Des rapports différents à la modélisation
Au Viêt Nam
L’interdisciplinarité, associée au rôle d’outil des mathématiques, est mentionnée explicitement dans le programme de mathématiques au secondaire :
Le programme est construit et développé selon les points de vue suivants :
Select essential and updated foundational mathematical knowledge that is systematic and tailored to the cognitive levels of students This knowledge should reflect interdisciplinary connections, incorporate didactic content, and highlight the role of mathematics as a valuable tool.
+ intensifier la pratique et l’application, enseigner les mathématiques reliées à la réalité
[…] (Extrait du programme du lycée, 2006)
Le premier objectif du programme est l’obtention des significations et des applications des connaissances mathématiques dans la vie et dans les autres disciplines
(Extrait du programme de la classe 11, 2006)
L’un des effets de cette injonction a été par exemple, d’introduire dans le programme de la classe
The two chapters on Combinations and Probability, along with Derivatives, are designed to equip students with essential mathematical tools needed for other disciplines such as biology and physics These concepts were previously taught in the 12th grade in earlier curricula.
In educational materials and classroom instruction, the integration of different subjects is often overlooked To illustrate this point, we spoke with Mr Nguyen Tran Trac, a physics teacher and author of physics textbooks, regarding the application of mathematical knowledge in solving physical problems Here is an excerpt from our interview.
+ pour les connaissances précédemment enseignées en mathématiques, l’enseignant de physique va les rappeler avant de les utiliser
In cases where mathematical concepts have not yet been taught, the physics teacher takes the initiative to instruct students For instance, when dealing with periodic functions in equations, the teacher guides students through the resolution process Understanding periodic functions is essential when studying the motion of a pendulum Therefore, the teacher first imparts the necessary mathematical knowledge before applying it to physical phenomena.
(Extrait de l’entretien avec M Nguyen Tran Trac)
The relationship between physics and mathematics education in Vietnam reveals that these subjects can be viewed as independent disciplines Physics teachers are tasked with imparting essential mathematical knowledge required for solving physical problems In this context, mathematical concepts are introduced in physics as ready-made tools to address various challenges in the field.
De plus, selon l’auteur de manuels avec lequel nous nous sommes entretenus, l’utilisation des problèmes de physique pour introduire des connaissances mathématiques n’est jamais envisagée
The physics teacher should directly present mathematical concepts to apply them in physics For instance, when studying the composition of two harmonic oscillations, the teacher introduces the topic to students through either algebraic methods or Fresnel's approach Students then utilize this knowledge to solve related exercises effectively.
The relationship between mathematics and physics is one-directional, with mathematics serving solely as a tool to be applied in specific cases determined by physicists From this perspective, the gap between mathematics and physics curricula may seem inconsequential However, this discrepancy often leads to confusion among students.
The issue of teaching modeling in Vietnam is not adequately addressed in the construction of curricula and textbooks While there are some instances of modeling found in the application of mathematical knowledge to real-world problems, these exercises are infrequently included in secondary math textbooks Often, they appear in supplementary sections or at the beginning of chapters, primarily serving to introduce new concepts According to the Kahane Commission report (2000), such limited exposure to modeling exercises can negatively impact the perception of mathematics and other disciplines.
This type of exercise serves merely as a pretext for engaging in mathematics without regard for relevance Such exercises are detrimental to the image of our discipline, as they present mathematics as a field where data is used without understanding its origin, calculations are performed without grasping their significance, and the meaningfulness of results is overlooked.
En France
Like many other countries, there is an institutional commitment to incorporate modeling into the teaching of mathematics and other scientific disciplines For instance, the orientation and program law for the future of education (April 23, 2005) emphasizes the necessity of a scientific approach as a required skill for students within the framework of scientific and technological culture This approach is described as follows:
- savoir observer, questionner, formuler une hypothèse et la valider, argumenter, modéliser de faỗon ộlộmentaire ;
- comprendre le lien entre les phénomènes de la nature et le langage mathématique qui s'y applique et aide à les décrire
L’institution franỗaise d’enseignement secondaire prộconise la mise en place d’un enseignement de la modélisation mathématique pour les raisons suivantes :
Mathematics is a demanding discipline that fosters critical thinking among students, requiring them to present compelling arguments and appreciate the certainties these arguments provide Additionally, students can explore the modeling capabilities of mathematics, while recognizing that models, surveys, and equations are not reality and must continually be tested against it.
(Document d’accompagnement des programmes des classes Terminale S, ES, p 5)
Toujours dans ce document d’accompagnement des programmes, on découvre une partie ô Mathộmatique et modộlisation ằ On trouve là des indices concernant l’enseignement de la modélisation au lycée :
In high school, students will be introduced to modeling through the study of simplified real-world situations This intentionally extreme simplification allows for the creation of a coarse model that can provide insights or predictions The challenge lies in maintaining meaning and consistency within the simplified problem.
In France, while modeling education is included in secondary school curricula, its actual implementation within the education system remains lacking The Kahane Commission report (2000) identifies several reasons for this discrepancy.
1) Les manuels n’ont pas encore pris la mesure de ces problèmes
Les exercices qu'on y trouve pèchent souvent de deux côtés :
- On ne discute pas l'origine des formules fournies (c'est souvent le cas pour les exercices issus de l'économie)
The model leads to an absurdity that goes unaddressed in the text, illustrating a population that follows an exponentially increasing law without any discussion of the implications as time progresses.
2) La rộalitộ est plus complexe qu’il n’y paraợt
Teachers often face the challenge that mathematically accessible models are overly simplistic and fail to accurately represent reality.
3) Les programmes sont mal adaptés
One notable example is the harmonic oscillator, a fundamental concept in high school physics, which relates to second-order linear differential equations with constant coefficients However, these equations are no longer included in the high school mathematics curriculum.
C’est ce genre de décalage, très fréquent, entre les programmes de mathématiques et de physique qui est souvent, pour les élèves, une cause d’incompréhension
Physicists employ mathematics to describe phenomena, as physical laws are expressed through mathematical formulas However, translating these concepts into the appropriate mathematical language can require adaptation and effort, varying in difficulty depending on the educational level, time period, and curriculum.
(Rapport de la commission Kahane (2000), p 38-39)
There is a significant issue in education regarding the teaching of modeling; either it is neglected by viewing the relationship between mathematics and other scientific disciplines as purely applicative (as seen in Vietnam), or it is recommended without providing teachers with the necessary resources to effectively teach it (as in France).
Quels effets ont ces conditions et ces contraintes différentes sur la vie dans l’enseignement secondaire de savoirs mathématiques issus d’un processus de modélisation ?
Nous décidons de travailler cette question en choisissant la périodicité et les fonctions périodiques comme savoirs mathématiques issus d’un processus de modélisation.
Le choix de la périodicité et des fonctions périodiques
Le phénomène (système) étudié dans un processus de modélisation mathématique peut être intra ou extra-mathộmatique et le terme ô modộlisation mathộmatique ằ ộvoque le plus souvent ce que
Chevallard (1989) qualifie d’emplois extra-mathématiques :
Mathematics is utilized in two primary contexts: first, in the study of mathematical objects, and second, in the mathematical examination of non-mathematical systems such as physical, biological, and social systems The latter is typically referred to as mathematical modeling.
The emphasis placed by secondary education institutions on understanding the connection between natural phenomena and the mathematical language that describes them encourages the study of non-mathematical phenomena, particularly periodic phenomena explored across various scientific disciplines.
Periodicity is a fundamental concept in modeling cyclical and oscillatory phenomena Historically, periodic functions, particularly trigonometric functions, have evolved as models for variable quantities that consistently and indefinitely return to the same state, typically in relation to time.
Functions play a crucial role as modeling tools, enabling the representation of variable or evolving phenomena that are prevalent in daily life and are subjects of study across various scientific disciplines such as finance, biology, physics, and economics This article will focus on functional modeling.
According to Krysinska and Schneider (2010), the functional modeling process in mathematics education primarily involves constructing or identifying a functional model through a parameterized formula, utilizing associated representations such as numerical tables, Cartesian graphs, or functional equations, and adapting this model to the specific characteristics of the phenomenon being studied.
La réforme de 1902 a introduit la notion de fonction dans l’enseignement des mathématiques en
France en arguant de sa nécessité pour l’enseignement de la physique et donc en se référant à des problèmes extra-mathématiques
D’un autre côté, le nouvel enseignement de la physique doit abandonner les méthodes descriptives et dogmatiques qui réduisaient l’expérimentation à la démonstration d’appareils
Experimental physics aims to derive laws from facts, which must be expressible mathematically Graphical representation reveals functional relationships between physical variables and constants based on experimental measurements The professor emphasizes the use of graphs to illustrate phenomena and instill crucial concepts of function and continuity in students' minds Consequently, the concept of function is naturally integrated into physics education, prompting the commission to introduce it into mathematics teaching as well This marks a significant innovation in the second cycle curriculum, where the study of functions in mathematics retains its practical, quasi-experimental roots, focusing on constructing graphs of simple functions using graph paper rather than relying on abstract definitions.
The concept of function serves as a crucial response to the challenges of modeling phenomena in physics However, students encounter several epistemological and didactic difficulties in functional modeling, as identified by Krysinska and Schneider (2010).
- percevoir la variation d’une grandeur dès que celle-ci est liée à une autre grandeur constante
- percevoir comme fonctions ou mờme grandeurs ô variables ằ des grandeurs qui ne varient pas dans le temps
- identifier et choisir une variable indépendante et exprimer la grandeur à optimiser en fonction de cette variable et d’elle seule (dans le domaine des problèmes d’optimisation)
- comprendre la signification de la notion de variable
The periodic function, both as a mathematical concept and a modeling tool, exhibits the general characteristics of numerical functions while also possessing unique features such as period, phase, and amplitude These distinct attributes contribute to the complexity of teaching and learning about periodic functions.
Questionnement initial et méthodologie de recherche
Enquête épistémologique
We begin our study with an epistemological investigation to analyze periodic phenomena in physical sciences and identify the mathematical models representing these phenomena This inquiry enables us to formulate hypotheses regarding the role and significance of various mathematical formalizations utilized in models of periodic phenomena in physics.
Les résultats de cette enquête seront les éléments de référence pour notre analyse institutionnelle
This study provides a synthesis of modeling processes in physics from both epistemological and didactic perspectives, supporting our selection of the specific modeling process utilized in our experimental setup.
Etude institutionnelle
Pour trouver des éléments de réponse aux questions initiales, nous utilisons le cadre théorique de la Transposition Didactique (Chevallard, 1991) et de la Théorie Anthropologique du Didactique
Didactic transposition highlights the challenge of legitimizing the knowledge being taught and examines the gap between the knowledge imparted and the references that validate it This gap arises from the constraints affecting the operation of the education system.
La Théorie Anthropologique du Didactique montre l’importance du rapport institutionnel à un objet de savoir et introduit la notion de praxéologie pour le caractériser :
What is lacking is the development of a method for analyzing institutional practices that allows for the description and study of the conditions under which they are realized Recent advancements in theorization address this gap A key concept that emerges is that of praxeological organization or praxeology.
Nous utilisons cette notion de praxéologie comme outil d’une analyse institutionnelle comparative entre les institutions d’enseignement des mathématiques et de la physique en
France et au Viêt Nam
By identifying the various types of tasks that implement periodicity in mathematics education and the associated praxeologies, this analysis reveals the institutional relationships between the two countries regarding periodicity, periodic functions, and the mathematical modeling of periodic phenomena It also helps us understand the reasons behind institutional choices, the constraints influencing these decisions, and the potential for modifications.
Cette analyse débouche sur des questions didactiques, relatives à la pratique de la démarche de modélisation chez l’élève, qui nous amènent à proposer un questionnaire aux élèves vietnamiens.
Questionnaire élèves
The questionnaire is designed around emblematic tasks identified in the institutional analysis of education in both countries A variation in didactic variable values may lead to a disruption of institutional contracts related to the modeling of periodicity specific to each institution.
The questionnaire is being tested in various high school classes in Vietnam Analyzing the students' responses allows us to deepen the institutional study by clarifying the relationship between students' roles and the concepts of periodicity and periodic functions, while also highlighting their challenges in engaging with the modeling process.
Les résultats obtenus nous fournissent ainsi des éléments complémentaires pour la conception d’une ingénierie didactique.
Ingénierie didactique
In our ongoing work, we aim to establish conditions, within institutional constraints, that enable students to engage in the modeling of periodic phenomena.
Pour cela, nous élaborons une ingénierie didactique pour introduire les fonctions périodiques comme modèles de phénomènes de co-variations périodiques
Pour ce faire, nous nous référons à la méthodologie d’ingénierie didactique ainsi décrite par Artigue (1989) :
Didactic engineering, regarded as a research methodology, is primarily defined by an experimental framework centered on didactic implementations in the classroom This involves the design, execution, observation, and analysis of teaching sequences.
The didactic engineering methodology is distinguished from other research types focused on classroom experiments by its specific framework and associated validation methods (Artigue, 1989, p 285-286).
La conception de cette ingénierie s’appuie sur les résultats de l’analyse épistémologique et de l’analyse institutionnelle comparative des systèmes d’enseignements :
The implementation and analysis of didactic engineering within one system not only raise questions about that specific system but also about another, facilitating the emergence of generic inquiries relevant to both Furthermore, this process highlights the broader issue of the reproducibility of engineering practices, applicable not only within the institution where they were developed but also in the other system.
Organigramme de la thèse
Questions initiales sur l’enseignement de la modélisation et de la périodicité
Chapitre1 Enquête épistémologique sur la modélisation mathématique de phénomènes périodiques temporels
Deux modèles mathématiques pour les phénomènes périodiques temporels (C et O)
Qu’est-ce qu’un processus de modélisation mathématique ?
Modélisation de phénomènes périodiques dans les enseignements mathématique et physique secondaires
Praxéologies concernant la périodicité et la modélisation Les modèles C et O dans les institutions
Faiblesse de l’articulation entre les deux modèles C et O Difficultés des élèves pour entrer dans un processus de modélisation
Nouvelles questions sur la praxis scolaire de la modélisation mathématique de la périodicité et hypothèses pour l’ingénierie didactique
Conception, expérimentation et analyse d’un processus de modélisation par des fonctions périodiques cherchant à articuler les deux modèles C et O
Conclusions et perspectives de recherche
Questions sur la praxis scolaire de la modélisation mathématique de la périodicité et sur l’articulation entre les deux modèles C et O
Enquête épistémologique sur la modélisation mathématique de phénomènes périodique temporels
Introduction
Sensevy & Mercier (1999) ộnoncent deux ô raisons d’ờtre ằ des mathộmatiques :
- faire des mathộmatiques, c’est fabriquer des modốles qui permettront de maợtriser des phénomènes dans la réalité ;
Mathematics involves utilizing tools to solve problems, as well as creating tools designed for problem-solving Additionally, it encompasses the study of these tools and the specific problems they can address.
The two fundamental reasons for the existence of mathematics illustrate its crucial role across all scientific disciplines Mathematics excels at creating models that represent and analyze the phenomena studied by various sciences, establishing strong connections with them This relationship is especially evident in physics, as noted by Klein (2000).
Mathematics undeniably provides physics with a unique pathway to unification, highlighting the universally acknowledged relationship between these two disciplines The foundational concepts of classical mechanics rely on differential and integral calculus, while electromagnetism is grounded in partial differential equations Furthermore, general relativity is built upon tensor calculus, and quantum physics utilizes vector spaces, illustrating the essential role of mathematics in the advancement of physical theories.
Selon lui, il y a trois types d’efficacité des mathématiques en physique liés à la modélisation :
Mathematics can predict experimental outcomes and replicate previously obtained data It offers numerical results that, within a certain acceptable margin of error, align with experimental findings.
- elles mettent en ộvidence des structures ô explicatives ằ ;
- elles permettent d’engendrer de nouvelles idées, de nouveaux concepts, des stratégies inédites ou des solutions originales à des problèmes anciens
Expressing and utilizing a physical property in digital form requires not only the use of mathematics to indicate the relationships between various measured quantities but also the ability to apply these relationships effectively According to Gruber and Benoit (1998), mathematics serves as both the language of physics and a fundamental way of thinking for physicists.
- c’est le langage de la nature que l’homme doit assimiler ;
- c’est le langage de l’homme pour traduire les faits de la nature
C’est cette deuxième interprétation, reliée du reste à la description moderne, qu’adopte
Heisenberg quand il affirme : ô Les formules mathộmatiques ne reprộsentent plus la Rộalitộ, mais la connaissance que nous possộdons ằ
Finalement, on remarquera que les mathématiques sont beaucoup plus qu’un langage : c’est la manière de penser du physicien (Gruber et Benoit (1998), p 8)
A son tour, la physique avec la résolution mathématique de ses problèmes conduit à développer des connaissances nouvelles et importantes en mathématiques
Through experimentation or theoretical analysis, physicists seek to establish mathematical relationships, known as laws, between physical quantities Mathematics serves as the natural language of physics, allowing these relationships to be expressed concisely Once a mathematical statement is established, it can be manipulated according to mathematical rules If the initial equations of an analysis are correct, mathematical logic can lead to new ideas and laws.
The strong epistemological connections between physical sciences and mathematics lead us to focus our investigation on the modeling of periodic phenomena within physics, despite other fields like life sciences and economics also examining what they define as periodic phenomena Our first question will be: what are the phenomena that physics studies and considers periodic?
Periodic phenomena are found across various branches of physics, including mechanics, electromagnetism, waves, and crystallography Among these, three significant types are commonly discussed in physics literature: circular motion, exemplified by the Earth's rotation around its axis and its orbit around the sun; oscillatory motion, as seen in a simple pendulum; and wave motion, illustrated by the vibrations of a string.
In physics, phenomena occur within a framework known as spacetime, where space is viewed as a three-dimensional continuum requiring three coordinates to pinpoint a location Time is treated as a linear continuum, directed by the arrow of time, which differentiates the future from the past Both length and time are fundamental concepts in this context.
In our investigation of motion, we will explore how progressive waves differ from circular and oscillatory motion by exhibiting dual periodicity in both time and space Specifically, at a given moment in time at a fixed point in space, waves oscillate in space over time.
Israel (1996) utilise le terme ô mathộmatique du temps ằ pour parler de l’outil mathộmatique de modélisation de phénomènes temporels :
The term "dynamic mathematics" or "mathematics of time" refers to the concept of mathematics as a tool primarily used to describe the evolution of phenomena over time, highlighting the changes that occur Technically, this implies that all variables involved depend on a parameter that specifically represents time.
Before delving into the specifics of the three movements mentioned in the physics treaties, it is essential to highlight the significance of time in physics and its connection to the concept of periodicity.
Le temps et la périodicité
II.1 Le temps en physique
Le temps apparaợt dans beaucoup de phộnomốnes ộtudiộs par la physique comme la cause du changement, de la variation des états Situation paradoxale pour le physicien, comme le souligne
Scientists across various disciplines grapple with the concept of time, which may seem paradoxical in the realm of physics While physics often aims to eliminate the notion of time, associating it with variability and instability, it simultaneously seeks to uncover relationships that remain unchanged Even when studying processes with histories or evolutions, physics focuses on identifying substances, forms, or laws that exist independently of time In its quest for a nearly divine perspective on nature, physics aspires to the immutable and invariant, yet it continually confronts the challenges posed by time.
What is time, and how is it represented? The periodic movement of clock hands around the circular face allows for the measurement of time Additionally, there is a connection between time and temporal periodic phenomena, highlighting the intricate relationship between our understanding of time and its manifestations in nature.
According to Feynman (1963), the term "time" was defined in the Webster dictionary as a "period," which is further described as "a time," leading to a circular definition that lacks clarity To gain a better understanding, let us explore his conception of time.
Perhaps it is best to accept that time is one of those concepts we may never fully define, as the dictionary suggests Instead, we can acknowledge what we already understand about it: the extent of our waiting.
La même idée est présente chez Klein (Ibid.) :
Time can be defined as what passes when nothing happens, the force behind all actions and changes, the sequence of events, or the ongoing process of becoming More playfully, it is the most convenient means nature has devised to prevent everything from occurring at once However, all these expressions already presuppose the concept of time and serve merely as metaphors, failing to capture its true essence.
1 Le Dictionnaire Webster est le nom donné à un type de dictionnaire de langue anglaise aux Etats-Unis et faisant autorité concernant l’anglais américain
When using an expression to define time, it inherently implies the concept of time itself However, physicists unanimously agree on the necessity of mathematically formalizing time.
Time is a mathematical tool that enables the equation of observable physical phenomena, providing valuable information This tool exists because there are beings capable of observing and measuring nature and its changes, as well as matter and movement, which are essential for transformations (Isoz, 2011)
Donc le problème qui doit nous préoccuper ici n’est pas de trouver une définition du temps mais de savoir comment les physiciens le représentent et comment ils le mesurent
II.2 La représentation du temps
Time is represented in physics as a real number, denoted by the parameter τ, which has a single dimension; one number is sufficient to specify a date This representation allows for a defined direction of flow, indicating that time is orientable Such a conceptualization of time implies that there exists only one time at any given moment and that this time is continuous.
Unlike the topology of space, the topology of time is quite limited, offering only two variations: the line and the circle These represent linear time, which progresses forward, and cyclical time, which loops back on itself While the cyclical concept has been favored in myths due to the enchanting nature of the circle, it is now largely abandoned in physics as it does not adhere to the principle of causality.
Time is traditionally represented as a linear progression, contrasting with its circular representation, and is depicted as a directed line consisting of infinitesimal moments known as the real number line R In physics equations, time functions as an independent real variable, typically illustrated on the horizontal axis, often referred to as the x-axis.
Time, represented mathematically as a real line, has an ordered structure where each point is necessarily positioned before or after another point.
The passage of time is represented by a timeline, traditionally marked by a small arrow that does not conform to the usual concept of a "time arrow." This arrow signifies that, although the direction of time's flow may be arbitrary, it can be defined Additionally, it emphasizes that time travel is impossible; on the timeline, one cannot go back or revisit the same moment.
This mathematical representation of time suggests that time is what organizes and orders the continuity of moments, thereby creating duration without necessarily implying change Rather than generating phenomenal novelty, time produces moments that are fundamentally original, even if they are receptive to repetitive phenomena.
This mathematical representation of time raises the question of distinguishing between the past and the future Does the mathematical concept of time differ from that of physics in this regard? This is the perspective presented by Israel (1996).
The mathematical parameter t is conventionally defined with the present at t = 0, the past as negative values t < 0, and the future as positive values t > 0 This definition significantly diverges from common intuitions about time, where the past and future are perceived as asymmetrical, making time inversion unimaginable In mathematical time, however, it is theoretically possible to reverse time, a concept that seems inconceivable in reality.
Des phénomènes périodiques temporels
To identify and analyze periodic temporal phenomena in physics, we select, in addition to Feynman's physics course, two other works that also serve as comprehensive treatises on general physics, providing a unified perspective on the subject.
Richard Feynman's "Lectures on Physics," originally delivered between 1961 and 1963, are a celebrated collection of course notes compiled by Robert B Leighton and Matthew Sands These lectures aim to provide a logical understanding of physics to undergraduate students, covering essential topics such as mechanics, electromagnetism, and quantum mechanics The lectures are known for their clarity and engaging style, making them a valuable resource for both students and researchers The French edition, translated by G Delacote in 1979, is divided into five volumes, reflecting the structure of the original American edition This work continues to be highly regarded and is often recommended for physics students at various levels.
In a 1905 article titled "On the Measurement of Time," published in "The Value of Science," Henri Poincaré, as cited by Barreau (1996), argued that the postulate regarding the measurement of time, which underpins all of physics, extends beyond the basic assertion that "the duration of two identical phenomena is the same." He emphasized that it also encompasses the idea that identical causes take the same amount of time to produce identical effects.
- ô Physique 4 ằ, Harris Benson, Adaptation de Marc Sộguin, Benoợt Villeneuve, Bernard
Marcheterre, Richard Gagnon, De Boeck, 2009, que nous noterons dans la suite [P1] ;
- ô Physique gộnộrale 5 ằ, Marcelo Alonso, Edward J Finn, Texte franỗais de Michel
Daune, Dunod, Paris, 2001, que nous noterons dans la suite [P2]
These works contain the fundamental principles of physical phenomena, their consequences, and limitations, highlighting the core ideas that form the foundation of contemporary physics This knowledge is essential for a variety of users.
Future physicists and engineers are not the only ones who need to grasp these fundamental concepts; anyone considering a scientific career, including students specializing in biology, chemistry, and mathematics, must achieve the same level of understanding.
The article discusses the translated edition of "University Physics" by Harris Benson, published globally with the permission of John Wiley & Sons, Inc in 1991 This textbook serves as an introductory resource for natural science students, aiming to provide a clear and accurate representation of fundamental physics concepts and principles.
L’édition originale de [P2] a été publiée au Etats-Unis par Addison-Wesley Publishing
The book "Fundamental University Physics, Second Edition" published by Addison-Wesley Publishing Company in 1967, is aimed at students pursuing scientific studies or engineering training According to the translators' preface, it presents physics as a holistic discipline rather than a mere collection of doctrines, rules, and laws Instead, it emphasizes physics as a mindset and a way of thinking about the world around us.
Dans le déroulement de notre étude, nous utiliserons également d’autres manuels de l’enseignement supérieur, plus spécialisés ou d’ambition plus restreinte que les trois traités
Dans les ouvrages [P1] et [P2], la définition d’un mouvement périodique s’appuie sur sa répétition régulière en fonction du temps :
Un mouvement périodique est un mouvement qui se répète à intervalles réguliers
Ou comme dans [P2], sa définition s’appuie sur une représentation graphique de fonction
Cependant, la répétition est toujours mise en avant (ici, la répétition d’un morceau du graphique)
A periodic motion with a period P is represented by the function x = f(t), where f(t) is a periodic function satisfying the condition f(t) = f(t + kP) As illustrated in Fig 12-35, the curve of f(t) repeats itself identically at regular intervals of P.
4 Tome I : Mécanique, Tome II : Electricité et magnétisme, Tome III : Ondes, optique et physique moderne
5 Tome I : Mécanique et thermodynamique, Tome II : Champs et ondes, Tome III : Physique quantique et physique statistique
The definitions presented in [P1] and [P2] at the beginning of the chapter on oscillatory motion highlight that oscillatory motion is a key subject for introducing periodic movements.
III.1 Les mouvements oscillatoires et le modèle O
III.1.1 Les fonctions périodiques en mathématiques
Some higher education textbooks focus on presenting mathematical methods for physical sciences, while many physics textbooks include sections summarizing the mathematical tools deemed essential for solving physical problems.
Drawing on various academic works, we aim to illuminate the concept of habitat and the significance of periodic functions in physics A periodic function is defined by the equality f(x + T) = f(x) for all real numbers x, as described by Schwartz (1965).
On appelle période d’une fonction f(x) tout nombre réel T tel que f(x + T) = f(x) Le nombre
Zero is always a period; the opposite of a period of f and the sum of two periods of f are also periods of f Therefore, the periods of f constitute a subgroup of the additive group R of real numbers, known as the group of periods of f.
Schwartz (1965) demonstrates that the periods of a function form a subgroup of R If f is continuous, its period group is a closed subgroup of R, as any limit T of a sequence of periods Tj of f remains a period of f The concept of the fundamental period is then introduced.
Or il n’y a que trois catégories de sous-groupes fermés de R :
- Le sous-groupe réduit à 0 Une fonction f n’ayant pas de période T ≠0 est dite non- périodique
- Le groupe R tout entier Une fonction ayant tout nombre réel T comme période est constante
- L’ensemble des multiples ℓT0 (ℓ entier > 0, < 0, ou = 0), d’un nombre T0 > 0
Si le groupe des périodes de f est de l’une des deux dernières catégories, f est dite périodique ; dans le troisième cas, T0 est appelée la période fondamentale de f
De plus, la notion de période peut être généralisée pour les fonctions sur R n (fonction de n variables) C’est un vecteur TJG tel que f(xG
Periodic functions are defined by the regular repetition of values at equal intervals known as the period Trigonometric functions are typically the first periodic functions introduced in secondary school mathematics Specifically, the most commonly studied periodic functions are sinusoidal functions.
III.1.2 Les fonctions sinusọdales en mathématiques
Dans le document de l’encyclopédie scientifique en ligne 6 , les rơles des fonctions sinusọdales sont soulignés comme suit :
Sine and cosine functions are essential in describing simple harmonic motion, a key concept in physics They are used to represent the one-dimensional projections of uniform circular motion, the movement of a mass on a spring, or as an approximation for the small-angle oscillations of a pendulum.
Modélisation mathématique de phénomènes physiques
IV.1 Ce qu’en disent les physiciens
Différentes références sur lesquelles nous nous appuyons sont citées par Smyrnaiou (2003) :
A crucial aspect of scientific activity involves the use of models, their modification, validation, and even the creation of new ones Consequently, many authors agree that modeling should be central to scientific education.
1990 ; Martinand, 1992, 1994 ; Lemeignan & Weil-Barais, 1993 ; Bliss, 1994; Kurtz dos
Santos & Ogborn, 1994 ; Mellar et al., 1994 ; Teodoro, 1994 ; Tiberghien, 1994 ; Hestenes,
1996 ; Jackson, Stratford, Krajcik, et Soloway, 1996; Dimitracopoulou et al., 1999 ; Komis et al., 1998 ; Gobet, 2000)
According to Von Neumann, as referenced by Smyrnaiou, science should focus on constructing models rather than attempting to explain or interpret phenomena He emphasizes that the essence of a model lies in its mathematical construction.
Sciences focus on creating mathematical models rather than explaining or interpreting phenomena These models, accompanied by verbal interpretations, aim to describe observed events Their validity is based solely on their expected functionality.
(John von Neumann, 1947, p 180-196, cité par Smyrnaiou (2003), p 17-18)
La modélisation mathématique est l’art (ou la science, selon le point de vue) de représenter
(ou de transformer) une réalité physique en des modèles abstraits accessibles à l’analyse et au calcul (Allaire (2005), p 1)
Henry (2001) évoque les registres de représentation pour caractériser les modèles mathématiques :
Ce modèle peut être représenté dans différents systèmes de signes : images, schémas, langages ou symbolismes, s’inscrivant dans différents registres de représentations, plus ou moins isomorphes
Mathematical language and symbolism serve as powerful tools for representation, enabling effective descriptions that can leverage general properties and algorithms We will refer to these as mathematical models.
The mathematical model illustrates physical properties and relationships through equations and graphs For instance, the quantitative model operates on measurable quantities, with the relationships governing these quantities expressed through algebraic formulas.
C’est ce que dit Israel (1996) en parlant du langage mathématique :
To translate a phenomenon into mathematical language, it is essential to identify one or more variables that describe the state of the phenomenon at a specific moment, particularly when dealing with dynamic phenomena that evolve over time.
Mathematics plays a crucial role in formulating fundamental laws and predicting new phenomena, as well as calculating their evolution Beyond mere description, it enhances our understanding of reality Cohen-Tannoudji (2002) emphasizes the importance of mastering the temporal evolution of a physical system.
In the realm of complexity, modeling serves to simplify intricate realities that cannot be fully understood due to their numerous degrees of freedom This is achieved through the use of a tangible or mathematical model, or through computer simulations, which allow for predictable or calculable evolution Anticipation, akin to extrapolation in spatial reasoning, relies on modeling as its foundational approach.
While not all physicists confine modeling to its mathematical forms—often citing analogical models—there is a consensus that modeling requires a cognitive process that connects two realms of knowledge: one pertaining to theory (and its mathematical models, if applicable) and the other to the real world.
IV.2 Le processus de modélisation de phénomènes physiques
Quel est le processus de modélisation en science physique ? Comment construit-on un modèle pour un phénomène physique ?
In many cases, a predefined model is not available, necessitating the selection of a relevant model from multiple options or the simplification of known models that are overly complex.
Cette idée est présente dans la définition suivante du modèle :
A model is an abstract construct designed to selectively and simplistically represent specific aspects of a technical object or a real process, which, due to its immense complexity, cannot be fully grasped in its entirety.
(Union des professeurs de sciences et techniques industrielles (1993), p 10)
In the modeling process, the goal is to create a simplified representation of reality that retains key characteristics while disregarding others based on the questions posed about that reality Furthermore, validating a model—assessing how closely theoretical results align with corresponding experimental values to select the most suitable model for the situation at hand—requires specialized knowledge of the phenomena being studied and involves iterative experimental adjustments.
When modeling, specific aspects of reality are selectively retained while others are arbitrarily disregarded The outcomes derived from a model are inherently linked to these choices, making it crucial to experimentally validate the model afterward There are instances where a model may prove unsuitable for its intended purpose, necessitating its reconstruction based on different assumptions.
Voici un exemple d’un modèle qui est valide pour certaines questions mais non valide pour d’autres :
The hypothesis of the rigid body, commonly employed in mechanics, assumes the neglect of object deformability This model is effective for analyzing the motion of rigid components subjected to moderate forces, but it fails to predict vibrational phenomena.
S’appuyant sur les points de vue développés par des didacticiens de la physique à partir de 1994 (Tiberghien, 1994 ; Tiberghien & Magalakaki, 1995 ; Tiberghien, 1996 ; Tiberghien & de Vries,
1997, Lacroix, 1996 ; Collet, 1996 ; Pateyron, 1997 ; Bécu-Robinault, 1997 ; Quintana-Robles,
1997 ; Guillaud 1998), Buty (2000) décrit le processus de modélisation en physique par quatre ensembles de choix :
- Il faut d’abord choisir la théorie applicable au champ des phénomènes étudiés […]
The phenomenon we are addressing is inherently complex and multifaceted; its modeling requires identifying relevant elements while disregarding others In this regard, the physicist constructs their experimental field.
Conclusion
We have highlighted the significance of time in the study of physical science and the interconnection between the concepts of time and periodicity through time measurement Sinusoidal functions play a crucial role in modeling temporal periodic phenomena According to physicists, these sinusoidal functions are the primary mathematical tools for modeling, defined through models C and O.
- Modèle O (oscillations harmoniques) avec deux registres (algébrique : x = A cos (ωt+φ) ou
- Modèle C (mouvements circulaires uniformes) caractérisé par une trajectoire circulaire et une vitesse constante avec deux registres (graphique : cercle ; algébrique : x = R cos θ, y
Nous allons maintenant enquêter sur les institutions d’enseignement des mathématiques et de la physique en France et au Viêt Nam avec les questions suivantes :
- Quels sont les habitats de la périodicité dans les programmes et les manuels scolaires en
France et au Viêt Nam ?
- Comment la périodicité y est-elle présentée ? Avec quelles significations ?
- Comment la modélisation des phénomènes périodiques est-elle enseignée en mathématiques et en physique ?
- Quelles places les fonctions sinusọdales occupent-elles dans l’étude de la périodicité et de phénomènes périodiques ?
Do the C and O models appear in textbooks, and if so, how are they presented? What is the relationship between these models? To what extent do the C and O models contribute to the mathematical modeling of periodic phenomena? Are they included among the intermediate models highlighted in our modeling process diagram?
Pour trouver des ộlộments de rộponses à ces questions, l’analyse ô institutionnelle ằ que nous mốnerons sera comparative des deux institutions d’enseignement, franỗaise et vietnamienne
Cette analyse est présentée dans le prochain chapitre.