Generalizing in the context of an early algebra intervention (La generalización en el contexto de una intervención algebraica temprana)

ABSTRACT

This study aims to understand the instructional interactions that foster students’ generalizing. By analysing video-recorded lessons from 13 different Grade 3 classrooms in which an early algebra intervention was implemented, activities related to generalizing and students’ generalizing activities were identified, and the nature of the relationships between these activities were explored. Namely, I consider the activities that potentially encourage students’ generalizing and the generalizing that students themselves do, and describe ways in which these activities interact. The findings have both practical and theoretical implications; in the former case, they provide insight into the design of instruction to foster generalizing, and in the latter case, they contribute to an ongoing conversation about students’ mathematics learning.     

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.     

Learners’ Difficulties with Quantitative Units in Algebraic Word Problems and the Teacher’s Interpretation of those Difficulties

This study examines 8th grade students’ coordination of quantitative units arising from word problems that can be solved via a set of equations that are reducible to a single equation with a single unknown. Along with Unit-Coordination, Quantitative Unit Conservation also emerges as a necessary construct in dealing with such problems. We base our analysis within a framework of quantitative reasoning (Thompson, 1988; 1989; 1993; 1995) and a theory of children’s units-coordination with different levels of units (Steffe 1994) that both encompass and are extended by these two constructs. Our data consist of videotaped classroom lessons, student interviews, and teacher interviews. Ongoing analyses of these data were conducted during the teaching sequence. A retrospective analysis using constant comparison methodology was then undertaken during which the classroom video, related student interviews, and teacher interviews were revisited many times in order to generate a thematic analysis. Our results indicate that the identification and coordination of the units involved in the problem situation are critical aspects of quantitative reasoning and need to be emphasized in the teaching-learning process. We also concluded that unit coordination and unit conservation are cognitive prerequisites for constructing a meaningful algebraic equation when reasoning quantitatively about a situation.     

Modelling activities integrating construction and simulation supported explanatory and evaluative reasoning

ABSTRACT

Model-based instruction offers numerous benefits to students, including increased content knowledge and critical thinking. This study explored the differences in the knowledge outcomes and reasoning processes employed by undergraduate students in an introductory biology lab as they constructed, revised, and simulated a computational model of a prokaryotic gene regulatory system. We analysed post-lesson conceptual models (n?=?335) and audio-recorded small group discussions (n?=?10) of students that either constructed and revised a computational model and then investigated a pre-constructed computational model or only investigated a pre-constructed computational model to determine the impact of constructing and revising a computational model. No significant differences were found between the mental models of students who constructed and revised a computational model and those who only simulated a pre-constructed computational model. However, this study demonstrated that constructing and revising a model before simulating it offered additional benefits to students by fostering deeper cognitive reasoning processes. Constructing and revising a model elicited more explanatory and evaluative reasoning and prompted students to discuss the underlying mechanisms of the biological system.     

Consistent and Inconsistent Pupils' Reasoning about Intensive Quantities: The Case of Density and Pressure

In this research we investigate whether the categorisation of physical quantities into intensive and extensive, according to scientific knowledge, is perceivable and understood by pupils aged 12–15 years. For this purpose we administered written questionnaires to 300 pupils, comprising four tasks for each intensive quantity: density and pressure. The tasks investigate whether pupils realise that intensive quantities do not depend on the amount of the system they refer to and the factors affecting pupils' reasoning and their consistency when studying phenomena involving fluids at rest or in transition. The analysis of responses led us to identify three models of consistent pupils' reasoning. It was also observed that a significant percentage of the pupils provide inconsistent answers, that is, they change their reasoning across tasks and use alternatives to scientific reasoning, influenced by phenomenological features of the tasks. According to our results we propose a number of recommendations, such as the introduction of intermediate teaching/learning steps to be followed as well as suitable experiments to be used towards the acquisition of the scientific model for density and pressure.     

Turkish elementary student teachers’ epistemological beliefs and moral reasoning

The purpose of the present study was to explore the relationships among Turkish elementary student teachers’ epistemological beliefs and moral reasoning, and to determine which types of epistemological beliefs elementary student teachers exhibit. The findings of the present study demonstrated that epistemological beliefs did not make a unique contribution to moral reasoning and student teachers’ epistemological beliefs developed as more or less independent beliefs. The factor analyses results suggest that multidimensional theory is more appropriate than unidimensional theory in explaining student teachers’ epistemological beliefs.     

维吾尔族学生汉语标点符号书写和使用偏误分析

本文对维吾尔族学生汉语标点符号书写情况进行了调查,概括维吾尔族学生常用汉语标点符号的使用和书写偏误类型,分析了各类偏误产生的原因,并如何改进维吾尔族学生汉语标点符号教学提出了建议。     

英语背景留学生的汉语比字句使用偏误分析

比字句在比较句中使用频率最高,句式结构及语用方法相对复杂,在比较句的对外汉语教学中,既是重点,也是难点。英语背景留学生使用“比”字句的偏误主要表现在两个方面:第一,比字句结构要素之间的位置关系次序失当;第二,结果项的构成成分失当。造成偏误的主要原因:一是母语的负迁移,二是目的语规则泛化。     

Students’ reasons for preferring teleological explanations

The teleological bias, a major learning obstacle, involves explaining biological phenomena in terms of purposes and goals. To probe the teleological bias, researchers have used acceptance judgement tasks and preference judgement tasks. In the present study, such tasks were used with German high school students (N?=?353) for 10 phenomena from human biology, that were explained both teleologically and causally. A sub-sample (n?=?26) was interviewed about the reasons for their preferences. The results showed that the students favoured teleological explanations over causal explanations. Although the students explained their preference judgements etiologically (i.e. teleologically and causally), they also referred to a wide range of non-etiological criteria (i.e. familiarity, complexity, relevance and five more criteria). When elaborating on their preference for causal explanations, the students often focused not on the causality of the phenomenon, but on mechanisms whose complexity they found attractive. When explaining their preference for teleological explanations, they often focused not teleologically on purposes and goals, but rather on functions, which they found familiar and relevant. Generally, students’ preference judgements rarely allowed for making inferences about causal reasoning and teleological reasoning, an issue that is controversial in the literature. Given that students were largely unaware of causality and teleology, their attention must be directed towards distinguishing between etiological and non-etiological reasoning. Implications for educational practice as well as for future research are discussed.     

Signs and meanings in students' emergent algebraic thinking: A semiotic analysis

The purpose of this article, which is part of a longitudinal classroom research about students' algebraic symbolizations, is twofold: (1) to investigate the way students use signs and endow them with meaning in their very first encounter with the algebraic generalization of patterns and(2) to provide accounts about the students' emergent algebraic thinking. The research draws from Vygotsky's historical-cultural school of psychology, on the one hand, and from Bakhtin and Voloshinov's theory of discourse on the other, and is grounded in a semiotic-cultural theoretical framework in which algebraic thinking is considered as a sign-mediated cognitive praxis. Within this theoretical framework, the students' algebraic activity is investigated in the interaction of the individual's subjectivity and the social means of semiotic objectification. An ethnographic qualitative methodology, supported by historic, epistemological research, ensured the design and interpretation of a set of teaching activities. The paper focuses on the discussion held by a small group of students of which an interpretative, situated discourse analysis is provided. The results shed some light on the students' production of (oral and written) signs and their meanings as they engage in the construction of expressions of mathematical generality and on the social nature of their emergent algebraic thinking. This revised version was published online in July 2006 with corrections to the Cover Date.     

网络流行语在大学生群体中泛化的四维审视简

网络流行语既具有传统语言的某些特质,又兼具网络语言的显著特征,备受网民青睐。近年来,伴随三网融合,网络流行语如雨后春笋,在大学生群体中不断泛化,并呈现出一些规律性特征,对大学生思维方式和价值观念产生了重要影响。基于网络流行语在大学生群体中泛化的表现形式、主要归因和实现途径,顺势而为,采取科学的因应措施,将进一步提升大学生舆情引领的科学性、时代性和实效性。     

代数式求值的研究

本文研究了代数式求值的十二种方法     

提高多层前向神经网络泛化能力的讨论

泛化能力是多层前向神经网络最重要的性能．泛化问题已成为目前神经网络领域的研究热点。本文从泛化理论现有提高神经网络泛化能力的方法等几个方面总结了当前神经网络结构优化与泛化能力研究的现状。神经网络泛化能力的提高可通过神经网络结构的优化和正则化等方法加以实现,并对提高网络泛化能力问题进行讨论。     

不同类型推理偏差和困难研究综述

心理学对推理困难的研究主要体现在演绎推理、归纳推理和类比推理等方面。不同类型的推理困难研究有自己的特点,但都有深厚的逻辑学渊源。国内学者对推理困难的研究比较零散,主要针对引起困难的原因进行探讨。文章从研究内容和研究方法等方面分析了推理困难研究的不足,对研究的发展趋势进行了预测。     

求逆阵及解线性方程组的迭代算法

本文给出求可逆方阵的逆矩阵和利用线性代数方程组AX=b的系数矩阵A的一个初始近似逆阵P求解方程组的迭代算法,这种算法具有迭代格式简单,能有效地控制求解过程中的舍入误差的影响,灵活确定迭代次数等特点。     

略论法律推理与逻辑推理

逻辑推理和法律推理的关系十分密切,法律推理是逻辑推理的一般方法在法律领域中广泛、综合的运用,逻辑推理是法律推理重要的理性工具之一。法律推理中的逻辑推理形式必须是有效的,有效的逻辑推理可以消解裁判中的任意性,避免谬误,帮助获得正当的裁判结论。     

Getting at the Mathematics: Sara’s Journal

In this paper, we discuss issues in planning and conducting research into mathematics learning. We emphasize two central themes: (a) the learners mathematics (especially the issues and ideas, in given problem situations, that learners choose to think about and to present) and (b) the kinds of knowledge that learners may be building (including their ideas about what mathematics is, and how people do, learn, use, communicate and understand it). While the first theme is at least partly mathematical, the second interweaves cognition and epistemology. Anchored in student data from an extended classroom teaching experiment in the mathematics of change, we focus on key choices needed to build narratives to help researchers capture, in detail, how students engage with mathematics in extended problem explorations.