The Mathematics Curriculum in Primary and Lower Secondary Grades

Mathematics is an important tool by which information can be organized, manipulated, and communicated. It also is an ever-expanding body of facts, skills, concepts, and strategies that may be used to solve a wide range of problems.

The National Curriculum Framework, together with a number of other policy documents published recently, emphasizes the need of a learning-outcomes based approach in today’s educational structure. Consequently, when implementing the syllabus, mathematics teachers emphasize both the utilitarian and aesthetic aspects of mathematics:

Utilitarian aspect—Mathematics is useful, equipping learners with the necessary knowledge to help them understand and interact with the world around them.
Aesthetic aspect—Mathematics is a beautiful subject with an evolving body of knowledge that is characterized by order, precision, conciseness, and logic.

In mathematics instruction, teachers should ensure that students do the following:

Understand and appreciate the role and purpose of mathematics in culture and society, in the past as well as the present
Apply mathematical knowledge and understanding to solve a wide range of standard and nonstandard problems, ideally from real life situations
Think and communicate mathematically (i.e., precisely, logically, and effectively)
Develop a positive attitude toward mathematics that fosters creativity, confidence, perseverance, and enjoyment of the subject
Develop the ability to work both independently and cooperatively when doing mathematics
Acquire a secure foundation for the further study of mathematics
Appreciate the interdependence of the different branches of mathematics
Appreciate the interdisciplinary nature of mathematics and its use in other areas of knowledge
Make efficient, creative, and effective use of appropriate technology in mathematics

The mathematics curriculum comprises a list of learning outcomes for each year at the primary and secondary levels. At the primary level, the lists are organized within four strands—Number and Algebra; Space and Shape; Measurement; and Data Handling—and students are expected to achieve specified attainment levels within each strand.

At the primary level, there are three attainment levels (Levels 4, 5, and 6) while at the secondary level there are four attainment levels (Levels 7, 8, 9, and 10). Each attainment level from Levels 4 to 8 (inclusive) spans two academic years. Level 9 is covered in one year, and Level 10 is aimed at talented and gifted students. Attainment Levels 1, 2, and 3 are designed for students with special education needs (Entry Levels). If students perform above or below the expected levels, adjustments are made accordingly. Exhibit 1 presents the curriculum attainment levels along with their corresponding academic years and age ranges.

Exhibit 1: Curriculum Attainment Levels with Corresponding Year and Age Ranges

Cycle	Attainment Level	Number of Academic Years	Ages
Primary
	Level 4	Years 1–2	5–7
	Level 5	Years 3–4	7–9
	Level 6	Years 5–6	9–11
Secondary
	Level 7	Years 7–8	11–13
	Level 8	Years 9–10	13–15
	Level 9	Year 11	15–16
	Level 10	Gifted and talented students	All ages

Because students in Malta are assessed at the fifth grade (Level 5), Exhibit 2 presents the learning objectives within the four mathematics curriculum strands for the later primary attainment levels covering Grade 5.

Exhibit 2: Curriculum Attainment Levels with Corresponding Year and Age Ranges

Strand	Attainment Level	Student Competencies
Number	Level 5	Read, write, and use numbers up to at least 1,000 Identify place value in four-digit numbers, and round two-digit and three-digit whole numbers to the nearest 10 or 100 Add and subtract two-digit and three-digit numbers mentally and using informal pencil and paper procedures (including column addition) Count by 100s and identify odd and even numbers to at least 100 Develop an understanding of multiplication as “repeated addition” and as an array, and understand and apply the zero and commutative properties of multiplication Develop an understanding of division as “grouping,” “repeated subtraction,” and “sharing” with and without remainders Recall multiplication facts for the 2, 3, 4, 5, 8, and 10 times tables, and derive division facts corresponding to the 2, 3, 4, 5, and 10 times tables Identify fractions and equivalent forms of simple fractions, understand mixed numbers, compare and order fractions, position fractions on a number line, make estimates, and solve and complete practical tasks and problems involving fractions
Number	Part of Level 6	Use written methods to add and subtract two or more numbers up to 1,000 involving decimals Understand decimal notation for tenths and hundredths, round numbers with one or two decimal places to the nearest integer, and order numbers with up to three decimal places Multiply and divide decimals by 10 and 100 and integers by 1,000 and explain the effect, and use written methods to multiply or divide a three-digit integer by a two-digit integer Find simple common multiples and factors and apply simple tests of divisibility Recognize odd and even numbers up to 1,000, square numbers, and triangular numbers Understand fractions, find fractions of numbers or quantities, reduce fractions to their simplest form, and solve problems Solve simple problems involving proportion, understand percentage as the number of parts in every 100, and find simple percentages of small whole number quantities
Algebra	Level 5	Identify patterns in numbers from 0 to 1,000 Count by 4s or by 100s and recognize odd and even numbers to at least 100 Describe (identify the rule in) and extend simple number sequences Understand that the same quantity can be written as an addition or a subtraction expression (e.g., 24=20+4 or that 24=30-6), understand the relationship between addition and subtraction, and provide a subtraction statement corresponding to a given addition statement, and vice versa Understand the principles of the commutative, associative, and distributive laws of multiplication Recognize that division is the inverse of multiplication, and that halving is the inverse of doubling Solve number sentences in one or two unknowns and translate a word problem into a number sentence Solve mathematical puzzles by recognizing patterns and relationships, and suggest extensions
Algebra	Part of Level 6	Develop the idea of continuity and understand that the number line is continuous Solve inequalities Extend number sequences, such as sequences of square numbers and triangular numbers Understand and use the relationships between the four arithmetic operations and the principles of arithmetic laws Factor numbers and use partitions (e.g., 87×6=(80×6)=(7×6)) Identify properties and rules regarding brackets and order of operations Check results of calculations with an equivalent calculation or with the inverse operation Translate more difficult word problems into number sentences and equations Draw graphs to display factual information, show mathematical relationships, and describe them in their own words
Shape, Space, and Measure	Level 5	Identify, describe, and sort two- and three-dimensional shapes referring to properties such as symmetry, the number or shape of faces, the number of sides or edges and vertices, whether sides or edges are the same length, and whether angles are right angles Recognize that a straight line is equivalent to two right angles and compare angles with a right angle Identify and sketch lines of symmetry in simple shapes, sketch the reflection of a simple shape across a mirror line along one edge, and recognize shapes with no lines of symmetry Recognize and use the four compass directions and make and describe right-angled turns Estimate measurements (to the nearest whole unit or half unit or in mixed units), compare length (km, m, cm), mass (kg, g), and volume (l, ml) using standard units, and suggest units and equipment for measurement Read simple scales to the nearest division, including using a ruler to draw and measure lines to the nearest centimeter Use units of time (second, minute, hour, day, week, month, and year), know the relationship between them, read a calendar, suggest suitable units to estimate and measure time, and read the time to 5 minutes on an analog clock and a 12-hour digital clock
Shape, Space, and Measure	Part of Level 6	Classify triangles (isosceles, equilateral, scalene) according to sides, angles, and lines of symmetry, and draw shapes with increasing accuracy Visualize three-dimensional shapes from two-dimensional drawings and identify different nets of solid shapes Recognize symmetry in regular polygons, complete symmetrical patterns with two lines of symmetry at right angles, and recognize the position of a shape after reflection across a mirror line Identify, estimate, and order acute and obtuse angles, use a protractor to measure and draw these angles, and check the sum of angles in a triangle and on a straight line Use the eight compass directions, and make and measure clockwise and counter-clockwise turns Know and use the relationships between familiar units of length, mass, and volume and make estimations in relation to everyday situations Understand perimeter and area of rectangles and other simple shapes (including compound shapes) Use timetables and read the time on a 24 hour digital clock and use notation for 24 hour time
Data Handling	Level 5	Read and interpret numerical data in lists, charts, frequency tables, pictograms, bar charts, and line graphs, and construct them Use scales on charts and graphs and understand intervals Collect, organize, and represent data to solve problems and complete practical tasks Use appropriate language such as “least common,” “most common,” “most favorite,” and “least favorite” to discuss and explain results
Data Handling	Part of Level 6	Solve problems by representing, extracting, and interpreting data in lists, tables, charts, graphs, pictograms, and diagrams, including those generated by a computer Construct and use charts, graphs, tables, and pictograms Perform experiments such as tossing a coin or rolling a die, record outcomes, and construct frequency charts, graphs, and tables Decide how best to organize and present findings (including the use of ICT where appropriate), and use precise mathematical language and vocabulary when discussing data Answer questions by collecting, selecting, and organizing relevant data, draw conclusions, and identify further questions Explore and calculate the mean of a simple set of data

Exhibit 3 presents the learning objectives for the lower secondary attainment levels covering eighth grade (Level 7 and part of Level 8) within the four mathematics curriculum strands.

Exhibit 3: Lower Secondary (Grade 8) Mathematics Learning Objectives

Strand	Attainment Level	Student Competencies
Number	Level 7	Use the four arithmetic operations (addition, subtraction, multiplication, and division) for calculating with integers, fractions, and decimals, including the correct order of operations and the use of brackets Use ratios to compare two or more quantities, find percent associated with a quantity, and find percent increase or decrease Understand and use positive exponents to represent squares and cubes and understand and use prime numbers, prime factors, least common multiples, and greatest common factors Use basic functions of a scientific calculator, round numbers to a specified number of decimal places, and carry out rough estimates
Number	Part of Level 8	Reverse percent changes and understand and calculate simple interest Solve problems involving direct and inverse proportions Apply the four arithmetic operations to mixed numbers and understand reciprocals Work with numbers in standard form and round numbers to a specified number of significant figures Use the number line to illustrate inequalities
Algebra	Level 7	Understand the use of letters to represent unknown values, simplify algebraic expressions by collecting like terms, multiplying a single term over a bracket, and taking out a common factor Construct and solve linear equations in one unknown; understand that the equation of a straight line describes the relationship between the x and y coordinates, generate ordered pairs, and plot them Draw and use graphs to convert between units Generate terms of a sequence
Algebra	Part of Level 8	Simplify fractions, factor expressions, and expand expressions written in factored form Solve linear and simultaneous equations in two unknowns by trial and error Understand and determine the term of a n^th sequence Write and manipulate more complex formulas Understand, interpret, and calculate the slope of a line and identify the slope and intercept in a linear equation Solve systems of two linear equations graphically Draw and use quadratic graphs to identify maxima and minima and solve quadratic equations and related problems
Shape, Space, and Measure	Level 7	Use a protractor to measure and draw angles up to 360° and solve problems involving basic angle facts, including parallel lines Use bearings to describe direction Draw basic constructions and identify geometric properties of triangles and quadrilaterals through line and rotational symmetry Classify quadrilaterals using their geometric properties Identify parts of a circle (center, radius, diameter, and circumference) Use formulas to find areas of triangles, parallelograms, and compound shapes Find the volume of compound shapes involving cubes and cuboids, and draw translations, reflections, and rotations
Shape, Space, and Measure	Part of Level 8	Convert units of area and volume Use angle properties of polygons and construct regular polygons Understand proof and use the Pythagorean theorem Calculate perimeter and area of circles, sectors, and segments, and surface area and volume of prisms and pyramids Understand and use the three basic trigonometric ratios (sine, cosine, and tangent) and solve problems involving angles of elevation, depression, and bearing Draw and interpret scale drawings Understand and prove the fundamental angle-circle theorems
Data Handling	Level 7	Collect data using observations, surveys, and experiments Understand, compute, and interpret the mean, mode, median, and range of a set of ungrouped data; compile and interpret frequency tables for ungrouped or grouped continuous and discreet data Find probability by experiment and compile sample spaces Use spreadsheets to construct bar graphs and pie charts, and compute the mean and range of sets of ungrouped data
Data Handling	Part of Level 8	Find the modal class, an estimate of the mean, and the class interval in which the median lies in grouped frequency distributions Understand that the probabilities of all mutually exclusive outcomes add up to 1 Use frequency tables and sample spaces