Idealisation and Galileo’s Pendulum Discoveries: Historical, Philosophical and Pedagogical Considerations

Galileo’s discovery of the properties of pendulum motion depended on his adoption of the novel methodology of idealisation. Galileo’s laws of pendulum motion could not be accepted until the empiricist methodological constraints placed on science by Aristotle, and by common sense, were overturned. As long as scientific claims were judged by how the world was immediately seen to behave, and as long as mathematics and physics were kept separate, then Galileo’s pendulum claims could not be substantiated; the evidence was against them. Proof of the laws required not just a new science, but a new way of doing science, a new way of handling evidence, a new methodology of science. This was Galileo’s method of idealisatioin. It was the foundation of the Galilean–Newtonian Paradigm which characterised the Scientific Revolution of the 17th century, and the subsequent centuries of modern science. As the pendulum was central to Galileo’s and Newton’s physics, appreciating the role of idealisation in their work is an instructive way to learn about the nature of science.     

论数学的经验性本质

数学的发展历史表明,数学就其整体、过程、趋势、源泉来说是经验的,演绎体系是数学经验知识的理论形态,是数学发展到一定阶段的特征.数学与经验的关联随着数学、科学和人类认识的发展而呈现出不同的阶段性特点.公理体系的二重性说明抽象和演绎无法解决数学的根本问题.强调数学的经验性本质,并不是否认它的演绎性,而是给广大数学学习者和研究者一个本体论的信念和方法论的启示.     

Mathematics as Liberal Education: Whitehead and the Rhythm of Life

In several of his works, Alfred North Whitehead (1861–1947) presents mathematics as a way of learning about general ideas that increase our understanding of the universe. The danger is that students get bogged down in its technical operations. He argues that mathematics should be an integral part of a new kind of liberal education, incorporating science, the humanities, and “technical education” (making things with one’s hands), thereby integrating “head-work and hand-work.” In order to appreciate the role mathematics plays in modern science, students should understand its diverse history which is capable of bringing abstract ideas to life. Moreover, mathematics can discern the alternating rhythms of repetition and difference in nature constituting the periodicity of life. Since these same rhythms are to be found in his theory of learning as growth, there appears to be a pattern linking Whitehead’s approach to mathematics and his educational philosophy.     

从数学角度看西方古代和近代科学发展的逻辑主线

西方科学发展的逻辑起点是古希腊数学。古希腊数学也是以研究数开始的,但由于古希腊文明的特质,推动古希腊数学从数的研究转到几何的研究,并由此建立了欧几里得几何公理系统,为其他数学问题和科学问题提供了模型和方法。但西方的代数是落后的。西方数学吸收了东方数学的代数成果并与几何结合起来,产生了笛卡尔解析几何。牛顿和莱布尼兹在解析几何的基础上发展出了微积分,牛顿在微积分的基础上建立起了近代经典力学大厦。     

Newton’s Cradle in Physics Education

Newton’s Cradle is a series of bifilar pendulums used in physics classrooms to demonstrate the rôle of the principles of conservation of momentum and kinetic energy in elastic collisions. The paper reviews the way in which textbooks use Newton’s Cradle and points out the unsatisfactory nature of these treatments in almost all cases. The literature which attempts to explain how the apparatus works is discussed and alternative strategies which make it possible to teach about the nature of models in science as well as to teach physics through use of the apparatus are suggested.     

Responding to a Relevance Imperative in School Science and Mathematics: Humanising the Curriculum Through Story

There has been a recent push to reframe curriculum and pedagogy in ways that make school more meaningful and relevant to students’ lives and perceived needs. This ‘relevance imperative’ is evident in contemporary rhetoric surrounding quality education, and particularly in relation to the junior secondary years where student disengagement with schooling continues to abate. This paper explores how teachers translate this imperative into their mathematics and science teaching. Interview data and critical incidents from classroom practice are used to explore how six teachers attempted to make the subject matter meaningful for their students. Four ‘Categories of Meaning Making’ emerged, highlighting key differences in how the nature of science and mathematics content constrained or enabled linkages between content and students’ lifeworlds. While the teachers demonstrated a commitment to humanising the subject at some level, this analysis has shown that expecting teachers to make the curriculum relevant is not unproblematic because the meaning of relevance as a construct is complex, subject-specific, and embedded in understanding the human dimensions of learning, using, and identifying with, content. Through an examination of the construct of relevance and a humanistic turn in mathematics and science literature I argue for an expanded notion of relevance.     

Newton’s Use of the Pendulum to Investigate Fluid Resistance: A Case Study and some Implications for Teaching About the Nature of Science

Books I and III of Newton’s Principia develop Newton’s dynamical theory and show how it explains a number of celestial phenomena. Book II has received little attention from historians or educators because it does not play a major role in Newton’s argument. However, it is in Book II that we see most clearly Newton both as a theoretician and an experimenter. In this part of the Principia Newton dealt with terrestrial rather than with celestial phenomena and described a number of experiments he carried out to establish the success of his theory in explaining the properties of fluid resistance. It demonstrates most clearly the activities of a scientist working at the forefront of knowledge and working with phenomena which he did not fully understand. In this paper the first of Newton’s set of experiments into fluid resistance are described and the theory which underlies his explanation is outlined. A number of issues arising from this portion of the Principia together with implications for teaching about the nature of science are discussed.     

Workshop on Friction: Understanding and Addressing Students’ Difficulties in Learning Science Through a Hermeneutical Perspective

Hermeneutics is useful in science and science education by emphasizing the process of understanding. The purpose of this study was to construct a workshop based upon hermeneutical principles and to interpret students’ learning in the workshop through a hermeneutical perspective. When considering the history of Newtonian mechanics, it could be considered that there are two methods of approaching Newtonian mechanics. One method is called the ‘prediction approach’, and the other is called the ‘explanation approach’. The ‘prediction approach’ refers to the application of the principles of Newtonian mechanics. We commonly use the prediction approach because its logical process is natural to us. However, its use is correct only when a force, such as gravitation, is exactly known. On the other hand, the ‘explanation approach’ could be used when the nature of a force is not exactly known. In the workshop, students read a short text offering contradicting ideas about whether to analyze a friction situation using the explanation approach or the prediction approach. Twenty-two college students taking an upper-level mechanics course wrote their ideas about the text. The participants then discussed their ideas within six groups, each composed of three or four students. Through the group discussion, students were able to clarify their preconceptions about friction, and they responded to the group discussion positively. Students started to think about their learning from a holistic perspective. As students thought and discussed the friction problems in the manner of hermeneutical circles, they moved toward a better understanding of friction.     

Defining Computational Thinking for Mathematics and Science Classrooms

Science and mathematics are becoming computational endeavors. This fact is reflected in the recently released Next Generation Science Standards and the decision to include “computational thinking” as a core scientific practice. With this addition, and the increased presence of computation in mathematics and scientific contexts, a new urgency has come to the challenge of defining computational thinking and providing a theoretical grounding for what form it should take in school science and mathematics classrooms. This paper presents a response to this challenge by proposing a definition of computational thinking for mathematics and science in the form of a taxonomy consisting of four main categories: data practices, modeling and simulation practices, computational problem solving practices, and systems thinking practices. In formulating this taxonomy, we draw on the existing computational thinking literature, interviews with mathematicians and scientists, and exemplary computational thinking instructional materials. This work was undertaken as part of a larger effort to infuse computational thinking into high school science and mathematics curricular materials. In this paper, we argue for the approach of embedding computational thinking in mathematics and science contexts, present the taxonomy, and discuss how we envision the taxonomy being used to bring current educational efforts in line with the increasingly computational nature of modern science and mathematics.     

Gödel’s proof

In 1931, the great Austrian mathematician Kurt Gödel proved “all consistent axiomatic formulations of number theory include undecidable propositions”. This discovery of Gödel and its proof had enormous repercussions in mathematics and computer science. The proof hinged upon the writing of a self-referential mathematical statement, in the same way as the liar’s paradox — I am lying — is a self-referential statement. In this three-part article, we describe Gödel’s discovery and his famous proof.     

探究如何实现小学数学与信息技术的整合

小学阶段开设数学课程能够增加小学生的数学知识储备,有利于小学生将所学的数学知识运用在实际生活中。因此,在新课改和现代化科学教学的背景下,将现代信息技术应用到数学教学中,对于提高学生学习数学的兴趣和提高教学质量有关键性的作用。     

Constructivism and empiricism: An incomplete divorce

The paper outlines the significant influence of constructivism in contemporary science and mathematics education, and emphasises the central role that epistemology plays in constructivist theory and practice. It is claimed that despite the anti-empiricism of much constructivist writing, in most forms its epistemology is nevertheless firmly empiricist. In particular it is subject-centered and experience-based. It is argued that its relativist, if not skeptical conclusions, only follow given these empiricist assumptions. Further it is suggested that such assumptions belong to Aristotelian science, and were effectively overthrown with the modern science of Galileo and Newton. Thus constructivism cannot provide understanding of post-Aristotelian science. Specializations: history, philosophy and science teaching.     

Hydraulics for Royal Gardens: Water Art as a Challenge for 18th Century Science and 21st Century Physics Teaching

Hydraulics is an engineering specialty and largely neglected as a topic in physics teaching. But the history of hydraulics from the Renaissance to the Baroque, merits our attention because hydraulics was then more broadly conceived as a practical and theoretical science; it served as a constant bone of contention for mechanics and mathematics; its obvious practical importance from raising water in mines to the playful fountains in royal gardens illustrates the social role of science like few others do. The playful character of historic hydraulics problems makes it also an appealing topic for modern science education.     

浅谈高等数学的教与学

数学是一门比较抽象的学科,是一切自然科学的基础。在当今的社会,科技的进步和发展越来越要求人们更好地掌握和利用数学,数学成为了人们不可或缺的必需品。本文阐述了学习高等数学的意义,其对象和特点,以及教师如何教学生如何学的问题。     

Scaffolded Inquiry-Based Instruction with Technology: A Signature Pedagogy for STEM Education

Inquiry-based instruction has become a hallmark of science education and increasingly of integrated content areas, including science, technology, engineering, and mathematics (STEM) education. Because inquiry-based instruction very clearly contains surface, deep, and implicit structures as well as engages students to think and act like scientists, it is considered a signature pedagogy of science education. In this article the authors discuss the nature of scaffolded inquiry-based instruction and how it can be applied to the use of emerging technologies, such as data mashups and cloud computing, so that students not only learn the content of STEM, but can also begin answering the critical socioscientific questions that face the modern era.     

Finding Common Ground: A Synthesis of Science and Mathematics Teacher Educators’ Experiences with Professional Growth

The seven articles that comprise this Special Issue examine the professional growth of mathematics and science teacher educators across different contexts and different foci of who is the teacher educator being studied. Despite these differences, a common thread running throughout these seven articles is the need for learning to be situated in collaboration with others. In this final article, we examine the contribution of these articles through two perspectives: that of the collaborative contexts supporting the professional growth of mathematics and science teacher educators, and the role of disciplinary knowledge as part of the purpose for teacher educators’ professional growth. We notice that collaboration can take on very different structures in supporting teacher educators’ professional learning due to the different purposes and roles of the teacher educators in the studies. We also notice that while collaboration figures as an important component in all of the studies, the disciplinary specific aspects of collaboration, i.e., how collaboration might be negotiated differently by teacher educators in mathematics and science, is still not well understood. Overall, these articles provide important insights that help to shed new light on the complex and multifaceted nature of teacher educators’ learning and growth and provide productive avenues for future research.

     

Plato, Pascal, and the dynamics of personal knowledge

Abstract educational practices are to be based on proven scientific knowledge, not least because the function science has to perform in human culture consists of unifying practical skills and general beliefs, the episteme and the techne (Amsterdamski, 1975, pp. 43–44). Now, modern societies first of all presuppose regular and standardized ways of organizing both our concepts and our institutions. The explanatory schemata resulting from this standardization tend to destroy individualism and enchantment. But mathematics education is in fact the only place in which to treat the human subject’s relationship with mathematics. And that is what mathematics education is all about: make the human subject grow intellectually and as a person by means of mathematics. At first sight, mathematics, in its formal guise, seems the opposite of philosophy, because philosophy constructs concepts (meanings), whereas mathematics deals with extensions of concepts (sets). We shall, however, turn this problem into an instrument, using the complementarity of intensions and extensions of theoretical terms as our main device for discussing the relationship between philosophy and mathematics education. The complementarity of the “how” and the “what” of our representations outlines, in fact, the terrain on which epistemology and education are to meet.     

The Law of Inertia: How Understanding its History can Improve Physics Teaching

The law of inertia is a problem in teaching due to the impossibility of showing the proposition experimentally. As we cannot do an experiment to verify the law, we cannot know if it is correct. On the other hand, we know that the science based upon it is successful. A study in the history of mechanics has shown that there are different foundations for the law but also that the law plays the same role in the science since Newton. To avoid a statement of which we cannot be sure, the present paper proposes to understand the law through its function in the theory. In this case, we do not have to say how a free body moves, but rather that the rectilinear and uniform motion is the motion of reference in Newtonian mechanics.     

In Pursuit of a Connected Way of Knowing: The Case of One Mathematics Teacher

In this paper, we offer illustrations of a mathematics teacher’s difficulties with content knowledge when trying to find connections between school mathematics and science; we do so by describing the development of this teacher’s thinking and learning in her pursuit of connections between the concepts of slope of a line and density of matter. The paper is based on a sub-study that is part of a larger Colombian project, PROMESA (Creating Science and Mathematics Connected Learning Experiences that Open Opportunities for the Promotion of Algebraic Reasoning), which incorporated a Professional Learning Programme (PLP) seeking to integrate school science and mathematics teachers into working teams, in order to create science and mathematics connected learning experiences that considered the promotion of algebraic reasoning. The ‘challenging questions’ that emerged for this teacher, during the workshops of the induction stage of the PLP, became the driving force for her continued engagement in learning mathematics content in a connected way, as opposed to the compartmentalised content-item thinking that she had experienced as a school student. We provide illustrations of first steps in the development of a teacher’s mathematical understanding, which can support growth of mathematical knowledge for teaching.     

Unfulfilled hopes in education for equity: redesigning the mathematics curriculum in a US high school

Mathematics education aimed at empowering students for economic and democratic participation must address two critical issues: the long‐standing function of mathematics as a gatekeeper, and the complicated nature of designing and implementing systematic reform at the school department level. The study reported here examines a curricular redesign implemented by teachers in one US high school department. The department was redesigning its curriculum to remedy high failure rates in targeted courses disproportionately populated by students of colour. Using a case‐study methodology, this study examines the process of curricular redesign and its influence on these students’ access to more advanced mathematics courses. It describes redesigned courses and department characteristics that aided or challenged the redesign process, and discusses the significance of the identified challenges in constructing placement policies that did little to increase students’ likelihood of taking additional, and more advanced‐ level, mathematics courses. The analysis revealed a department that, despite its intentions, implemented a curriculum design which perpetuated inequities. This study discusses the teachers’ expectations of their students and perspectives about the nature of mathematics as a partial explanation for the department’s failure. The resigned curriculum failed to promote mathematics course‐taking because it created more defined tracks with less rigorous courses for students in low‐level courses.