Orientamento scolastico individualizzato
MATHEMATICS
GENERAL OUTLINES AND SKILLS
At the end of the scientific high school course, the student will know the elementary concepts and methods of mathematics, both internal to the discipline itself and relevant for the description and prediction of phenomena, in particular of the physical world. He will be able to frame the various mathematical theories studied in the historical context within which they developed and will understand their conceptual meaning.
The student will have acquired a historical-critical view of the relationships between the main themes of mathematical thought and the philosophical, scientific and technological context. In particular, he will have acquired the meaning and scope of the three main moments that characterize the formation of mathematical thought: mathematics in Greek civilization, the infinitesimal calculus that was born with the scientific revolution of the seventeenth century and that led to the mathematization of the physical world, the turning point that started from Enlightenment rationalism and that led to the formation of modern mathematics and to a new process of mathematization that involved new fields (technology, social, economic, biological sciences) and that has changed the face of scientific knowledge.
Hence the groups of concepts and methods that will be the focus of the study:
This articulation of themes and approaches will form the basis for establishing conceptual and methodological links and comparisons with other disciplines such as physics, natural and social sciences, philosophy and history.
At the end of the course, the student will have deepened the characteristic procedures of mathematical thinking (definitions, proofs, generalizations, formalizations), will know the basic methodologies for the construction of a mathematical model of a set of phenomena, will be able to apply what has been learned for the solution of problems, also using computer tools of geometric representation and calculation. These operational skills will be particularly accentuated in the scientific high school course, with particular regard to the knowledge of infinitesimal calculus and basic probabilistic methods.
The computer tools available today offer suitable contexts for representing and manipulating mathematical objects. The teaching of mathematics offers numerous opportunities to become familiar with these tools and to understand their methodological value. The path, when this proves appropriate, will encourage the use of these tools, also in view of their use for data processing in other scientific disciplines. The use of computer tools is an important resource that will be introduced in a critical way, without creating the illusion that it is an automatic means of solving problems and without compromising the necessary acquisition of mental calculation skills.
The wide range of content that will be addressed by the student will require the teacher to be aware of the need for a good use of the available time. Without prejudice to the importance of the acquisition of techniques, dispersion in repetitive technicalities or sterile cases that do not contribute significantly to the understanding of the problems will be avoided. The in-depth study of the technical aspects, although greater in the scientific high school than in other high schools, will never lose sight of the objective of an in-depth understanding of the conceptual aspects of the discipline. The main indication is: a few fundamental concepts and methods, acquired in depth.
SPECIFIC LEARNING OBJECTIVES
FIRST TWO-YEAR PERIOD
Arithmetic and algebra
The first two years will be dedicated to the transition from arithmetic to algebraic calculus. The student will develop his/her skills in calculation (mental, with pen and paper, using instruments) with integers, with rational numbers both in writing as a fraction and in decimal representation. In this context, the properties of operations will be studied. The study of the Euclidean algorithm for the determination of the GCD will allow to deepen the knowledge of the structure of integers and of an important example of algorithmic procedure. The student will acquire an intuitive knowledge of real numbers, with particular reference to their geometric representation on a straight line. The demonstration of the irrationality of and other numbers will be an important opportunity for conceptual deepening. The study of irrational numbers and the expressions in which they appear will provide a significant example of the application of algebraic calculus and an opportunity to address the issue of approximation. The acquisition of radical calculation methods will not be accompanied by excessive manipulative technicalities.
The student will learn the basic elements of literal calculus, the properties of polynomials and the operations between them. He will be able to factor simple polynomials, will be able to perform simple cases of division with remainder between two polynomials, and will deepen the analogy with the division between integers. Here, too, the acquisition of the ability to calculate will not involve excessive technicalities.
The student will acquire the ability to perform calculations with literal expressions both to represent a problem (by means of an equation, inequalities or systems) and to solve it, and to demonstrate general results, in particular in arithmetic.
He will study the concepts of vector, linear dependence and independence, scalar and vector product in plane and space, as well as the elements of matrix calculus. It will also deepen the understanding of the fundamental role that the concepts of vector and matrix algebra have in physics.
Geometry
The first two years will aim to learn the fundamentals of Euclidean plane geometry. The importance and meaning of the concepts of postulate, axiom, definition, theorem, proof will be clarified, with particular regard to the fact that, starting from Euclid's Elements, they have permeated the development of Western mathematics. In line with the way it has presented itself historically, the Euclidean approach will not be reduced to a purely axiomatic formulation. Particular attention will be paid to the Pythagorean theorem in order to understand both its geometrical aspects and its implications in number theory (introduction of irrational numbers), insisting above all on the conceptual aspects. The student will acquire the knowledge of the main geometric transformations (translations, rotations, symmetries, similarities with particular regard to Thales' theorem) and will be able to recognize the main invariant properties. It will also study the fundamental properties of circumference. The realization of elementary geometric constructions will be carried out both by traditional tools (in particular the ruler and compass, emphasizing the historical significance of this methodology in Euclidean geometry), and by means of computer programs of geometry.
The student will learn to make use of the Cartesian coordinate method, in a first phase limiting himself to the representation of points, lines and bundles of lines in the plane and properties such as parallelism and perpendicularity. The study of quadratic functions will be accompanied by the geometric representation of conics in the Cartesian plane. The intervention of algebra in the representation of geometric objects will not be separated from the deepening of the conceptual and technical scope of this branch of mathematics.
Circular functions and their elementary properties and relationships, theorems that allow the solution of triangles and their use in other disciplines, in particular in physics, will also be studied.
Relationships and functions
The aim of the study will be the language of sets and functions (domain, composition, inverse, etc.), also to construct simple representations of phenomena and as a first step to the introduction of the concept of mathematical model. In particular, the student will learn to describe a problem with an equation, an inequality or a system of equations or inequalities; to obtain information and derive the solutions of a mathematical model of phenomena, also in the context of operations research or decision theory.
The study of functions of the type f(x) = ax + b, f(x) = ax2 + bx + c and the representation of lines and parabolas in the Cartesian plane will allow to acquire the concepts of solution of equations of first and second degree in one unknown, of the associated inequalities and of systems of linear equations in two unknowns, as well as the techniques for their graphical and algebraic solution.
The student will study the functions f(x) = x, f(x) = a/x, piecewise linear functions, circular functions both in a strictly mathematical context and as a function of the representation and solution of application problems. You will learn the elements of the theory of direct and inverse proportionality. The contemporary study of physics will offer examples of functions that will be the subject of a specific mathematical treatment, and the results of this treatment will serve to deepen the understanding of physical phenomena and related theories.
The student will be able to easily switch from one representation register to another (numerical, graphic, functional), also using computer tools for data representation.
Data and predictions
The student will be able to represent and analyze a set of data in different ways (also using computer tools), choosing the most suitable representations. He/she will be able to distinguish between qualitative, discrete quantitative and continuous quantitative traits, work with frequency distributions and represent them. The definitions and properties of mean values and measures of variability will be studied, as well as the use of computational tools (calculator, spreadsheet) to analyze data collections and statistical series. The study will be carried out as much as possible in connection with other disciplines, even in areas where data are collected directly from students. The student will be able to derive simple inferences from statistical diagrams. He will learn the notion of probability, with examples taken from classical contexts and with the introduction of notions of statistics. The concept of mathematical model will be rigorously explored, distinguishing its conceptual and methodical specificity from the classical physics approach.
Elements of Computer Science
The student will become familiar with computer tools, in order to represent and manipulate mathematical objects and will study the methods of representation of elementary textual and multimedia data. A fundamental topic of study will be the concept of algorithm and the elaboration of algorithmic solution strategies in the case of simple and easy modeling problems; and, also, the concept of computable function and computability and some simple related examples.
SECOND TWO-YEAR PERIOD
Arithmetics and algebra
The study of the circumference and the circle, of the number p, and of contexts in which exponential growths appear with the number e, will allow us to deepen the knowledge of real numbers, with regard to the theme of transcendental numbers. On this occasion, the student will study the formalization of real numbers also as an introduction to the problem of mathematical infinity (and its connections with philosophical thought). The topic of approximate calculus will also be addressed, both from a theoretical point of view and through the use of calculation tools. The definition and calculus properties of complex numbers, in algebraic, geometrical and trigonometric form, will be studied.
Geometry
The conic sections will be studied from both a geometric, synthetic and analytical point of view. In addition, the student will deepen the understanding of the specificity of the two approaches (synthetic and analytical) to the study of geometry. It will study the properties of the circumference and the circle and the problem of determining the area of the circle, as well as the notion of geometric place, with some significant examples. The study of geometry will continue with the extension to space of some of the themes of plane geometry, also in order to develop geometric intuition. In particular, the reciprocal positions of lines and planes in space, parallelism and perpendicularity, as well as the properties of the main geometric solids (in particular polyhedra and rotational solids) will be studied.
Relations and functions
A topic of study will be the problem of the number of solutions of polynomial equations. The student will acquire the knowledge of simple examples of numerical sequences, also defined by recurrence, and will be able to deal with situations in which arithmetic and geometric progressions occur. It will deepen the study of the elementary functions of analysis and, in particular, of the exponential and logarithm functions. He/she will be able to build simple models of exponential growth or decrease, as well as periodic trends, also in relation to the study of other disciplines; All this in both a discrete and continuous context. Finally, the student will learn to analyze both graphically and analytically the main functions and will be able to operate on compound and inverse functions. An important topic of study will be the concept of rate of change of a process represented by a function.
Data and predictions
The student, in increasingly complex areas, whose study will be developed as much as possible in connection with other disciplines and in which data can be collected directly from students, will learn to make use of conditional and marginal double distributions, the concepts of standard deviation, dependence, correlation and regression, and sample. It will study conditional and compound probability, Bayes' formula and its applications, as well as the basic elements of combinatorial calculus. In relation to the new knowledge acquired, it will deepen the concept of mathematical model.
FIFTH YEAR
In the final year, the student will deepen the understanding of the axiomatic method and its conceptual and methodological usefulness also from the point of view of mathematical modeling. Examples will be taken from the context of arithmetic, Euclidean geometry or probability, but it is left to the teacher's choice of which subject area to privilege for the purpose.
Geometry
The introduction of Cartesian coordinates in space will allow the student to study lines, planes and spheres from an analytical point of view.
Relations and functions
The student will continue the study of the fundamental functions of analysis also through examples taken from physics or other disciplines. He will acquire the concept of limit of a sequence and a function and will learn how to calculate limits in simple cases. The student will acquire the main concepts of infinitesimal calculus - in particular continuity, derivability and integrability - also in relation to the problems in which they arose (instantaneous velocity in mechanics, tangent of a curve, calculation of areas and volumes). No special training in calculus techniques will be required, which will be limited to the ability to derive already known functions, simple products, quotients and compositions of functions, rational functions and the ability to integrate integer polynomial functions and other elementary functions, as well as to determine areas and volumes in simple cases. Another important topic of study will be the concept of differential equation, what is meant by its solutions and their main properties, as well as some important and significant examples of differential equations, with particular regard to Newton's equation of dynamics. Above all, it will be a question of understanding the role of calculus as a fundamental conceptual tool in the description and modeling of physical or other phenomena. In addition, the student will become familiar with the general idea of optimization and its applications in numerous fields.
Data and predictions
The student will learn the characteristics of some discrete and continuous probability distributions (such as the binomial distribution, the normal distribution, the Poisson distribution). In relation to the new knowledge acquired, also in the context of the relationship of mathematics with other disciplines, the student will deepen the concept of mathematical model and develop the ability to build and analyze examples.
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