nevertheless her concept of cardinality is fairly highly developed. She knows that in counting she must match number words and objects one to one, the number word sequence she uses is stable, and her insight into the irrelevance of order of enumeration when counting, which she finds by trial, is a sign of the robustness of her cardinal concept. She also understands the relationships of equivalence and order of sets, and she solves arithmetical problems by counting on or down, which means that she understands the number words as cardinal and at the same time as sequence numbers. Errors occur in complex situations, where several components have to be considered. But her concept of cardinality is also incomplete: she has special difficulties concerning counting out objects bundled in tens. The same problems occur when she uses multidigit numbers: she does not see a ten-unit as composed of ten single unit items, that is to say, she replaces the hierarchic structure of the number sequence by a concatenated one. These difficulties must be interpreted as a consequence of her special weakness concerning synthetic thinking and simultaneous performing, as similar patterns can be seen in her spatial perception and in her speech. In the syntactic structure of her utterances, too, the combination of simple entities to complicated units is replaced by a mere concatenation. This means that due to brain dysfunction her behavior is determined by a particular pattern which repeatedly appears intrapersonally, and which is characteristic of some mentally retarded persons though not of all of them. Evidently mathematical thinking is also not a determined system, but a variable one. Mentally retarded students may therefore have great difficulties concerning some areas and at the same time make better progress in others. In particular, difficulties in counting objects are no obstacle to knowledge of cardinality.     

Risk assessment for infectious disease and its impact on voluntary vaccination behavior in social networks

Achievement of the herd immunity is essential for preventing the periodic spreading of an infectious disease such as the flu. If vaccination is voluntary, as vaccination coverage approaches the critical level required for herd immunity, there is less incentive for individuals to be vaccinated; this results in an increase in the number of so-called “free-riders” who craftily avoid infection via the herd immunity and avoid paying any cost. We use a framework originating in evolutionary game theory to investigate this type of social dilemma with respect to epidemiology and the decision of whether to be vaccinated. For each individual in a population, the decision on vaccination is associated with how one assesses the risk of infection. In this study, we propose a new risk-assessment model in a vaccination game when an individual updates her strategy, she compares her own payoff to a net payoff obtained by averaging a collective payoff over individuals who adopt the same strategy as that of a randomly selected neighbor. In previous studies of vaccination games, when an individual updates her strategy, she typically compares her payoff to the payoff of a randomly selected neighbor, indicating that the risk for changing her strategy is largely based on the behavior of one other individual, i.e., this is an individual-based risk assessment. However, in our proposed model, risk assessment by any individual is based on the collective success of a strategy and not on the behavior of any one other individual. For strategy adaptation, each individual always takes a survey of the degree of success of a certain strategy that one of her neighbors has adopted, i.e., this is a strategy-based risk assessment. Using computer simulations, we determine how these two different risk-assessment methods affect the spread of an infectious disease over a social network. The proposed model is found to benefit the population, depending on the structure of the social network and cost of vaccination. Our results suggest that individuals (or governments) should understand the structure of their social networks at the regional level, and accordingly, they should adopt an appropriate risk-assessment methodology as per the demands of the situation.     

Mary Wynne Warner (1932-1998)

Mary Warner, as she was mainly known in the mathematical world,died in April 1998. At a time when few women mathematiciansreached the top in their profession, she succeeded in doingso through her ability and determination. Her research contributionswere commemorated at a recent international conference on fuzzytopology, the field in which she was one of the pioneers andrecognized as one of the leading figures for the past thirtyyears. She was also an outstanding teacher. But to understandher achievements properly it is necessary to know somethingof her life.     

Challenges of maintaining cognitive demand during the limit lessons: understanding one mathematician’s class practices

In this study, we examined five limit lessons using Mathematical Tasks Framework to understand students’ opportunities to learn cognitively challenging tasks and maintain cognitive demand during limit lessons. Our analysis of Dr A’s five lessons shows that students rarely had opportunities to maintain or increase cognitive demand. There are two main factors that shaped her instructional practices, students and time. These two factors greatly influenced how she selects and implements limit tasks in her classes. To serve her students’ needs of knowing more rules, formulas and procedures, she selected and discussed those simple tasks a lot. Although Dr A thinks challenging tasks and asking demanding questions can be potentially good instructional practices, she thinks these instructional practices would not serve her students well. With these factors, we made possible recommendations to have more student-centred teaching.     

Olga Taussky,a torchbearer for mathematics and teacher of mathematicians

Olga Taussky-Todd's mathematical and personal life (1906-1995), her achievements and obstacles, her scientific reasoning and teaching all have served as inspiration to many mathematicians. We describe her role in the mathematics world of the previous century as a torchbearer for mathematics and mathematicians, bearing the “torch of scientific truth” that burns inside of mathematics and its applications. Besides her many deep math contributions – too many to elaborate – she excelled at distilling and presenting mathematical concepts and ideas in her work and gave us many visionary papers and math talks. By sharing her mathematical vision freely she has inspired many of us. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Practicing Reform‐Based Science Curriculum in an Urban Classroom: A Hispanic Elementary School Teacher's Thinking and Decisions

This study explores the thinking and decisions of Vera (pseudonym), a Hispanic elementary teacher, while she enacted a reform‐based science curriculum in an urban school in the southern United States. Vera's thinking, decisions, experiences, and practices were documented over a 2‐year period. Using the data collected from semistructured interviews, participant observations and classroom documents, a rich and complex case study of Vera is developed in this paper. This case study describes how Vera makes curricular choices from reform‐based science curricula such as the LiFE curriculum; how she enacts those choices to empower poor urban minority students; how Vera believes that preparing students for the high‐stakes test is empowering because it ensures continued schooling for students; how, for Vera, teaching connected science using students' lived experiences is a risky act; and how she uses negotiation in her science teaching.     

Practicing Reform‐Based Science Curriculum in an Urban Classroom: A Hispanic Elementary School Teacher's Thinking and Decisions

This study explores the thinking and decisions of Vera (pseudonym), a Hispanic elementary teacher, while she enacted a reform‐based science curriculum in an urban school in the southern United States. Vera's thinking, decisions, experiences, and practices were documented over a 2‐year period. Using the data collected from semistructured interviews, participant observations and classroom documents, a rich and complex case study of Vera is developed in this paper. This case study describes how Vera makes curricular choices from reform‐based science curricula such as the LiFE curriculum; how she enacts those choices to empower poor urban minority students; how Vera believes that preparing students for the high‐stakes test is empowering because it ensures continued schooling for students; how, for Vera, teaching connected science using students' lived experiences is a risky act; and how she uses negotiation in her science teaching.     

A Preservice Secondary Teacher's Moves to Protect Her View of Herself as a Mathematics Expert

This paper discusses the experience of a preservice secondary mathematics teacher during lesson study. Although the preservice teacher was a strong undergraduate mathematics student, she used compensation “moves” to deflect attention away from her insecurities about her conceptual understanding of secondary mathematics. She feared being labeled as “dumb” and redirected conversations in order to protect her identity as a knower of mathematics. This paper investigates the culture in which preservice teachers develop confidence in their personal mathematics knowledge and how that confidence may influence behavior.     

Teacher listening: The role of knowledge of content and students

In this research report we consider the kinds of knowledge needed by a mathematician as she implemented an inquiry-oriented abstract algebra curriculum. Specifically, we will explore instances in which the teacher was unable to make sense of students’ mathematical struggles in the moment. After describing each episode we will examine the instructor's efforts to listen to the students and the way that these efforts were supported or constrained by her mathematical knowledge for teaching. In particular, we will argue that in each case the instructor was ultimately constrained by her knowledge of how students were thinking about the mathematics.     

Complexity theory and collaboration: An agent-based simulator for a space mission design team

In this paper, we investigate how complexity theory can benefit collaboration by applying an agent-based computer simulation approach to a new form of synchronous real-time collaborative engineering design. Fieldwork was conducted with a space mission design team during their actual design sessions, to collect data on their group conversations, team interdependencies, and error monitoring and recovery practices. Based on the fieldwork analysis, an agent-based simulator was constructed. The simulation shows how error recovery and monitoring is affected by the number of small group, or sidebar, conversations, and consequent noise in the room environment. This simulation shows that it is possible to create a virtual environment with cooperating agents interacting in a dynamic environment. This simulation approach is useful for identifying the best scenarios and eliminating potential catastrophic combinations of parameters and values, where error recovery and workload in collaborative engineering design could be significantly impacted. This approach is also useful for defining strategies for integrating solutions into organizations. Narjès Bellamine-Ben Saoud is an Associate Professor at the University of Tunis and Researcher at RIADI-GDL Laboratory, Tunisia. After Computer Science engineering diploma (1993) of the ENSEEIHT of Toulouse, France, she received her PhD (1996), on groupware design applied to the study of cooperation within a space project, from the University of Toulouse I, France. Her main research interests concern studying complex systems particularly by modeling and simulating collaborative and socio-technical systems; developing Computer Supported Collaborative Learning in tunisian primary schools; and Software Engineering. Her current reserach projects include modeling and simulation of emergency rescue activities for large-scale accidents, modeling of epidemics and study of malaria, simulation of collabration artifacts. Gloria Mark is an Associate Professor in the Department of Informatics, University of California, Irvine. Dr. Mark received her Ph.D. in Psychology from Columbia University in 1991. Prior to UCI, she was a Research Scientist at the GMD, in Bonn, Germany, a visiting research scientist at the Boeing Company, and a research scientist at the Electronic Data Systems Center for Advanced Research. Dr. Mark’s research focuses on the design and evaluation of collaborative systems. Her current projects include studying worklife in the network-centric organization, multi-tasking of information workers, nomad workers, and a work in a large-scale medical collaboratory. Dr. Mark is widely published in the fields of CSCW and HCI, is currently the program co-chair for the ACM CSCW’06 conference and is on the editorial board of Computer Supported Cooperative Work: The Journal of Collaborative Computing, and e-Service Qu@rterly.     

Alicia Boole Stott,a geometer in higher dimension

In this paper we present the life and work of Alicia Boole Stott, an Irish woman who made a significant contribution to the study of four-dimensional geometry. Although she never studied mathematics, she taught herself to “see” the fourth dimension and developed a new method of visualizing four-dimensional polytopes. In particular, she constructed three-dimensional sections of these four-dimensional objects, which resulted in a series of Archimedean solids. The presence in the University of Groningen of an extensive collection of these three-dimensional models, together with related drawings, reveals a collaboration between Boole Stott and the Groningen professor of geometry, P.H. Schoute. This collaboration lasted more than 20 years and combined Schoute's analytical methods with Boole Stott's unusual ability to visualize the fourth dimension. After Schoute's death in 1913 Boole Stott was isolated from the mathematical community until about 1930, when she was introduced to the geometer H.S.M. Coxeter, with whom she collaborated until her death in 1940.     

一种惩罚机制下一次性n人囚徒困境的合作性

为解决一次性n人囚徒困境中局中人如何走出困境的问题,引进了背叛惩罚函数及其严厉度和参与人的背叛愿意度等概念,并用数学论证法证明了如下结果:(1)参与人的背叛愿意度都不超过1.(2)背叛愿意度越大,这个参与人越愿意背叛;(3)背叛愿意度为0零时,这个参与人是否背叛其赢得一样;(4)当背叛愿意度取负数时,其绝对值越大,参与人的合作积极性越大.得到博弈结果的判定法:(1)计算各参与人的背叛愿意度.(2)若至少有一个参与人愿意背叛,则全体参与人都背叛.(3)若全体参与人都愿意合作,则合作成功.例子表明,本结果在理论上可有效地解决中局中人如何走出困境和在给定惩罚机制下博弈结果的预测问题.