Modelling Mathematical Reasoning in Physics Education

Many findings from research as well as reports from teachers describe students’ problem solving strategies as manipulation of formulas by rote. The resulting dissatisfaction with quantitative physical textbook problems seems to influence the attitude towards the role of mathematics in physics education in general. Mathematics is often seen as a tool for calculation which hinders a conceptual understanding of physical principles. However, the role of mathematics cannot be reduced to this technical aspect. Hence, instead of putting mathematics away we delve into the nature of physical science to reveal the strong conceptual relationship between mathematics and physics. Moreover, we suggest that, for both prospective teaching and further research, a focus on deeply exploring such interdependency can significantly improve the understanding of physics. To provide a suitable basis, we develop a new model which can be used for analysing different levels of mathematical reasoning within physics. It is also a guideline for shifting the attention from technical to structural mathematical skills while teaching physics. We demonstrate its applicability for analysing physical-mathematical reasoning processes with an example.     

类比思维在物理教学中的应用

类比是一种重要的思维方法,也是学习能力、创新能力的一个重要组成部分.文章从类比的概念谈起,阐述了这种思维方法,在科学探索及物理教育中的作用.     

Evaluating an Automatically Scorable, Open-Ended Response Type for Measuring Mathematical Reasoning in Computer-Adaptive Tests

The first generation of computer-based tests depends largely on multiple-choice items and constructed-response questions that can be scored through literal matches with a key. This study evaluated scoring accuracy and item functioning for an open-ended response type where correct answers, posed as mathematical expressions, can take many different surface forms. Items were administered to 1,864 participants in field trials of a new admissions test for quantitatively oriented graduate programs. Results showed automatic scoring to approximate the accuracy of multiple-choice scanning, with all processing errors stemming from examinees improperly entering responses. In addition, the items functioned similarly in difficulty, item-total relations, and male-female performance differences to other response types being considered for the measure.     

Mathematical Reasoning in School Tasks

An earlier study (Lithner 1998) treated the question “what are the main characteristics and background of undergraduate students' difficulties when trying to solve mathematical tasks?” This paper will focus on, and extend, the part of the earlier study that concerns task solving strategies. The results indicate that focusing on what is familiar and remembered at a superficial level is dominant over reasoning based on mathematical properties of the components involved, even when the latter could lead to considerable progress.     

Intuitions and Schemata in Mathematical Reasoning

The present paper is an attampt to analyze the relationship between intuitions and structural schemata. Intuitions are defined as cognitions which appear subjectively to be self-evident, immediate, certain, global, coercive. Structural schemata are behavioral-mental devices which make possible the assimilation and interpretation of information and the adequate reactions to various stimuli. Structural schemata are characterized by their general relevance for the adaptive behavior. The main thesis of the paper is that intuitions are generally based on structural schemata. The transition from schemata to intuitions is achieved by a particular process of compression described in the paper.This revised version was published online in September 2005 with corrections to the Cover Date.     

Conceptual Understanding of Causal Reasoning in Physics

College students often experience difficulties in solving physics problems. These difficulties largely result from a lack of conceptual understanding of the topic. The processes of conceptual learning reflect the nature of the causal reasoning process. Two major causal reasoning methods are the covariational and the mechanism‐based approaches. This study was to investigate the effects of different causal reasoning methods on facilitating students’ conceptual understanding of physics. 125 college students from an introduction physics class were assigned into covariational group, mechanism‐based group, and control group. The results show that the mechanism‐based group significantly outperformed the other two groups in solving conceptual problems. However, no significant difference was found in all three groups performance on solving computational problems. Speculation on the inconsistent performance of the mechanism‐based group in conceptual and computational problem solving is given. Detailed analyses of the results, findings, and educational implications are discussed     

数学理智与培养策略

数学理智水平是影响初中学生数学学习成绩的一个重要因素。指出了数学理智的基本内涵、心理表征和教学价值,创设探究情境、重视反思教学、正确归因是培养初中学生数学理智的有效策略。     

试论数学推理过程的逻辑性——兼论什么是有逻辑的推理

论证了数学推理、进而论证了一类逻辑推理的本质是推理的过程具有传递性,包括关系传递性和性质传递性,并且用数学的语言和符号确切地表述了这两种传递性.因为数学推理的对象是数学命题,数学命题又隐含着数学研究对象的定义,因此,限定了数学定义和数学命题的基本形态.为了数学教育的需要,基于所表述的两种传递性,进一步论证了常用的数学论证方法为什么会是有效的.研究结论对于数学教育的发展是重要的,对于逻辑学的发展也是有意义的.     

几类数学物理方程中的变量代换

变量代换在数学物理方程中是不可或缺的,变量代换并不能作为方程求解的通用方法,但是对于一些特殊的方程,变量代换还是有着重要的作用.本文总结了数学物理方程这门课程中的几类变量代换,分别对于不同类型的方程,给出了比较重要的变量代换方法,并对一些结果作了补充和推广.     

物理教学中演示实验的处理

物理教学离不开演示实验.演示实验既是学生完成一个单元学习的关键点,也是教师对学生进行有效指导的切入点,学生通过完成演示实验的学习,理解概念、掌握规律、培养技巧,教师通过演示实验的教学,对学生进行思维方案、学习态度和科学实践精神的培养,因而在教学中演示实验所处的地位相当重要.     

Themes and Interplay of Beliefs in Mathematical Reasoning

Upper secondary students’ task solving reasoning was analysed with a focus on arguments for strategy choices and conclusions. Passages in their arguments for reasoning that indicated the students’ beliefs were identified and, by using a thematic analysis, categorized. The results stress three themes of beliefs used as arguments for central decisions: safety, expectations and motivation. Arguments such as ‘I don’t trust my own reasoning’, ‘mathematical tasks should be solved in a specific way’ and ‘using this specific algorithm is the only way for me to solve this problem’ exemplify these three themes. These themes of beliefs seem to interplay with each other, for instance in students’ strategy choices when solving mathematical tasks.     

Stimulating Mechanistic Reasoning in Physics Using Student-Constructed Stop-Motion Animations

Journal of Science Education and Technology - This article reports on a case study that aims to help students develop mechanistic reasoning through constructing a model based stop-motion animation...     

Electromagnetic Concepts in Mathematical Representation of Physics

Our paper deals with the use of mathematics when studying the physics of electromagnetism. We have concentrated on common electromagnetic concepts (magnetic field and flux) and their associated mathematical representation and arithmetical tools. Our studies showed that most students do not understand the significant aspects of physical situations. Students have difficulty in using relationships and models specific to magnetic phenomena (the construction of relationships between concepts, and the use of mathematical formalism).     

匈牙利培养学生数学推理能力的经验及借鉴

匈牙利正在着手进行新一轮数学课程改革,在改革过程中,匈牙利非常强调学生数学推理能力的培养.匈牙利的做法可为当前我国进行的数学教育改革所借鉴。     

构建数学模型解决物理问题

本文着重探讨了在中学物理教学中注重数学模型与数学方法教学的必要性,构建数学模型的基本途径,数学方法在解决物理问题中的具体应用,数学的解与物理的解的统一等。使学生能自觉地运用数学知识、思想、方法去考虑和处理物理问题,从而有效地培养学生应用数学思想与方法解决物理问题的能力。     

结合数学学科开展研究性学习的实践与探索

结合数学学科开展研究性学习，首先，要注意目标定位与内容的选择。其次，组织形式以小组合作方式为主。第三，在评价方式上，注重形成性评价和总结性评价相结合，以形成性评价为主；以肯定性评价为主；强调差异性评价；突出自我评价。结合数学学科开展研究性学习，能有效地改变学生学习数学的方式，促进素质教育向更高层次发展。     

怎样做好物理演示实验

实验是物理学的基础,是检验物理理论的标准.演示实验是指为配合教学内容由教师操作表演示范的实验.它是深受学生欢迎的实验形式,是教师施展教学艺术的独特方法.它能化抽象为具体,化枯燥为生动,把要研究的物理现象清楚地展示在学生面前.能引导学生观察,并进行思考,配合讲授使学生认识物理概念和规律,达到"事半功倍"的效果.     

数学物理方法的现代化教学

将现代化的教学于段应用到大学物理专业的基础必修课——数学物理方法这门课程中,采取多媒体课件、引入计算机软件和网络资源平台等现代化教学模式与传统教学模式相结合,取长补短,使课程不再枯燥,极大地调动了学生学习的积极性.     

重视物理实验题中数学图像问题的讨论

物理和数学是紧密联系的,应用数学处理物理问题的能力是高考要求的五种能力之一,近几年的高考实验题均对该能力提出了较高的要求.     

数学推理:发展学生核心素养

“推理”是贯穿于学生数学学习始终的一项活动。引导学生进行数学推理,要让学生做到“推之有源”“推之有据”“推之有理”。归纳推理有助于引导学生合情猜想,类比推理有助于引导学生合理迁移,演绎推理有助于引导学生合法论证。推理教学能有效提升学生的数学学习能力,发展学生的数学核心素养。