Mathematics is a critical part of much scientific research. Physics in particular weaves math extensively into its instruction beginning in high school. Despite much research on the learning of both physics and math, the problem of how to effectively include math in physics in a way that reaches most students remains unsolved. In this paper, we suggest that a fundamental issue has received insufficient exploration: the fact that in science, we don’t just use math, we make meaning with it in (...) a different way than mathematicians do. In this reflective essay, we explore math as a language and consider the language of math in physics through the lens of cognitive linguistics. We begin by offering a number of examples that show how the use of math in physics differs from the use of math as typically found in math classes. We then explore basic concepts in cognitive semantics to show how humans make meaning with language in general. The critical elements are the roles of embodied cognition and interpretation in context. Then, we show how a theoretical framework commonly used in physics education research, resources, is coherent with and extends the ideas of cognitive semantics by connecting embodiment to phenomenological primitives and contextual interpretation to the dynamics of meaning-making with conceptual resources, epistemological resources, and affect. We present these ideas with illustrative case studies of students working on physics problems with math and demonstrate the dynamical nature of student reasoning with math in physics. We conclude with some thoughts about the implications for instruction. (shrink)

In order to show how formal analogies between different physical systems play an important conceptual work in physics, this paper analyzes the evolution of Einstein’s thoughts on the structure of radiation from the point of view of the formal analogies he used as “lenses” to “see” through the “black box” of Planck’s blackbody radiation law. A comparison is also made with his 1925 paper on the quantum gas where he used the same formal methods. Changes of formal points of view (...) are most of the time taken for granted or passed over in silence in studies on the mathematization of physics as if they had no special significance. Revisiting Einstein’s classic papers on the nature of light and matter from the angle of the various theoretical tools he used, namely entropy and energy fluctuation calculations, helps explain why he was in a unique position to make visible the particle structure of radiation and the dual nature of light and matter. Finally, this case study calls attention to the more general question of the surprising creative power of formal analogies and their frequent use in theoretical physics. This aspect of intellectual creation can be useful in the teaching of physics. (shrink)

In a famous 1960 paper, Wigner discussed “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” I suggest that the effectiveness of mathematics in producing successful new theories and surprising discoveries is even more unreasonable than Wigner claimed. In this paper, I present several historical case studies to support the claim that mathematics is often responsible for instigating scientific revolutions. However, that does not mean that mathematics is always the key to the universe, and other cases where mathematization was not (...) successful are discussed in order to problematize a naïve Platonism. (shrink)

For more than a century the notion of a pre-established harmony between the mathematical and physical sciences has played an important role not only in the rhetoric of mathematicians and theoretical physicists, but also as a doctrine guiding much of their research. Strongly mathematized branches of physics, such as the vortex theory of atoms popular in Victorian Britain, were not unknown in the nineteenth century, but it was only in the environment of fin-de-siècle Germany that the idea of a pre-established (...) harmony really took off and became part of the mathematicians’ ideology. Important historical figures were in this respect David Hilbert, Hermann Minkowski and, somewhat later, Albert Einstein. Roughly similar ideas can be found also among British theorists, among whom Arthur Eddington, Arthur Milne, and Paul Dirac are singled out. Although largely limited to the period 1870–1940, the paper also considers Max Tegmark’s recent hypothesis of the universe being a one-to-one reflection of mathematical structures. (shrink)

In physics education, equations are commonly seen as calculation tools to solve problems or as concise descriptions of experimental regularities. In physical science, however, equations often play a much more important role associated with the formulation of theories to provide explanations for physical phenomena. In order to overcome this inconsistency, one crucial step is to improve physics teacher education. In this work, we describe the structure of a course that was given to physics teacher students at the end of their (...) master’s degree in two European universities. The course had two main goals: To investigate the complex interplay between physics and mathematics from a historical and philosophical perspective and To expand students’ repertoire of explanations regarding possible ways to derive certain school-relevant equations. A qualitative analysis on a case study basis was conducted to investigate the learning outcomes of the course. Here, we focus on the comparative analysis of two students who had considerably different views of the math-physics interplay in the beginning of the course. Our general results point to important changes on some of the students’ views on the role of mathematics in physics, an increase in the participants’ awareness of the difficulties faced by learners to understand physics equations and a broadening in the students’ repertoire to answer “Why?” questions formulated to equations. Based on this analysis, further implications for physics teacher education are derived. (shrink)

This paper examines physics and mathematics textbooks published in France at the end of the 1950s and at the beginning of the 1960s for children aged 11–15 years old. It argues that at this “middle school” level, textbooks contributed to shape cultural representations of both disciplines and their mutual boundaries through their contents and their material aspect. Further, this paper argues that far from presenting clearly delimited subjects, late 1950s textbooks offered possible connections between mathematics and physics. It highlights that (...) such connections depended upon the type of schools the textbooks aimed at, at a time when educational organization still differentiated pupils of this age. It thus stresses how the audience and its projected aptitudes and needs, as well as the cultural teaching traditions of the teachers in charge, were inseparable from the diverse conceptions of mathematics and physics and their relationships promoted through textbooks of the time. (shrink)

This article discusses the role of mathematics during physics lessons in upper-secondary school. Mathematics is an inherent part of theoretical models in physics and makes powerful predictions of natural phenomena possible. Ability to use both theoretical models and mathematics is central in physics. This paper takes as a starting point that the relations made during physics lessons between the three entities Reality, Theoretical models and Mathematics are of the outmost importance. A framework has been developed to sustain analyses of the (...) communication during physics lessons. The study described in this article has explored the role of mathematics for physics teaching and learning in upper-secondary school during different kinds of physics lessons. Observations are from three physics classes led by one teacher. The developed analytical framework is described together with results from the analysis of the 7 lessons. The results show that there are some relations made by students and teacher between theoretical models and reality, but the bulk of the discussion in the classroom is concerning the relation between theoretical models and mathematics. The results reported on here indicate that this also holds true for all the investigated organizational forms lectures, problem solving in groups and labwork. (shrink)

We study the interconnection between Physics and Mathematics in concrete instances, departing from the usual expression for the Coulomb electric field, produced by a point-like charge. It is scrutinized by means of six epistemology-intensive questions and radical answers are proposed, intended to widen one’s understanding of the subject. Our interventions act along two complementary directions. One of them regards ontology, since questions induce one to look closely at the electric charge, from different perspectives, promoting reflections about its nature and reinforcing (...) the corresponding concept. Formal manipulations rely on the identification of concepts with symbols, and the other direction concerns the spatial extension of mathematical structures. Our questions and their somewhat unusual answers help disclosing information which is not present in many textbooks, and show that Mathematics can be used as an efficient epistemological tool in Physics teaching. (shrink)

The language of physics is mathematics, and physics ideas, laws and models describing phenomena are usually represented in mathematical form. Therefore, an understanding of how to navigate between phenomena and the models representing them in mathematical form is important for a physics teacher so that the teacher can make physics understandable to students. Here, the focus is on the “experimental mathematization,” how laws are established through quantifying experiments. A sequence from qualitative experiments to mathematical formulations through quantifying experiments on electric (...) current, voltage and resistance in pre-service physics teachers’ laboratory reports is examined. The way students reason and justify the mathematical formulation of the measurement results and how they combine the treatment and presentation of empirical data to their justifications is analyzed. The results show that pre-service physics teachers understand the basic idea of how quantifying experiments establish the quantities and laws but are not able to argue it in a justified manner. (shrink)

In this paper, we discuss the history of the concept of function and emphasize in particular how problems in physics have led to essential changes in its definition and application in mathematical practices. Euler defined a function as an analytic expression, whereas Dirichlet defined it as a variable that depends in an arbitrary manner on another variable. The change was required when mathematicians discovered that analytic expressions were not sufficient to represent physical phenomena such as the vibration of a string (...) and heat conduction. The introduction of generalized functions or distributions is shown to stem partly from the development of new theories of physics such as electrical engineering and quantum mechanics that led to the use of improper functions such as the delta function that demanded a proper foundation. We argue that the development of student understanding of mathematics and its nature is enhanced by embedding mathematical concepts and theories, within an explicit–reflective framework, into a rich historical context emphasizing its interaction with other disciplines such as physics. Students recognize and become engaged with meta-discursive rules governing mathematics. Mathematics teachers can thereby teach inquiry in mathematics as it occurs in the sciences, as mathematical practice aimed at obtaining new mathematical knowledge. We illustrate such a historical teaching and learning of mathematics within an explicit and reflective framework by two examples of student-directed, problem-oriented project work following the Roskilde Model, in which the connection to physics is explicit and provides a learning space where the nature of mathematics and mathematical practices are linked to natural science. (shrink)

The process of the mathematization of physical situations through differential calculus requires an understanding of the justification for and the meaning of the differential in the context of physics. In this work, four different conceptions about the differential in physics are identified and assessed according to their utility for the mathematization process. We also present an empirical study to probe the conceptions about the differential that are used by students in physics, as well students’ perceptions of how they are expected (...) to use differential calculus in physics. The results support the claim that students have a quasi-exclusive conception of the differential as an infinitesimal increment and that they perceive that their teachers only expect them to use differential calculus in an algorithmic way, without a sound understanding of what are they doing and why. These results are related to the lack of attention paid by conventional physics teaching to the mathematization process. Finally, some proposals for action are put forward. (shrink)