England

The Mathematics Curriculum in Primary and Lower Secondary Grades

In 2015, students who participated in the fourth grade TIMSS mathematics assessments (Year 5) were taught under both the 1999 and the 2013 versions of the national curriculum; students who participated in the eighth grade mathematics assessments (Year 9) were taught under the 1999, 2007, and 2013 versions successively. In 2014−2015, students were taught under the newly introduced national curriculum described in this chapter.

The national curriculum for mathematics aims to ensure that all students:

• Become fluent in the fundamentals of mathematics, through varied and frequent practice with increasingly complex problems over time, so that students develop conceptual understanding and the ability to recall and apply knowledge rapidly and accurately
• Reason mathematically by following a line of inquiry, conjecturing relationships and generalizations, and developing an argument, justification, or proof using mathematical language
• Can solve problems by applying their mathematics to a variety of routine and nonroutine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions

The programs of study for mathematics are set out year by year for Key Stages 1 and 2.  Year 5 (international Grade 4) students are in their penultimate year of Key Stage 2. A description follows of the focus of the upper Key Stage 2 (Years 5 and 6) program of study and provides the specifics of the Year 5 curriculum.

The principal focus of mathematics instruction in upper Key Stage 2 is to ensure that students extend their understanding of the number system and place value to include larger integers. This should help students develop connections between multiplication and division with fractions, decimals, percentages, and ratio. During Year 5, students should develop their ability to solve a wider range of problems, using increasingly complex properties of numbers and arithmetic, and problems demanding efficient written and mental methods of calculation. With this foundation in arithmetic, students are introduced to the language of algebra as a means of solving a variety of problems. Instruction in geometry and measures should consolidate and extend knowledge developed in number, and also should ensure that students classify shapes with increasingly complex geometric properties and learn the vocabulary needed to describe them. Students should read, spell, and pronounce mathematical vocabulary correctly.

The content areas and main curriculum elements at Key Stage 2 (Year 5) include:

• Number—Number and place value
  • Read, write, order, and compare numbers to at least 1 million, and determine the value of each digit
  • Count forward or backward in steps of powers of 10 for any given number up to 1 million
  • Read Roman numerals to 1,000 (M) and recognize years written in Roman numerals
• Number—Addition and subtraction
  • Add and subtract whole numbers with more than four digits, using formal written methods (columnar addition and subtraction)
  • Add and subtract numbers mentally with increasingly large numbers
• Number—Multiplication and division
  • Identify multiples and factors, including finding all factor pairs of a number, and common factors of two numbers
  • Read, write, and order whole numbers, recognizing that the position of a digit gives its value; use the symbols <, >, and =; multiply and divide whole numbers and those involving decimals by 10, 100, and 1,000
• Number—Fractions (including decimals and percentages)
  • Compare and order fractions whose denominators are all multiples of the same number
  • Identify, name, and write equivalent fractions of a given fraction, represented visually, including tenths and hundredths
• Measurement
  • Understand and use approximate equivalences between metric units and common imperial units such as inches, pounds, and pints
  • Measure and calculate the perimeter of composite rectilinear shapes in centimeters and meters

Due to word limits, this is a sample of the curriculum content. Full details are available online.⁸
At Key Stage 3, the national curriculum for mathematics aims to ensure that all students:

• Become fluent in the fundamentals of mathematics, through varied and frequent practice with increasingly complex problems over time, so that students develop conceptual understanding and the ability to recall and apply knowledge rapidly and accurately
• Reason mathematically by following a line of inquiry, conjecturing relationships and generalizations, and developing an argument, justification, or proof using mathematical language
• Can solve problems by applying their mathematics to a variety of routine and nonroutine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions

The program of study is organized into distinct domains, and students build on learning achieved at Key Stage 2 and connections across mathematical ideas to develop fluency, mathematical reasoning, and competence in solving increasingly sophisticated problems with good written and mental arithmetic. The use of equipment such as calculators is recommended only as a supporting tool near the end of Key Stage 2.

The curriculum at Key Stage 3 for mathematics includes the following subject content:

• Number
  • Understand and use place value for decimals, measures, and integers of any size
  • Order positive and negative integers, decimals, and fractions; use the number line as a model for ordering the real numbers; use the symbols =, ≠, <, >, ≤, ≥
  • Define percentage as “number of parts per 100,” interpret percentages and percentage changes as fractions or as decimals, interpret these multiplicatively, express one quantity as a percentage of another, compare two quantities using percentages, and work with percentages greater than 100 percent
• Algebra
  • Use and interpret algebraic notation, including ab in place of axb, 3y, in place of y+y+y and 3xy, a²,  in place of axa, and a³ in place of axaxa
  • Reduce a given linear equation in two variables to the standard form y-mx+c; calculate and interpret gradients and intercepts of graphs of standard form linear equations numerically, graphically, and algebraically
  • Recognize geometric sequences and appreciate other sequences that arise
• Ratio, proportion, and rates of change
  • Change freely between related standard units (e.g., time, length, area, volume/capacity, and mass)
  • Use scale factors, scale diagrams, and maps
  • Express one quantity as a fraction of another, where the fraction is less than 1 and greater than 1
• Geometry and measures
  • Derive and apply formulas to calculate and solve problems involving perimeter and area of triangles, parallelograms, and trapezoids, and volume of cuboids (including cubes) and other prisms (including cylinders)
  • Apply the properties of angles at a point, angles at a point on a straight line, and vertically opposite angles
  • Interpret mathematical relationships both algebraically and geometrically
• Probability
  • Record, describe, and analyze the frequency of outcomes of simple probability experiments involving randomness, fairness, and equally and unequally likely outcomes, using appropriate language and the 0−1 probability scale
  • Understand that the sum of the probabilities of all possible outcomes equals 1
  • Generate theoretical sample spaces for single and combined events with equally likely, mutually exclusive outcomes, and use these to calculate theoretical probabilities
• Statistics
  • Describe, interpret, and compare observed distributions of a single variable through appropriate graphical representation involving discrete, continuous, and grouped data, and appropriate measures of central tendency (mean, mode, median) and spread (range, consideration of outliers)
  • Construct and interpret appropriate tables, charts, and diagrams, including frequency tables, bar charts, pie charts, and pictograms for categorical data, and vertical line (or bar) charts for ungrouped and grouped numerical data
  • Describe simple mathematical relationships between two variables (bivariate data) in observational and experimental contexts and illustrate using scatter graphs⁹

Due to word limits, this is a sample of the curriculum content. Full details are available online.¹⁰