Children's Perspectives on the Engagement Potential of Mathematical Problem Tasks

This article examines elementary students' perspectives on the engagement potential of particular mathematical problems and students' views on their general classroom problem activities. Third-, fifth-, and seventh-grade children from different reform-oriented classrooms were individually interviewed about (a) how they would improve their classroom problem-solving activities and (b) the problems they find the most and least potentially engaging when presented with a range of routine and nonroutine problems. The children requested more relevant, meaningful, and interesting problem experiences in their classrooms, and the fifth and seventh graders requested more representational materials. The children's criteria for determining potentially engaging and nonengaging problems primarily pertained to problem structure and perceived cognitive demands. The nonroutine examples that focused on important reasoning processes and did not involve computation had the greatest engagement potential, while the computational problems had the least appeal.     

Young Children's Composition of Geometric Figures: A Learning Trajectory

The purpose of this research is to chart the mathematical actions-on-objects young children use to compose geometric shapes. The ultimate goal is the creation of a hypothetical learning trajectory based on previous research, as well as instrumentation to assess levels of learning along the developmental progression underlying the trajectory. We tested both the developmental progression and the instrument through a series of studies, including formative studies (including action research by 8 teachers) and a summative study involving 72 children ages 3 to 7 years. Results provide strong support for the validity of the developmental progression's levels and suggest that children move through these levels of thinking in developing the ability to compose 2-dimensional figures. From lack of competence in composing geometric shapes, they gain abilities to combine shapes-initially through trial and error and gradually by attributes-into pictures, and finally synthesize combinations of shapes into new shapes (composite shapes).     

数学建模竞赛的发展与数学建模教师的作用

本文简要回顾数学建模竞赛活动的起源和发展,介绍国内外特别是国外学者关于提高数学建模教学质量的主要观点,并结合作者自身的经验和体会,探讨教师在数学建模教学中的作用。     

The Role of Visual Representations in Advanced Mathematical Problem Solving: An Examination of Expert-Novice Similarities and Differences

Expert mathematicians are contrasted with undergraduate students through a two-part analysis of the potential and actual use of visual representations in problem solving. In the first part, a classification task is used to indicate the extent to which visual representations are perceived as having potential utility for advanced mathematical problem solving. The analysis reveals that both experts and novices perceive visual representation use as a viable strategy. However, the two groups judge visual representations likely to be useful with different sets of problems. Novices generally indicate that visual representations would likely be useful mostly for geometry problems, whereas the experts indicate potential application to a wider variety of problems. In the second part, written solutions to problems and verbal protocols of problem-solving episodes are analyzed to determine the frequency, nature, and function of the visual representations actually used during problem solving. Experts construct visual representations more frequently than do novices and use them as dynamic objects to explore the problem space qualitatively, to develop a better understanding of the problem situation, and to guide their solution planning and enactment of problem-solving activity. In contrast, novices typically make little use of visual representations.     

The Role of Visual Representations in Advanced Mathematical Problem Solving: An Examination of Expert-Novice Similarities and Differences

Expert mathematicians are contrasted with undergraduate students through a two-part analysis of the potential and actual use of visual representations in problem solving. In the first part, a classification task is used to indicate the extent to which visual representations are perceived as having potential utility for advanced mathematical problem solving. The analysis reveals that both experts and novices perceive visual representation use as a viable strategy. However, the two groups judge visual representations likely to be useful with different sets of problems. Novices generally indicate that visual representations would likely be useful mostly for geometry problems, whereas the experts indicate potential application to a wider variety of problems. In the second part, written solutions to problems and verbal protocols of problem-solving episodes are analyzed to determine the frequency, nature, and function of the visual representations actually used during problem solving. Experts construct visual representations more frequently than do novices and use them as dynamic objects to explore the problem space qualitatively, to develop a better understanding of the problem situation, and to guide their solution planning and enactment of problem-solving activity. In contrast, novices typically make little use of visual representations.     

非逻辑创造思维在数学发现中的作用

借助实例介绍归纳法、类比法、联想以及减弱条件提出猜想这些非逻辑创造思维方法在一些重大数学发现中所发挥的作用.     

OntoMathPRO: An Ontology of Mathematical Knowledge

Doklady Mathematics - The article describes OntoMathPRO, the first Linked Open Data ontology of professional mathematical knowledge. The ontology is designed to represent mathematical concepts. The...     

理想化抽象在数学建模中的作用

理想化抽象是以抽象的理想形态表现研究对象的性质,以理想实验和理想模型为表现形式,是数学研究中采用的重要抽象方法,特别是在建立数学理论模型和数学应用模型时起到重要的作用。     

An Emergent Framework: Views of Mathematical Processes

The findings reported in this paper were generated from a case study of teacher leaders at a state‐level mathematics conference. Investigation focused on how participants viewed the mathematical processes of communication, connections, representations, problem solving, and reasoning and proof. Purposeful sampling was employed to select nine participants who were then interviewed and observed as they presented a session at the conference. Participants' statements revealed differences in their views of mathematical processes. The analysis led to an emergent framework for views of mathematical processes that includes three levels: participatory, experiential, and sense‐making. Implications are shared for mathematics methods instructors, professional learning, and research. Discussion also relates the framework to the Standards for Mathematical Practice.     

经管类专业学生数学学习心理调查

针对湖南科技学院经管类专业学生的问卷调查显示,在他们中间存在着学习动机不明确、学习方法不科学和学习效果不明显的问题.进一步分析调查结果,寻找问题产生的原因,给出解决问题的方案:采取措施加强学生对数学课程重要性的认识,实行分层次教学,优化教师知识结构,提高教师的教学水平.     

Using Student Reasoning to Inform the Development of Conceptual Learning Goals: The Case of Quadratic Functions

Despite the proliferation of mathematics standards internationally and despite general agreement on the importance of teaching for conceptual understanding, conceptual learning goals for many K-12 mathematics topics have not been well-articulated. This article presents a coherent set of five conceptual learning goals for a complex mathematical domain, generated via a method of systematic empirical analysis of students' reasoning. Specifically, we compared the reasoning of pairs of students who performed differentially on the same task and inferred the pivotal intermediate conceptions that afforded one student deeper engagement with the task than another student. In turn, each pivotal intermediate conception framed an associated conceptual learning goal. While the empirical analysis of student reasoning is typically used to understand how students learn, we argue that such analysis should also play an important role in determining what concepts students should learn.     

Of Our Own Nation: John Wallis's Account of Mathematical Learning in Medieval England

In A treatise of algebra both historical and practical (London 1685), John Wallis wrote the first survey of the state of mathematical learning in medieval England, and discussed with particular care the arrival and significance of the Hindu–Arabic numeral system. This paper offers a detailed commentary on Wallis's account in relation to the sources he used and the 17th-century Oxford context in which he wrote. The paper also supplements Wallis's treatment where possible with some of the findings of modern scholarship. It therefore provides on the one hand an overview of the spread of mathematical learning into medieval England, and on the other an insight into late 17th-century historiography. Wallis pioneered several new historiographical methods and can perhaps be claimed as the first modern historian of mathematics. Copyright 2001 Academic Press.Dans A treatise of algebra both historical and practical (Londres 1685) John Wallis a été le premier à écrire sur l'etat des études de mathématiques dans l'Angleterre du Moyen Age, et il a discuté avec un soin particulier de l'arrivée et du sens du système numérique hindou–arabe. Cet exposé offre un commentaire détaillé sur l'essai de Wallis en ce qui concerne les sources utilisées et le contexte d'Oxford au dix-septième siècle, époque à laquelle il écrivit. Cet exposé ajoute en plus au traitement de Wallis certaines découvertes d'erudition moderne. Il fournit donc d'une part un aperçu général de l'étendue des études de mathématiques dans l'Angleterre moyenâgeuse et d'autre part un aperçu dans l'historiographie à la fin du dix-septième siècle. Wallis à été le pionier de plusiers méthodes d'historiographie et pourrait être acclamé comme le premier historien moderne de mathématiques. Copyright 2001 Academic Press.MSC classifications: 01A35, 01A45, 01A85.     

Mathematical Analysis: An Inverse Problem Arising in Convective-Diffusive Flow

A particular system constructed and analysed for the combinedconvective flow and diffusive movement of concentrations througha packed column gives rise to a novel inverse problem analogousto ‘the sideways heat equation’, which may be ill-posed.It is possible to find an explicit formula for the solute concentrationat the top of the column in terms of that at the bottom of thecolumn. This is used in the simulation of biological experimentsinvolving hormonal stimulation of perifused cells in elutionchromatography. The formula obtained for the concentration atthe top of the column involves an infinite series of time derivativesof increasing order of the concentration at the bottom of thecolumn. While this presents possible complications from a computationalperspective, the practical use of this formula is possible byuse of splines to fit experimental data. Provided a priori estimatescan be made on the inlet data, such as bounded L₂-norm and rapiddecay on the higher frequencies of its spectrum, the problemcan be stabilized.