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The Mathematics Curriculum in Primary and Lower Secondary Grades

In 2012, the MIUR published the new Indicazioni nazionali per il curricolo della scuola dell’infanzia e del primo ciclo di istruzione, or national curricular guidelines for preprimary school and for the first cycle of education.¹²

This document represents a framework for schools in the design and implementation of local primary and lower secondary curricula. Schools are free to determine content and methods of instruction autonomously, provided they are consistent with the learning objectives established by the Indicazioni. Learning objectives at the primary level correspond with goals that students are expected to achieve by the end of their third and fifth years of primary education.¹³ Exhibits 1 and 2 summarize the content areas and learning objectives in the mathematics curriculum at the primary level.¹⁴

Exhibit 1: Mathematics Learning Objectives at the End of Grade 3

Content Area	Objectives and Expectations
Numbers	Count objects or events, either aloud or mentally, in increasing or decreasing order, and in intervals of two and three Read and write whole numbers, accounting for place value; compare and sort numbers and plot them on a number line Mentally perform simple operations with whole numbers and verbalize calculation procedures Know multiplication tables for numbers up to 10; perform operations with whole numbers using common written algorithms Read, write, and compare decimal numbers (written without units, in monetary units, or as simple measurements), represent them on a number line, and perform simple addition and subtraction
Space and Shapes	Be aware of one’s position and be able to estimate distance and volume within a specified space Report the position of objects in physical space, relative to oneself and to other people or objects, using appropriate terms (e.g., above and under, in front of and behind, left and right, and inside and outside) Perform a simple procedure given a verbal description or drawing, describe the direction of one’s movement in a defined space, and instruct others to follow a given direction Recognize, name, and describe geometric figures Draw geometric shapes and construct material models in space
Relations, Data, and Chance	Classify numbers, figures, and objects according to one or more properties using contextual information and appropriate representations Explain the criteria used in ordering and classifying Read and represent relations and data with diagrams, charts, and tables Measure quantities (e.g., length, time) using arbitrary units or conventional instruments (e.g., ruler, clock)

Exhibit 2: Mathematics Learning Objectives at the End of Grade 5

Content Area	Objectives and Expectations
Numbers	Read, write, and compare decimal numbers Perform arithmetic operations with confidence, evaluating and choosing a method of calculation (i.e., mental, written, or calculator) based on the context of a problem Perform division of natural numbers with remainders, and identify multiples and divisors of a number Estimate the result of arithmetic operations Work with fractions and recognize equivalent fractions Use decimal numbers, fractions, and percentages to describe everyday situations Interpret negative integer numbers within concrete contexts Represent numbers on a number line and use graded scales in science and technology contexts Understand systems of numerical notation that are or were used in places, times, and cultures other than the present
Space and Shapes	Describe, classify, and identify significant elements and symmetries of geometric  figures, in a way that allows others to reproduce them Reproduce a figure, based on a given description and using appropriate instruments (e.g., graph paper, ruler, compass, squares, and geometry software) Use the Cartesian plane to locate points Construct and use material models on a plane and in space as a support to visualizing them in the abstract Recognize rotated, translated, and reflected figures Compare and measure angles using properties and instruments Use and distinguish concepts of perpendicular, parallel, horizontal, and vertical Reproduce a figure according to scale (e.g., on graph paper) Determine the perimeter of figures Determine the area of rectangles, triangles, and other figures by decomposition Recognize two-dimensional representations of three-dimensional objects, and identify different perspectives of the same object (e.g., top view, front view)
Relations, Measurement, Data, and Chance	Represent relations and data, and use the representations to obtain information and make judgments and decisions Use notions of arithmetical mean, mode, and frequency, where appropriate to the type of available data Represent the structure of problems with graphs and tables Understand the common units of measurement for length, angles, area, volume, time, and mass or weight, and use them to make measurements and estimates Convert between units of measurement (including monetary units) Consider a pair of events, and explain which is more likely or recognize that both events are equally probable Recognize and describe patterns in a sequence of numbers or figures

The objectives and expectations of the mathematics curriculum at the lower secondary level are summarized in Exhibit 3.¹⁵

Exhibit 3: Mathematics Learning Objectives at the End of Grade 8

Content Area	Objectives and Expectations
Numbers	Perform arithmetic operations with and compare natural numbers, whole numbers, fractions, and decimals mentally or using common written algorithms, calculators, and calculation tables, and evaluate which method is most suitable in a specific context Estimate the results of arithmetic operations and check the plausibility of calculations Represent numbers on a number line, use graded scales in science and technology contexts, and describe ratios and quotients using both decimals and fractions Use equivalent fractions and decimal numbers to denote the same rational number in different ways, and understand the advantages and disadvantages of different numerical representations in context Calculate percentages, and interpret a percentage increase in a given quantity as a multiplication by a decimal number Identify multiples and divisors of natural numbers, and multiples and divisors common to several numbers Understand the meaning and applications of the least common multiple and greatest common divisor in mathematics and in other practical contexts Find prime factors of natural numbers and understand their applications Use positive whole exponents correctly, understanding their meaning, and use properties of exponents to simplify calculations and notation Understand square root as the inverse operation of squaring a number Estimate square roots using multiplication Know that the square root of 2 is an irrational number Perform simple mental calculations using associative and distributive properties to simplify arithmetic operations Express a sequence of operations with simple algorithms as a solution to a problem Correctly use rules governing order of operations with and without brackets Correctly express large numbers using scientific notation
Space and Shapes	Reproduce figures and geometric shapes from descriptions, using suitable tools (e.g., ruler, square, compass, and geometry software) Represent points, segments, and figures on a Cartesian plane Know the definitions and significant properties of plane figures (e.g., triangles, quadrilaterals, regular polygons, and circles) Describe complex figures and geometric constructions Reproduce geometrical figures and drawings according to a given description Recognize similar plane figures in various contexts and reproduce figures to scale Understand the Pythagorean theorem and its applications in mathematics and the real world Calculate the area of simple figures by breaking them down into more elementary figures (e.g., triangles) or using the most common formulas Estimate the area of a figure consisting of curves by rounding Understand π , and some ways to approximate it Understand formulas used to find the area of a circle and the length of its circumference Know and use the main geometric transformations and their invariants Represent three-dimensional objects and plane figures in various ways (e.g., drawings on a plane) Visualize three-dimensional objects from two-dimensional representations Calculate volume of the most common three-dimensional figures and estimate volume of everyday objects Solve problems using geometric properties of figures and solids
Relations and Functions	Construct, interpret, and transform formulas containing variables to express relationships and properties Express proportionality using equivalent fractions (and vice versa) Use the Cartesian plane to represent relations and functions, to understand functions such as y=ax, y=a/x, y=ax2, y=2n and their graphs, and to link y=ax and y=a/x to the concept of proportionality Explore and solve problems using first-degree equations
Measurement, Data, and Chance	Represent data sets in several forms, including software spreadsheets; compare data to inform decisions using frequency and relative frequency distributions, as well as arithmetical mean, mode, and median; evaluate variability of a data set calculating range, for example Identify elementary events in simple random situations, assign a probability to them, calculate the probability of an event, and break it down into separate elementary events Recognize pairs of complementary, incompatible, or independent events