Italy: Description of the Advanced Mathematics Programs and Curriculum
Students assessed in TIMSS Advanced 2015 have been taught according to the 2010 National Guidelines for upper-secondary schools (Licei, Istituti Tecnici, Istituti Professionali). Only students of Liceo Scientifico (high schools specializing in science education) and of Istituto Tecnico–Settore Tecnologico (technical high school–technology sector) participated in the TIMSS Advanced 2015 advanced mathematics assessment. In fact, these are the only tracks with elements of advanced mathematics in their curricula, with an appropriate lesson time (4 hours per week, for 33 weeks in a school year).
The mathematics curriculum for upper-secondary school includes four main areas:
- Arithmetic and Algebra
- Geometry
- Relations and Functions
- Statistics and Probability
Liceo Scientifico focus on the study of the link between the scientific and the humanistic traditions. They promote the acquisition of knowledge and methods of mathematics, physics, and the natural sciences. At the conclusion of the program of study, students should be able to:
- Understand the formal language of mathematics, know how to use typical mathematical procedures, and know the basic content of the theories underling the mathematical description of reality
- Use data processing tools critically and in-depth; understand the methodological value of information technology in formalizing and modelling complex processes; and identify procedures that lead to conclusions and judgments about real-world systems modelled by data
- Understand the fundamental structures of mathematical argumentation and demonstrate mathematical processes through the mastery of the language of formal logic; use mathematical argumentation and formal logic to identify and to solve problems of various kinds
- Know how to use computational and representation tools for modelling and solving problems
At the end of the course, students of Liceo Scientifico must know the basic concepts and methods of mathematics and apply them to describe and predict phenomena in the physical world. They can situate mathematical theories in historical context and understand their conceptual meaning. The Liceo Scientifico five-year curriculum is divided into three parts by grade. The topics taught at each grade are listed below.
| Grade | Topics |
|---|---|
| Grades 9 and 10 | Arithmetic: integer, rational and real numbers; algebra, polynomials, algebraic equations of first and second-degree, inequalities, simultaneous equations
Functions: Linear functions f(x)=ax+b, quadratic functions f(x)=ax2+bx+c, f(x)=|x|, and f(x)=a/x Euclidean geometry and Cartesian plane geometry: geometric transformations in the plane, circles, circumference, and π (pi), introduction to trigonometric functions and to vectors Descriptive statistics: average values, variance, standard deviation Classic probability, probability theorems |
| Grades 11 and 12 | Analytic geometry
Conics Spatial geometry: planes, lines, polyhedra, pyramids, solids of rotation (cylinder, cone, sphere), areas and volumes of elementary solids Trigonometry: triangles, law of sines and law of cosines law Trigonometric functions: trigonometric equations and inequalities Exponential functions: exponential equations and inequalities Logarithmic functions, e (base of natural logarithms), logarithmic equations and inequalities Arithmetic and geometric sequences and series Mathematical induction Complex numbers: algebraic, geometric and trigonometric forms and representations; sums, products, and quotients; complex solutions of quadratic equations Combinatorics Statistics: regression and correlation Conditional probability, Bayes’ theorem |
| Grade 13 | Limit of a series
Functions Limits: finite or infinite limits, limits of sums, products, quotients or composite functions, asymptotes Continuity on an interval Differentiation: numerical derivatives; the derivatives of common functions; derivatives of sums, products, quotients, and composite functions; applications of derivatives; the relationship between differentiability and continuity; the fundamental theorems of differential calculus; maxima and minima Integration: integration on an interval, the relationship between the definite integral and the area under a curve, the antiderivative of a function, applications of definite integrals Differential equations and applications, particularly in physics Analytic spatial geometry: coordinates, equations of planes, lines, spheres Probability: discrete and continuous random variables, probability distributions (Bernoulli, Poisson, normal), variance, standard deviation |
The topics taught at each grade at Istituti Tecnici–Settore Tecnologico are essentially the same as those listed above, but have a more applicative orientation at Grades 11, 12 and 13. Also it should be noted that some topics of calculus (function limits, continuity, derivatives) are taught in Grade 12, instead of Grade 13 as in Liceo Scientifico, while the integral calculus is taught in Grade 13 as in Liceo Scientifico. In these technical institutes, at Grades 12 and 13, many mathematical topics that serve specific technological applications are taught, such as partial derivatives, Fourier series, Taylor’s formula, spherical trigonometry, etc.
In this type of high school, at the end of the five-year course, the study of mathematics helps students achieve the following learning outcomes:
- Mastery of formal language and demonstration procedures of mathematics
- Possession of the mathematical, statistical, and probability tools necessary for the understanding of scientific disciplines and the ability to work in the field of applied science
- Understanding of the place of mathematics in the history of science