Abstract
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.
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From Galileo (1638)
From Newton (1687)
From d’ Alembert (1743)
From Carnot (1824)
Based on Figure 24, p. 110 in Dingrando et al. (2007)
Based on Figure 2.50, p. 41 in Hamper (2009)
Based on the figure of question 13, p. 269 in Johnston et al. (2015)
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Notes
The term ‘mathematics of high school physics’ refers to mathematics used in high school physics textbooks and high school physics classes, while the term ‘high school mathematics’ refers to mathematics presented in high school mathematics textbooks and taught in high school mathematics classes.
That is, physics that was formulated in the seventeenth and eighteenth centuries.
According to scholastics, these qualities determined the ‘form’ of a body, which was changeable, whereas its ‘substance’ was unchanging (Lewi 2006).
The same view had been stated some years earlier by the Aristotelian Jesuits Alesandro Picolomini and Benedict Pereira (Marshall 2011).
The idea that the world had a hidden mathematical structure had long been endorsed in Greek philosophy and science. Archimedes, however, used mathematics instrumentally (without metaphysical assertions) to analyze certain parts of the physical world in a systematic (and fruitful) way.
The figure is based on Heath’s translation of the Works of Archimedes (Archimedes 2010, p. 190). On the other hand, in Hiebert’s edition of the same Works (Archimedes 1913), which is based on an older manuscript, this figure contains triangles instead of rectangles. These manuscripts have been reproduced many times through the centuries, and the accuracy of their figures is questionable.
Mathematical procedures for the direct determination of ‘velocity’ as a quotient of space divided by time were at that time forbidden (Ravetz 1961).
It seems, however, that some experiments preceded theoretical analysis, guiding it (Mahoney 2012).
According to Cohen (2010), apart from the essentially Aristotelian agenda and the Aristotelian logical rigor, Galileo adopted one more Aristotelian characteristic: the continual trading between theoretical analysis and experience.
In his Principia, Newton redefined mechanics from a science of the manual arts and simple machines to ‘the science, expressed with exact propositions and demonstrations, of the motions that result from any forces whatever and of the forces that are required from any motions whatever’ (Newton 1999, p. 382).
The parallelogram rule for adding forces.
Newton uses the ‘method of first and ultimate ratios’ (i.e. elements of calculus) to show that the rectilinear figures coincide with the curvilinear ones, and their rectilinear perimeters with curvilinear lines (Newton 1999, pp. 433–434).
In fact, Varignon determined ‘force accélératrice’—force per unit mass, numerically equivalent to acceleration—as y = dv/dt = dx/dt 2 (Varignon 1700), whereas Euler wrote Newton’s second law (or his ‘new principle of mechanics’) in the form: 2Md2 x = ∓Pdt 2, where P was the net force and the coefficient ‘2’ was balancing his units (Euler 1752).
According to Gingras, quantification, which is related to measurements of properties through graduated instruments, has to be distinguished from mathematization, which concerns the formulation of abstract geometric or algebraic expressions, such as the law of refraction of light or the second Newton’s law (Gingras 2001).
Carnot gave the meaning of ‘motive power’ in a footnote:
We use here the expression motive power to express the useful effect that a motor is capable of producing. This effect can always be likened to the elevation of a weight to a certain height. It has, as we know, as a measure, the product of the weight multiplied by the height to which it is raised (Carnot 1960, p. 5).
Caloric, according to the dominant theory of the era, was another name for heat, which was regarded as an imponderable and conserved fluid.
It is striking that in his Physics, Aristotle does not define ‘speed’ as an abstract concept, but only the swifter body in relation to the slower:
… if one thing is faster than another, it will cover a greater distance in an equal amount of time, and it will take less time to traverse an equal distance, and it will take less time to traverse a greater distance. Some people take these properties to define ‘faster’ (Aristotle 2008, 232a25–232a26).
Only a small subset of the system’s properties is represented. Giere (2006) compares physical models with maps. Maps and models, he points out, represent partly and selectively only certain elements of the physical system they are referring to. There are geophysical maps that depict geophysical data, political maps that depict political divisions, cities, highways, etc. Maps and models represent parts of the world approximately (absolutely exact maps do not exist), using agreed design conventions.
It is a simple geometric figure on which many conventional imagistic elements are incorporated. These are the arrows denoting the forces’ vectors, the shading denoting the unshakable support, etc.
Contrary to what the figure shows, the suspending ball is regarded as a point mass.
Redish (2005) and Redish and Kuo (2015) argue, convincingly enough, that physicists are blending mathematics and physics when they are handling the mathematics of physics, loading physical meaning onto mathematics. Hence, the series of operations on models might be more complex, in reality, than what is presented here. Moreover, this might add one more difficulty to students’ understanding of high school physics mathematics.
Usually, in high school experiments, a lot of effort is made to reduce the complexity of the physical system, e.g. by minimizing disturbing factors like friction. So, the physical systems are modified to resemble their models as close as possible.
This is a second denotation of the term ‘mathematization’. In the second (‘historical’) part of this paper, the term was used, in a broader way, to denote all procedures to transform physics from a qualitative, philosophical study of nature to a quantitative and exact description and examination of it via mathematics.
Of course, teaching practice has always the problem of the limited time along with the amount of information students are asked to handle. Consequently, teachers cannot analyze every implicit assumption every time the class solves a problem. On the other hand, by organizing around a concept (‘model’) all these simplifying procedures (explicit or implicit) that are always used when a theory is applied probably will save time and effort, and students might better understand how physics works. This, however, is a matter of empirical research and instructional practice, not just a matter of theoretical contemplation.
Context-based physics approaches (Taasoobshirazi and Carr 2008; Löffler and Kauertz 2014), which also work with real world situations and authentic problems, have much in common with modeling instruction projects. Although they usually do not introduce their models explicitly, their focus is shifted from teaching and applying theoretical statements (given in the form of physics formulas) to explain certain key phenomena—a shift from content to context. Stinner’s Large Context Problems are designed with the same tenets (see for example Stinner 2007a, b).
The term parametric equation denotes here an equation with parameters, as it is usually used in high school algebra textbooks.
In his 1685 A treatise of Algebra, John Wallis tried to explain Viète’s terminology. According to Wallis, the term ‘species’ operates in a similar way with the particular names the English lawyers use to indicate indefinitely any person in any situation, e.g. John-on-Oaks and John-a-Dawn. So, Viète’s ‘species’ and symbols represent indefinitely any number and any quantity (Wallis 1685; Klein 1998).
In fact, Klein’s views were rather overlooked for many years, until the historian of mathematics Sabetai Unguru brought them to the fore as an additional argument against the case of geometric algebra (Unguru 1975). His article initiated a controversy between new and traditional historians of mathematics (van der Waerden 1976; Weil 1978; Unguru 1979; Rowe 1996), which, apart from geometric algebra, spread itself into the ontological status of mathematical concepts and propositions, as well as their connection to the symbolic systems used to represent and handle them. For example, it was discussed whether mathematical concepts were ‘eternal, unchanging, unaffected by the idiosyncratic features of the culture in which they appear’ (Unguru 1979, p. 555) or the products of historical processes, constructed by certain mathematicians within certain cultures, and shaped by the symbolic systems invented to represent them. The details of this controversy, however, are beyond the scope of this paper, and they will not be examined. For a philosophical analysis of these matters see Stenlund (2015).
Zetetics is one of the components of Viète’s analytic art (i.e. algebra), and consists in finding the equations or other mathematical relations to solve a problem (Viète 1983).
These empirical researches, however, were selected because they are related to the former theoretical statements. They are not, and could not be, a full survey of educational research on these mathematical concepts.
There is, however, no sharp distinction between them. When school mathematics is applied to an authentic situation of the real world, mathematical symbols stand for entities and procedures of this worldly situation, and acquire concrete meaning too. But in school physics, symbols in equations stand for physical magnitudes and have physical meaning from the very beginning.
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Kanderakis, N. The Mathematics of High School Physics. Sci & Educ 25, 837–868 (2016). https://doi.org/10.1007/s11191-016-9851-5
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DOI: https://doi.org/10.1007/s11191-016-9851-5