Tables turned: Physics in the service of mathematics

E T Bell, the famous author of ‘Men of Mathematics’, has described mathematics as the ‘Queen of Arts and Servant of Science’. What he meant is that mathematics serves science by entering into the picture as soon as a proper mathematical model is set up by the scientist, and then after a purely mathematical analysis of the model, the final mathematical step is interpreted scientifically. The purpose of the present article is to convince the readers that sometimes the roles of science and mathematics are reversed, and a mathematical problem is interpreted as a physics problem; the laws of physics are utilized for a physical analysis, and the final result of the physical analysis is interpreted mathematically. We shall illustrate this by means of few examples.     

The Mathematics of High School Physics

In the seventeenth and eighteenth centuries, mathematicians and physical philosophers managed to study, via mathematics, various physical systems of the sublunar world through idealized and simplified models of these systems, constructed with the help of geometry. By analyzing these models, they were able to formulate new concepts, laws and theories of physics and then through models again, to apply these concepts and theories to new physical phenomena and check the results by means of experiment. Students’ difficulties with the mathematics of high school physics are well known. Science education research attributes them to inadequately deep understanding of mathematics and mainly to inadequate understanding of the meaning of symbolic mathematical expressions. There seem to be, however, more causes of these difficulties. One of them, not independent from the previous ones, is the complex meaning of the algebraic concepts used in school physics (e.g. variables, parameters, functions), as well as the complexities added by physics itself (e.g. that equations’ symbols represent magnitudes with empirical meaning and units instead of pure numbers). Another source of difficulties is that the theories and laws of physics are often applied, via mathematics, to simplified, and idealized physical models of the world and not to the world itself. This concerns not only the applications of basic theories but also all authentic end-of-the-chapter problems. Hence, students have to understand and participate in a complex interplay between physics concepts and theories, physical and mathematical models, and the real world, often without being aware that they are working with models and not directly with the real world.     

理论术语的经验主义解释

由于理论术语所指称实体或过程的不可观察性,超越了能够以经验术语去表述的可观察范围,因此它们在许多经验主义者的眼里是没有意义的,或者被认为完全可以通过"对应规则"用经验术语定义。然而,用可观察的性质显式地定义那些被用来命名不可观察的实体或过程的术语是有问题的。科学理论一方面要通过经验得到辩护,另一方面要给予我们关于经验现象的解释性说明。调和经验主义的认知策略与科学理论的解释或说明功能之间的关系是解决问题的关键。     

High school students’ approaches to learning physics with relationship to epistemic views on physics and conceptions of learning physics

Background and purpose : Knowing how students learn physics is a central goal of physics education. The major purpose of this study is to examine the strength of the predictive power of students’ epistemic views and conceptions of learning in terms of their approaches to learning in physics. Sample, design and method : A total of 279 Taiwanese high school students ranging from 15 to 18?years old participated in this study. Three questionnaires for assessing high school students’ epistemic views on physics, conceptions of learning physics and approaches to learning physics were developed. Step-wise regression was performed to examine the predictive power of epistemic views on physics and conceptions of learning physics in terms of their approaches to learning physics. Results and conclusion: The results indicated that, in general, compared to epistemic views on physics, conceptions of learning physics are more powerful in predicting students’ approaches to learning physics in light of the regression models. That is, students’ beliefs about learning, compared with their beliefs about knowledge, may be more associated with their learning approaches. Moreover, this study revealed that the higher-level conceptions of learning physics such as ‘Seeing in a new way’ were more likely to be positively correlated with the deep approaches to learning physics, whereas the lower-level conceptions such as ‘Testing’ were more likely to positively explain the surface approaches, as well as to negatively predict the deep approaches to learning physics.     

Scientific investigations,metaphorical gestures,and the emergence of abstract scientific concepts

Where anthropological and psychological studies have shown that gestures are a central feature of communication and cognition, little is known about the role of gesture in learning and instruction. Drawing from a large database on student learning, we show that when students engage in conversations in the presence of material objects, these objects provide a phenomenal ground against which students can enact metaphorical gestures that embody (give a body to) entities that are conceptual and abstract. In such instances, gestures are often subsequently replaced by an increasing reliance upon the verbal mode of communication. If gestures constitute a bridge between experiences in the physical world and abstract conceptual language, as we conjecture here, our study has significant implications for both learning and instruction.     

注重认知规律,加强概念教学

物理概念既是物理学大厦的砖石,又是对物理世界的数学描述．初中阶段,加强概念教学,至关重要．初中阶段的物理概念教学应从学生认知规律出发,注重概念引入,注重实验操作,注重启发引导,注重实际应用等,理论联系实际,使学生真正理解和掌握物理概念．     

Epistemic schemes and epistemic states. A study of mathematics convincement in elementary school classes

The paper introduces an interpretative framework that contains a characterization of epistemic schemes (constructs that are used to explain how class agents themselves are able to gain convincement in or promote convincement of mathematical statements) and epistemic states (a person’s internal states, such as convincement or certainty related to the person’s beliefs and to the schemes that explain them); a taxonomy for the epistemic schemes is also proposed. On the basis of the interpretative framework, an analysis is made of an excerpt of a regular elementary school class, a school level at which explicit mathematics reasoning rarely arises. The paper contends that teachers and students use schemes based on reasons in order to make mathematical statements credible, but that they also resort–perhaps unconsciously–to epistemic schemes that are governed by extra-rational considerations.     

What Are Models and Why Do We Need Them?

Educators and philosophers tout the virtues of the `new' view of theories. Unfortunately, there is no agreed-on name for the new view, but a recently a favored term has been `model based'. I address what a model might be in the context of science education. I am concerned about when and why we need to make the transition from implicit mental models to explicit external models.I explore two theories/models in physics. One draws on misunderstandings of rotating objects. The second concerns concerning density and flotation.I offer two morals:`Models' in physics are often mathematical, and more attention needs to be devoted to integrating mathematics and scienceModels are required in physics when we need an equation in which two variables occur and which is not linearly additive.     

A Transformative Model for Undergraduate Quantitative Biology Education

The BIO2010 report recommended that students in the life sciences receive a more rigorous education in mathematics and physical sciences. The University of Delaware approached this problem by (1) developing a bio-calculus section of a standard calculus course, (2) embedding quantitative activities into existing biology courses, and (3) creating a new interdisciplinary major, quantitative biology, designed for students interested in solving complex biological problems using advanced mathematical approaches. To develop the bio-calculus sections, the Department of Mathematical Sciences revised its three-semester calculus sequence to include differential equations in the first semester and, rather than using examples traditionally drawn from application domains that are most relevant to engineers, drew models and examples heavily from the life sciences. The curriculum of the B.S. degree in Quantitative Biology was designed to provide students with a solid foundation in biology, chemistry, and mathematics, with an emphasis on preparation for research careers in life sciences. Students in the program take core courses from biology, chemistry, and physics, though mathematics, as the cornerstone of all quantitative sciences, is given particular prominence. Seminars and a capstone course stress how the interplay of mathematics and biology can be used to explain complex biological systems. To initiate these academic changes required the identification of barriers and the implementation of solutions.     

Representing the Electromagnetic Field: How Maxwell��s Mathematics Empowered Faraday��s Field Theory

James Clerk Maxwell ??translated?? Michael Faraday??s experimentally-based field theory into the mathematical representation now known as ??Maxwell??s Equations.?? Working with a variety of mathematical representations and physical models Maxwell extended the reach of Faraday??s theory and brought it into consistency with other results in the physics of electricity and magnetism. Examination of Maxwell??s procedures opens many issues about the role of mathematical representation in physics and the learning background required for its success. Specifically, Maxwell??s training in ??Cambridge University?? mathematical physics emphasized the use of analogous equations across fields of physics and the repeated solving of extremely difficult problems in physics. Such training develops an array of overlearned mathematical representations supported by highly sophisticated cognitive mechanisms for the retrieval of relevant information from long term memory. For Maxwell, mathematics constituted a new form of representation in physics, enhancing the formal derivational and calculational role of mathematics and opening a cognitive means for the conduct of ??experiments in the mind?? and for sophisticated representations of theory.     

卡西尔艺术哲学引论

卡西尔一方面把艺术与人类全部文化活动联系起来加以考察,认为艺术是一种文化符号形式;另一方面,卡西尔又以艺术的形式为核心来逐层揭示艺术的本质与功能,从而形成了他独特的艺术观.     

How do students learn to apply their mathematical knowledge to interpret graphs in physics?

This paper describes a laboratory-based program in physics designed to help students build effective links between the mathematical equations used to solve problems in mechanics and the real world of moving objects. Through the analysis of straight line graphs derived from their own data students have been able to achieve a considerable development towards a concept of slope, or gradient, and how it relates to the concept of proportionality, but they continue to demonstrate a great resistance to applying their mathematical knowledge to physics. A model designed to help us apply current research ideas to this problem is described. The work described in this paper was carried out at Dickson College, a government senior secondary college (Years 11 and 12) in the Australian Capital Territory, where the author taught physics and biology.     

Toward preparing students for change: A critical discussion of the contribution of the history of physics in physics teaching

In this paper I place physics teaching, and the inclusion of the history of physics into teaching, within a wide context. I start from the conviction that there are considerable changes ahead in the life circumstances of people in western industrial societies. This expectation should influence our aims of education generally, and in particular the aims of physics teaching. The paper does not offer final solutions, but analyses the situation and thereby argues for a change in perspective in physics teaching. The main idea is that physics teaching has to solve the problem of balancing seemingly incompatible needs, for example, conveying a stock of stable, dependable physics knowledge to students, and on the other hand to train them to see their physics knowledge within varying contexts of change. It is argued that the history of physics can be of high value in solving this problem.This article was originally published in: F. Bevilacqua and P.J. Kennedy (eds.): 1983, Proceedings of the Conference on Using History of Physics in Innovatory Physics Education, Pavia University.     

Engineering problem-solving knowledge: the impact of context

Employer complaints of engineering graduate inability to ‘apply knowledge’ suggests a need to interrogate the complex theory-practice relationship in twenty-first century real world contexts. Focussing specifically on the application of mathematics, physics and logic-based disciplinary knowledge, the research examines engineering problem-solving processes as enacted by recent graduates in a range of industrial settings. Theoretically situated in the sociology of education, the Bernsteinian concept of knowledge structures and Legitimation Code Theory epistemic relations are utilised to surface the disciplinary basis of problem solving in different sociotechnical contexts. It is argued that the relationship between the ‘what’ and the ‘how’ of the problem gives rise to significantly different practice ‘codes’ between which successful engineering problem-solvers are required to shift. This paper presents two contrasting case studies which demonstrate the impact of the environment on code-shifting practices. Findings suggest that engineering curricula need to facilitate a more conceptual grasp of contextual complexities.     

The Anthropology of Meaning

Meaning is one of the recent terms which have gained great currency in mathematics education. It is generally used as a correlate of individuals' intentions and considered a central element in contemporary accounts of knowledge formation. One important question that arises in this context is the following: if, in one way or another, knowledge rests on the intrinsically subjective intentions and deeds of the individual, how can the objectivity of conceptual mathematical entities be guaranteed? In the first part of this paper, both Peirce's and Husserl's theories of meaning are discussed in light of the aforementioned question. I examine their attempts to reconcile the subjective dimension of knowing with the alleged transcendental nature of mathematical objects. I argue that transcendentalism, either in Peirce's or Husserl's theory of meaning, leads to an irresolvable tension between subject and object. In the final part of the article, I sketch a notion of meaning and conceptual objects based on a semiotic-cultural approach to cognition and knowledge which gives up transcendentalism and instead conveys the notion of contextual objectivity.     

中学物理教学中物理模型教学探究

在物理教学中,许多物理定律和规律都是把实际的研究对象或物理过程抽象为理想化的物理模型,然后研究物理模型所涉及的物理量及其相互关系。由于物理试题是根据物理模型编拟出来的,所以解题时必须首先正确还原＂物理模型＂,并且能清晰地认识物理模型的本质特征。