The Mathematics Curriculum in Primary and Lower Secondary Grades

The current mathematics curriculum for fourth grade was implemented in 2012,⁹ and the curriculum for eighth grade was implemented in 2013;¹⁰ TIMSS 2015 was administered in Chile in 2014. Exhibits 1 and 2 present the content areas and fundamental objectives of the mathematics curriculum for Grades 4 and 8.

Exhibit 1: Mathematics Curriculum, Grade 4

Content Area	Fundamental Objectives
Numbers and Operations	Represent and describe numbers from 0 to 10,000 Describe and apply mental arithmetic strategies Demonstrate understanding of addition and subtraction up to 1,000 Apply the properties of 0 and 1 for multiplication and the property of 1 for division Demonstrate understanding of the multiplication of three-digit numbers by one-digit numbers Demonstrate understanding of division with double-digit dividends and single-digit divisors Solve routine and non-routine problems in everyday contexts that involve money, selecting and using the proper operations Demonstrate understanding of fractions with denominators 100, 12, 10, 8, 6, 5, 4, 3, and 2 Add and subtract fractions with the same denominator (100, 12, 10, 8, 6, 5, 4, 3, and 2) in concrete and pictorial ways in a problem solving context Identify, describe, and represent proper fractions and mixed numbers up to 5 in concrete, pictorial, and symbolic ways, in a problem solving context Describe and represent decimals (tenths and hundredths) Add and subtract decimals (with place value to the hundredth) in a problem solving context
Algebra and Patterns	Identify and describe number patterns in tables that involve an operation, manually and/or using educational software Solve equations and inequalities for values from 0 to 100 in one step, using addition and subtraction, checking the results in a pictorial and symbolic manner, and applying the inverse operation (addition or subtraction)
Geometry	Describe the absolute location of an object on a simple map with informal coordinates (e.g., with letters and numbers) and the relative location in relation to other objects Determine the views of three-dimensional figures from the front, the side, and above Demonstrate understanding of line of symmetry Move, rotate, and mirror two-dimensional figures Construct angles using a protractor and compare them
Measuring	Read and record various time measurements on analog and digital clocks, using the concepts of a.m., p.m., and 24 hours Convert among units of time in the context of problem solving (i.e., the number of seconds in a minute, the number of minutes in an hour, the number of days in a month, and the number of months in a year) Measure length in standardized units (m, cm) and convert between units (e.g., from m to cm and vice versa) in the context of problem solving Demonstrate understanding of the concept of area of rectangles and squares Demonstrate understanding of the concept of volume
Data and Probabilities	Conduct surveys, analyze the data, and compare with the results of random tests, using charts and graphs Perform randomized experiments, tabulate the results, and represent them in graphs constructed manually and/or using educational software Read and interpret pictographs and simple bar graphs with scale, and report findings

Exhibit 2: Mathematics Curriculum, Grade 8

Content Area	Fundamental Objectives
Numbers	Demonstrate understanding of multiplication and division of integers Represent them in concrete, pictorial, and symbolic ways Apply procedures used in multiplication and division of natural numbers Apply the rule of order of operations Solve routine and nonroutine problems Use multiplication and division of rational numbers in the context of problem solving Represent them on the number line Use different number sets (fractions, decimals, and integers) Explain multiplication, division, and the process of forming powers of numbers with natural number bases and exponents up to 3, in concrete, pictorial, and symbolic ways Demonstrate understanding of the square roots of natural numbers Estimate them intuitively Represent them in concrete, pictorial, and symbolic ways Apply them in geometrical situations and in daily life Solve problems involving percent changes in different contexts, using pictorial representations and recording the process symbolically (e.g., annual savings interest)
Algebra and Functions	Demonstrate understanding of operations and algebraic expressions Represent them in pictorial and symbolic ways Relate them to the area of squares and rectangles and the volume of parallelepipeds Determine factored forms Demonstrate understanding of the concept of function in terms of linear change Use tables Use machine metaphors Establish rules for x and y Graph them manually and/or by using educational software (e.g., Cartesian plane, Venn diagrams) Model situations in everyday life and in other academic subjects, using linear equations of the form ax = b ;x/a = b ;a ≠ 0;ax + b = c ;x/a + b = c ;x = b + cx ; a(x + b) = c ;ax + b = cx + d (a, b, c, d, e) Solve linear inequalities with rational coefficients in the context of problem solving, using graphic representations and symbolic representations, generated manually and/or by using educational software Demonstrate understanding of affine functions: Generalize as the sum of a constant with a linear function Move linear functions in the Cartesian plane Determine the constant change from one interval to another, graphically and symbolically, manually and/or by using educational software Link them to simple interest Use them to solve everyday life problems and in other academic subjects
Geometry	Develop formulas for finding the surface area and volume of prisms with different bases and cylinders Estimate surface area and volume Use nets to find the surface area of prisms Apply the formula for volume of a cube (base height) to different prisms and cylinders Apply these formulas to geometric problems and problem solving in daily life Explain in concrete, pictorial, and symbolic ways the validity of the Pythagorean theorem, and apply the theorem to geometric problems and solving problems in daily life, manually and/or by using educational software Describe the position and transformations (i.e., translations, rotations, and reflections) of two-dimensional figures in a coordinate plane, manually and/or by using educational software, involving: Vectors used in translations The axes of the Cartesian plane (e.g., lines of reflection) Points in a plane of rotation Perform rotations, translations, and reflections in the Cartesian plane and in space, manually and/or by using educational software; apply these transformations to the lines of symmetry of polygons and polyhedra, and in geometric problems related to art
Probability and Statistics	Demonstrate an understanding of statistical measures of position (i.e., percentiles and quartiles) Identify the population above or below the percentile Represent them in diagrams, manually and/or by using educational software Use them to compare populations Evaluate the way data is presented Compare information from the same data set displayed in different types of graphs to determine the strengths and weaknesses of each Justify the choice of graph for a given situation and its corresponding data set Detect manipulation of graphics to represent data Explain the multiplication principle of probability In specific situations Represent it in regular tables and trees, generated manually and/or by using educational software Use it to calculate the probability of compound events