Portugal

The Mathematics Curriculum in Primary and Lower Secondary Grades

The mathematics curriculum for fourth grade students assessed in TIMSS 2015 was adopted in 2007. However, in the 2013–2014 school year, a new mathematics curriculum was implemented progressively starting with Grades 1 and 3. The new curriculum introduces some required adjustments to the 2007 curriculum in order to account for new orientations and curricular goals that emphasize the development of students’ mathematical abstraction, the comprehension of mathematical concepts and their applications (specifically working with fractions), as well as the promotion of teachers’ autonomy in their teaching practices.⁸

The 2007 mathematics curriculum is divided into the three cycles of basic education that correspond to primary (first cycle: Grades 1 to 4, and second cycle: Grades 5 to 6) and lower secondary (third cycle: Grades 7 to 9) education. The general objectives of the mathematics curriculum and its development during the three cycles are to:

• Promote the knowledge of basic mathematics facts and procedures
• Develop mathematical comprehension and reasoning
• Read and interpret different mathematical representations
• Develop, organize, and communicate mathematical ideas and reasoning
• Solve problems using mathematical knowledge and critically discuss the solutions reached
• Establish connections between different mathematical concepts and relationships, as well as relationships with nonmathematical contexts
• Develop autonomous mathematical work strategies
• Appreciate mathematics and its importance to other school subjects and daily life⁹

The curriculum emphasizes three major skills required of all mathematical learning: mathematical reasoning, resolution of mathematical problems, and communication of mathematical results and conclusions.

The first cycle mathematics curriculum (Grades 1 to 4) has three major domains: Numbers and Operations; Geometry and Measurement; and Organization of Data and Data Processing. Its main purpose is to develop students’ sense of number, numbers and operations comprehension, and written and mental calculations, as well as the use of these skills and knowledge to solve problems in different contexts. By the end of this cycle, students should:

• Understand and be able to use the properties of natural and rational non-negative numbers
• Understand the decimal numeration system
• Understand operations with natural and rational numbers, and apply them in the decimal system
• Be able to appreciate orders of magnitude of numbers, and understand the effect and reasonableness of number operations and its results
• Develop mental and written mathematical calculus skills and communicate results in numerical contexts

These learning objectives were combined for Grades 1 to 2 and Grades 3 to 4. Exhibit 1 describes the content and objectives that students are expected to master by the end of both grade ranges.¹⁰

Exhibit 1: Mathematics Topics and Objectives that Students are Expected to Learn in the First Cycle of Primary Education (Grades 1–4)

	Grades 1–2	Grades 3–4
Numbers and Operations
Understand the notion of natural numbers, use numerical relations and the decimal system, represent numbers, read and write numbers to at least 1,000, decompose numbers and solve problems using numerical relations Learn the concepts underlying the four arithmetic operations (addition, subtraction, multiplication, and division), and use these operations to solve real life problems Construct and investigate patterns in number sequences Develop a basic understanding of non-negative rational numbers as fractions (identify and represent fractions)		Develop an understanding of and use numerical relations, recognize multiples and factors of numbers, and read and write numbers up to 1 million Develop strategies for mental calculations for the four basic operations using their properties, and develop and understand methods and algorithms for basic calculations (multiplication and division by 10, 100, 1,000, etc.) Further develop pattern recognition in number sequences and develop skills to solve problems with proportionalities Deepen understanding of non-negative rational number as fractions, and develop skills to work with fractions and decimal numbers
Geometry and Measurement
Develop spatial orientation with emphasis on the visualization and understanding of the properties of geometric figures, including position and location relative to other objects Understand the notion of measurement, its process, and the different magnitudes of quantities involving money, length, area, mass, capacity, volume, and time		Further develop the ability to visualize geometric figures and their properties and operations in space, read maps, plants, and drawings, and locate objects in a coordinates system Further develop the capacity for measurement and use of mathematical operations with money, length, mass, capacity, area, volumes, and time, and their estimation and conversion
Organization of Data and Data Processing
Read and interpret data in tables and graphs, classify data using Venn and Carroll diagrams, and construct absolute frequency tables, graphs, and pictograms		Read and represent data, understand random events and their properties, construct data summaries in absolute and relative frequency data tables, construct and understand bar graphs, and identify the mode in a data set

The second cycle mathematics curriculum (Grades 5 to 6) further develops the mathematical concepts and skills learned in the first cycle and introduces Algebra as a fourth major domain. The main objective is to foster in students the concept and comprehension of number and number operations, the capacity to do mental and written mathematical calculations promoting the use of this knowledge, and the capacity to solve mathematical problems in different contexts. The general objectives aim to deepen the understanding and use of mathematical concepts and skills learned in the previous cycle. Exhibit 2 describes the content and objectives that students are expected to master by the end of the second cycle.¹¹

Exhibit 2: Mathematics Topics and Objectives that Students are Expected to Learn in the Second Cycle of Primary Education (Grades 5–6)

Grades 5–6

Numbers and Operations

Understand natural numbers, including prime and composite numbers and their decomposition, least common multiple and largest common divisor of two numbers, and powers of numbers and their operations Understand, represent in the numerical line, and do calculations with integers, non-negative rational numbers, representation and numerical operations, approximate values and percentages

Geometry

Geometric solids (prism, sphere, pyramid) and their models and plane representations Geometric figures in the plane, straight lines, semi-lines and line segments, angles’ amplitude and measurement, polygons, circles and circumference’s properties and construction Reflection, rotation, and translation, axial and rotational symmetries Area, perimeter, and volume

Algebra

Develop algebraic reasoning and skills to represent mathematical and nonmathematical conditions with the use of mathematical symbols Relations and regularities, numerical expression and calculus properties, direct proportionality

Organization of Data and Data Processing

Develop the skills to understand and produce statistical information, as well as to use statistical results and information to solve problems and make supported and informed decisions Data representations and interpretation, absolute and relative frequency tables, bar graphs, circular plots, line plots, and stem-and-leaf graphs Arithmetic mean, extreme values, and range

Lower secondary education corresponds to the third cycle of the Portuguese compulsory basic education (Grades 7 to 9). The mathematics curriculum is built upon the knowledge and skills acquired in the previous two cycles of study and has as its main objective to further develop mathematical knowledge and skills, plus introduce real numbers, their properties, and operations. Exhibit 3 describes the content and objectives that students are expected to master by the end of the third cycle of compulsory basic education.¹²

Exhibit 3: Mathematics Topics and Objectives that Students are Expected to Learn in the Third Cycle (Lower Secondary Education, Grades 7–9)

Grades 7–9

Numbers and Operations

Further develop and understand the notion of integers and rational numbers and their properties and operations Powers, quadratics, and cubic roots Notion of real numbers and their rational and nonrational parts, their representation on the real number line; understand and do more complicated operations in the domain of real numbers; intervals of real numbers; solve problems and examine patterns using real numbers

Geometry

Understand, classify, and do operations with triangles and quadrilaterals; understand their properties and properties of angles; geometric solids areas and volumes; parallel and perpendicular lines; and planes, circumferences, and angles Understand and construct geometric planes, circles, circumferences, bisectors, spherical surfaces, and mediator planes Construct inscribed and circumscribed circles and polygons Identify and construct congruent and similar figures Recognize and use vectors in geometric transformations (translation, reflection, and rotation) Demonstrate and use the Pythagorean theorem Understand and use right angle trigonometry, trigonometric ratios and their relations, and second-order equations with one variable

Algebra

Develop algebraic reasoning and language and solve problems involving algebraic procedures using sequences and patterns Understand and solve first-degree equations with one unknown, literal equations, and polynomials; understand and solve second-degree equations with one unknown and two equations systems with two unknowns; and understand and solve first-order inequalities Understand and solve first-degree equations with one unknown, literal equations, and polynomials; understand and solve second-degree equations with one unknown and two equations systems with two unknowns; and understand and solve first-order inequalities

Organization of Data and Data Processing

Further develop the use of statistical data analysis to make informed decisions Develop skills for statistical planning, formulate questions and plan adequate data collection Further emphasis on the organization, analysis, and interpretation of data, construct and interpret graphical data representations, and understand the concepts and estimate measures of central tendency and dispersion Choose the appropriate statistics for different data types, compare data distributions, and discuss critically inferences to populations from the analysis of samples Understand the notion of random events and random experiments, and understand and do probability calculations using relative frequencies and the Laplace rule