Convergence of sequences and series: Interactions between visual reasoning and the learner's beliefs about their own role

This paper examines part of a set of students who were followed during their first-term, first-year studies in formal definition-based real analysis at a British university. It explores the approaches to problems about convergence of sequences and series made by students who have a tendency to include visual imagery in their reasoning. We explore links between the students' mathematical behavior in solving these problems and their perception of their roles as learners. We develop a theory in which the tendency to visualize, coupled with the students' view of their role, can be used to account for their mathematical behavior. This revised version was published online in July 2006 with corrections to the Cover Date.     

Knowledge development and restructuring in the domain of medicine: The role of theory and practice

How does the knowledge of the medical student, clerk, intern, and registrar develop and how do formal and informal, classroom and experiential learning contribute to this process? These questions were investigated in two experiments, focusing on knowledge restructuring rather than knowledge acquisition. The experiments showed that practical experience plays an important role in knowledge restructuring. The process was, however, not as continuous as was expected. Notably, advanced students appeared to have considerable knowledge about conditions in patients and their environments that can predispose to disease. However, they rarely applied it in clinical reasoning. Contrary to what was found in expert physicians, advanced students' knowledge about enabling conditions seems not yet to be integrated into their other knowledge about diseases.     

Science education through informal education

To develop the pedagogic efficiency of informal education in science teaching, promoting a close cooperation between institutions is suggested by Monteiro, Janerine, de Carvalho, and Martins. In their article, they point out effective examples of how teachers and educators work together to develop programs and activities at informal education places such as science museums. Their study explored and discussed the viability and relevancy of school visits to museums and possibilities to enhance the connection between students’ visits in informal contexts and their learning in schools. Given that students learn science by crossing the boundaries of formal and informal learning contexts, it is critical to examine ways of integrated and collaborative approach to develop scientific literacy to help students think, act and communicate as members of problem solving communities. In this forum, we suggest the importance of students’ lifeworld contexts in informal learning places as continuum of Monteiro, Janerine, de Carvalho, and Martins’ discussion on enhancing the effectiveness of informal learning places in science education.     

Translation of algebraic equations and its relation to formal operational reasoning

A large proportion of science major college students are unable to translate even simple sentences into algebraic equations. Given the following sentence: There are six times as many students (S) as professors (P) at this university, most students write the following equation: 6S = P, referred to as the reversal error. In order to overcome the reversal error students need to operate in a hypothetico-deductive manner, i.e., performing a hypothetical operation that makes the group of professors six times larger than it really is (S = 6P). The objective of this study is to investigate the relation between student ability to translate sentences into equations, equations into sentences, and student performance in the following variables: formal operational reasoning, proportional reasoning, and introductory freshmen-level chemistry course. The results obtained show that as the student ability to translate sentences into equations and equations into sentences increases, their mean scores in Chemistry I, formal operational, and proportional reasoning increases. This study has found support for the hypothesis that students who lack formal operational reasoning skills (hypothetico-deductive reasoning) may experience more problems in the translation of algebraic equations.     

A schematic–Theoretic view of problem solving and development of algebraic thinking

This study explored the problem-solving schemas developed by 7th-grade pre-algebra students as they participated in a teaching experiment that was designed to help students develop effective schemas for solving algebraic problem situations involving contexts of (1) growth and change and (2) size and shape. This article describes the qualities and types of schemas that students developed through examples of problems used to develop schemas, examples of students' reasoning and writing, and excerpts from student interviews. Findings from the study indicated that there is a link between the type of generalizations students construct and the schemas they are forming. By using these schemas students recognize, extend, and generalize patterns and quantitative relationships both verbally and symbolically. This revised version was published online in July 2006 with corrections to the Cover Date.     

论学生非正式群体的必要性

学生群体分正式群体和非正式群体,其中非正式群体在学生发展中具有特殊意义。对非正式群体的向往与参与是学生正常的心理需求;学生与非正式群体间个体的交往有助于完善自己的个性;这种交往还有利于促进学生的社会化。     

Decision Making and Sources of Knowledge: How Students Tackle Integrated Tasks in Science,Technology and Mathematics

This article reports on students' decision making processes and sources of knowledge in an integrated teaching and learning setting. The study was conducted in a Year 9 classroom as students undertook a 10-week solar-powered boat project and were exposed to related concepts from science, technology and mathematics. Data collection involved detailed case studies of three pairs of students, interviews, classroom observation and analysis of the artefacts and portfolios produced by the students. Students were found to access knowledge from a variety of sources, including teacher's notes from formal instruction, informal interactions with the teacher, observation of and interaction with other students, as well as sources outside the classroom. However, the utility of the knowledge sources was influenced by the nature of the task. When students were performing open-ended tasks, they drew on a wider variety of knowledge sources than when they were performing less open tasks. Moreover, subject discipline-based sources often were not as helpful in solving open tasks. The study leads to several important implications for designing teaching and learning in integrated curriculum settings.     

The balance model: Hindrance or support for the solving of linear equations with one unknown

The use of concrete models for teaching students how to solve equations is often debated in scientific literature. This article aims to examine the balance model and to identify the issues that divide scientists. We based our reflections on the results of an empirical study and analysis of the various arguments put forward by supporters and opponents of the model. We describe learning situations that were the subject of the empirical study, which involved forty students in two 8th-grade classes. The aim was to teach the formal solving method, which involved performing the same operations on both sides of the equation using, notably, the balance model. Analysis of students' reasoning showed that the presence of negative numbers gave rise to many errors. The difficulties presented by negative numbers were reviewed, eight months later, during an interview with five students, chosen from those who took part in the experiment. Within that context, we discuss the relevance of the balance model and analyse the arguments put forward by researchers who either defend or reject its use.This revised version was published online in September 2005 with corrections to the Cover Date.     

Individual differences in the analysis of informal reasoning fallacies

After decades of research into formal or logical fallacies of reasoning, psychologists have only recently begun to examine the informal reasoning fallacies that are routinely present in critical discussions, debates, and other forms of argumentation. The present study considers several possible influences on an ability to identify and analyze these fallacies. College students completed measures of deductive reasoning, personal epistemology, and knowledge of specific argumentation norms and analyzed arguments containing fallacies such as argument from ignorance, begging the question, and slippery slope. Results indicated that effective analysis of informal fallacies was associated with some aspects of deductive reasoning—especially an ability to overcome belief bias—and with higher-order epistemic beliefs, as well as a commitment to argumentation norms for critical discussion. Results are discussed in terms of argumentation research and implications for pedagogical treatments of the fallacies are noted.     

论证的形式模型概述及其在哲学上的应用

论证是一个用理由支持观点以及回应其所受攻击的过程.在过去的几十年里,论证已经成为哲学和人工智能研究领域中的一个重要主题.在哲学方面,十九世纪50年代和60年代图尔敏和佩雷尔曼对形式逻辑的批判,促进了非形式逻辑这一学科的产生,它研究推理和论证的非形式模型.在人工智能方面,论证的形式模型也发展成为常识推理和多主体冲突解决的基本模型.本文将讨论后一领域中所研究发展的形式模型如何能够用以澄清一些哲学领域、以及非形式逻辑领域中的理论问题和争议.本文的一个重要观点是,图尔敏和佩雷尔曼时代的形式逻辑只关注数学化的推理,但那些非数学化的推理形式其实同样也能够被形式化.     

Balancing chemical equations: The role of developmental level and mental capacity

Why are some students able to learn to use the trial and error method to balance chemical equations while others are not? To test the hypothesis that formal reasoning is required to balance even simple one-step equations, while formal reasoning and a sufficiently large mental capacity are required to balance more complex many-step equations, a sample of science students was tested to determine level of intellectual development, mental capacity, and degree of field dependence/field independence. Students were then given classroom instruction in using trial and error to balance equations. As predicted, a posttest revealed significant correlations between developmental level and equation balancing ability for both simple and complex equations. Also, as predicted, mental capacity correlated significantly with complex equations but not with simple equations. Field dependence/field independence played no significant role in performance. Educational implications are drawn.     

Reasoning strategies of students in solving chemistry problems as a function of developmental level,functional M‐capacity and disembedding ability

Achievement in science depends among other factors on hypothetico‐deductive reasoning ability, that is, developmental level of the students. Recent research indicates that the developmental level of students should be studied along with individual difference variables, such as Pascual‐Leone's M‐capacity (information processing) and Witkin's Cognitive Style (disembedding ability). The purpose of this study is to investigate reasoning strategies of students in solving chemistry problems as a function of developmental level, functional M‐capacity and disembedding ability. A sample of 109 freshman students were administered tests of formal operational reasoning, functional M‐capacity, disembedding ability and chemistry problems (limiting reagent, mole, gas laws). Results obtained show that students who scored higher on cognitive predictor variables not only have a better chance of solving chemistry problems, but also demonstrated greater understanding and used reasoning strategies indicative of explicit problem‐solving procedures based on the hypothetico‐deductive method, manipulation of essential information and sensitivity to misleading information. It was also observed that students who score higher on cognitive predictor variables tend to anticipate important aspects of the problem situation by constructing general figurative and operative models, leading to a greater understanding. Students scoring low on cognitive predictor variables tended to circumvent cognitively more demanding strategies and adopt others that helped them to overcome the constraints of formal reasoning, information processing and disembedding ability.     

Influences on Students' Mathematical Reasoning and Patterns in its Development: Insights from a Longitudinal Study with Particular Reference to Geometry

We report some findings of the Longitudinal Proof Project, which investigated patterns in high-attaining students' mathematical reasoning in algebra and in geometry and development in their reasoning, by analyses of students' responses to three annual proof tests. The paper focuses on students' responses to one non-standard geometry item. It reports how the distribution of responses to this item changed over time with some moderate progress that suggests a cognitive shift from perceptual to geometrical reasoning. However, we also note that many students made little or no progress and some regressed. Extracts from student interviews indicate that the source of this variation from the overall trend stems from the shift over the three years of the study to a more formal approach in the school geometry curriculum for high-attaining students, and the effects of this shift on what students interpreted as the didactical demands of the item.     

谈如何利用非正式组织缓解90后大学生心理困境

非正式组织作为正式组织的伴生物,不可避免地、客观地存在于学生之中,尤其是在90后大学生这个特殊的群体中,更是发挥着其特有的作用。作为教师应该正确认识非正式组织,在实践工作中充分发挥其积极作用,抑制其消极作用,正确缓解90后大学生心理困境。     

大学生非正式组织及教育管理对策

大学生非正式组织的存在,对于高校的教育管理工作有重要影响。本文将非正式组织理论运用于高校组织的管理实践,分析了大学生非正式组织的类型、特征和对正式组织的影响,并最终提出了发挥非正式组织在学生工作中的积极作用,控制其消极作用的教育管理对策。     

过程哲学视角下的课外补习教育思考

课外补习作为补差培优的一种教育方式,是正规学校教育的有益补充,有其存在的合理性,但家长盲目地采取课外补习的方式教育孩子,导致课内、课外教育功能失衡,丧失课外补习的调节功能,影响学生正规的学校教育生活和秩序.本文以怀特海的过程哲学为理论基础,以课外补习热现象为切入点,深入分析造成此现象的社会价值观、教育意识和制度监管方面的原因,并寻找走出课外补习热误区的哲学基础和制度保证.     

Mapping Trade-Offs in Teachers' Integration of Technology-Supported Inquiry in High School Science Classes

This paper explores how two teachers concurrently enacting the same technology-based inquiry unit on evolution structured activity and discourse in their classrooms to connect students' computer-based investigations to formal domain theories. Our analyses show that the teachers' interactions with their students during inquiry were quite similar, and each teacher used whole-class discussions as a major vehicle for connecting students' understanding to formal domain theories. Each teacher, however, structured the discourse in these discussions quite differently. We interpret these differences as each teacher navigating a set of trade-offs to balance, on the one hand, opportunities for students to actively develop their own ideas, and on the other, their concerns that students develop normative understandings. We identify several dimensions of trade-offs that emerge from our data, and suggest how teachers' choices on these dimensions shape students' opportunities for learning.     

The Interplay of Teacher and Student Actions in the Teaching and Learning of Geometric Proof

Proof and reasoning are fundamental aspects of mathematics. Yet, how to help students develop the skills they need to engage in this type of higher-order thinking remains elusive. In order to contribute to the dialogue on this subject, we share results from a classroom-based interpretive study of teaching and learning proof in geometry. The goal of this research was to identify factors that may be related to the development of proof understanding. In this paper, we identify and interpret students' actions, teacher's actions, and social aspects that are evident in a classroom in which students discuss mathematical conjectures, justification processes and student-generated proofs. We conclude that pedagogical choices made by the teacher, as manifested in the teacher's actions, are key to the type of classroom environment that is established and, hence, to students' opportunities to hone their proof and reasoning skills. More specifically, the teacher's choice to pose open-ended tasks (tasks which are not limited to one specific solution or solution strategy), engage in dialogue that places responsibility for reasoning on the students, analyze student arguments, and coach students as they reason, creates an environment in which participating students make conjectures, provide justifications, and build chains of reasoning. In this environment, students who actively participate in the classroom discourse are supported as they engage in proof development activities. By examining connections between teacher and student actions within a social context, we offer a first step in linking teachers' practice to students' understanding of proof.     

科学教育的重要途径—非正规学习

非正规学习在培养学生科学素质方面具有正规学习不可替代的重要作用。了解非正规学习的特点、规律和怎样影响人的科学素质的形成和发展对当前我国政府倡导的大力普及科学教育,提高全民,特别是未成年人的科学素质具有重大意义。非正规科学学习不仅能为学生提供丰富的认知冲突和社会互动机会,使他们建立起更加完善的知识结构,而且还能提高他们的科学论证能力。非正规学习是发展学生科学素质的必经之路。