The Mathematics Curriculum in Primary and Lower Secondary Grades

France has a national curriculum that covers mathematics and science instruction at the fourth grade; it has been renewed 10 times over a 90-year period. The curriculum is national and compulsory for all teachers and students, and it governs teacher practice. Teachers are responsible for building a coherent progression through the curriculum, adapting the pace of the curriculum to suit their students’ abilities and needs, defining instructional strategies, and evaluating students.

The primary school curriculum was implemented in France officially at the beginning of the 2008–2009 school year. A new curriculum was presented and published in the Bulletin Officiel (official bulletin) on November 26, 2015, a new reference text that applies to learning in both primary and lower secondary schools and will be implemented at the beginning of the 2016–2017 school year (September 2016). Students assessed in TIMSS 2015 were not impacted by this new curriculum. The following descriptions include information on the 2008 curriculum,¹⁰ which was in effect throughout the education of students assessed in TIMSS 2015.

The 2008 mathematics curriculum for Grades 1 and 2 (second cycle of primary school) may be summarized as follows. The study of mathematics develops imagination, rigor, and accuracy as well as an appreciation of critical reasoning. The knowledge of numbers and operations is a priority objective in Cycle 2 (Grades 1 and 2). Instruction in problem solving follows a learning progression and helps build operation sense. Students practice mental operations at least 15 minutes per day and begin to acquire arithmetic facts. Students memorize mathematical procedures and develop an understanding of the associated concepts.

Numbers and Calculation—Students learn to count in the decimal number system up to 1,000. They enumerate collections to develop number sense, and they learn to sequence, compare, and order numbers. Students memorize and use addition and multiplication tables (by 2, 3, 4, and 5), learn the operations of addition, subtraction, and multiplication, and learn to solve problems using these operations. Exercises in grouping and sharing provide an introduction to learning division with numbers up to 100. Daily exercises in mental arithmetic allow students to develop a more thorough knowledge of numbers and their properties.
Geometry—Students develop their spatial sense in two and three dimensions. They learn to recognize and describe plane figures and solids. They use instruments and learn techniques for reproducing or tracing plane figures. They learn specific vocabulary.
Quantities and Measures—Students learn and compare units of length (mm, cm, m, and km), mass (kg and g), capacity (liter), time (hour, half hour), and currency (euro, euro cent). They begin to solve problems involving length, mass, time, and currency.
Organization and Management of Data—Students gradually learn to use common representations of data (e.g., tables and graphs).

The Common Base of Knowledge and Skills is a second reference text, in addition to the curriculum. Article 9 in the Law and Policy for the Future of the School¹¹ describes the principle of a “common base of knowledge and skills,” specifying the following: Compulsory education must guarantee for every student the means of acquiring a common base of knowledge and skills, which is indispensable to achieving success in school, pursuing training, building a personal and professional future, and making life in society a success.

The common base of knowledge and skills does not replace the primary and lower secondary programs. Its uniqueness lies in the aim to give meaning to the fundamental school culture, in taking the student’s point of view, and in building essential bridges between disciplines and programs. It defines fundamental knowledge and skill objectives in compulsory education that will prepare students for success after graduation.

In terms of curriculum, the common base of knowledge and skills has been reviewed, and a new version will be implemented in September 2016. The following descriptions include information on the 2008 curriculum, which was in effect throughout the schooling of students assessed in TIMSS 2015.

The common base¹² is organized according to seven skill domains, five of which comprise part of the current teaching curriculum: French Language Skills; Skills in a Living Foreign Language; Basic Skills in Mathematics and in Scientific and Technological Culture; Mastery of Common Techniques in Information and Communication; and Humanist Culture. The remaining two domains are Social and Civic Skills, and Autonomy and Initiative. All seven skill domains are designed as a combination of fundamental knowledge relevant to the times, the ability to implement this knowledge in various situations, and vital attitudes in life, including openness to others, interest in seeking the truth, respect for oneself and others, and curiosity and creativity.

The Common Base of Knowledge and Skills states that in mathematics, at the end of the second cycle of primary school, students should be able to:

Write, name, compare, and order natural numbers up to 1,000
Calculate using addition, subtraction, and multiplication
Divide whole numbers up to 100, by 2 and 5 (where the exact quotient is an integer)
Reproduce and use tables of addition and multiplication by 2, 3, 4, and 5
Perform mental operations (i.e., addition, subtraction, and simple multiplication)
Locate an object, or locate an object in relation to another object, give its position, and describe its movement
Recognize, name, and describe plane figures and regular solids
Use a ruler and set square to trace squares, rectangles, and right triangles with precision
Use units of measurement and estimate measurements
Be precise and diligent when making drawings, measurements, and calculations
Solve simple problems

The 2008 mathematics curriculum for Grades 3 to 5 (third cycle of primary school) may be summarized as follows. The practice of mathematics develops a taste for research and reasoning; imagination and the capacity for abstraction; and rigor and accuracy.

From Grades 3 to 5, in the four content areas of the curriculum—Numbers and Calculation; Geometry; Quantities and Measures; and Organization and Management of Data—students enrich their knowledge, acquire new tools, and continue to learn how to solve problems. They strengthen their mental mathematics skills and acquire new arithmetic facts. Students learn mathematical procedures and develop an understanding of the associated concepts.

The mastery of basic elements of mathematics helps students in everyday life and prepares them for further studies in lower secondary school. Student objectives in mathematics in Grades 3 to 5 comprise the following:

Numbers and Calculation—Students study numbers systematically up to 1 billion, but may encounter higher numbers
- Natural whole numbers
  - Principles of decimal place value
  - Reading and writing numbers in figures and in words
  - Comparing and ordering numbers, locating numbers on a graduated line, and using the signs > and <
  - Arithmetic relationships between commonly used numbers (e.g., double, half, quadruple, quarter, triple, and third) and the concept of multiples
- Decimal numbers and fractions
  - Simple and decimal fractions—Writing fractions, locating fractions between two consecutive integers, writing fractions as the sum of an integer and a fraction less than 1, and calculating the sum of two decimal fractions or the sum of two fractions with the same denominator
  - Decimal numbers—Reading and writing decimals, determining decimal place value, converting between decimals and fractions, comparing, ordering, and locating decimals on a number line, and rounding decimals to the nearest whole number, the nearest 10^th, and the nearest 100^th
- Calculation
  - Mental—Learning addition and multiplication tables; daily exercises in mental calculation using the four operations help students develop number sense
  - Written—The mastery of an arithmetic technique for each of the four operations is essential
  - Calculator—Students learn to use calculators depending on the computational complexity of the problems they are solving
- Solving problems in real life can deepen students’ knowledge of numbers, strengthen their understanding and mastery of operations, and help them develop a rigorous work ethic and an appreciation of reasoning
Geometry—The main objective of teaching geometry in Grades 3 to 5 is to allow students to move progressively from a perceptive recognition of objects to study based on the use of instruments for tracing and measuring
- Geometric relationships and geometric properties—Alignment, perpendicularity, parallelism, equality of lengths, axial symmetry, middle of a segment
- The use of instruments and techniques—Ruler, square, compass, tracing paper, graph paper, dotted paper, and folding paper
- Plane figures—Squares, rectangles, diamonds, parallelograms, triangles, special triangles, and circles
  - Describing, reproducing, and constructing plane figures
  - Learning vocabulary specific to plane figures (e.g., side, top, angle, diagonal, axis of symmetry, center, radius, and diameter)
  - Enlarging and reducing plane figures, linked with proportionality
- Regular solids (i.e., cubes, rectangular parallelepipeds, cylinders, prisms, and pyramids)
  - Recognizing regular solids and studying certain patterns
  - Learning vocabulary specific to regular solids (e.g., vertex, edge, and face)
- Reproducing and constructing various geometric configurations requires students to apply their knowledge of regular figures and gives students the opportunity to use specific vocabulary and approaches to measuring and drawing
Quantities and Measures
- Length, mass, and volume—Measurement, estimation, legal units of the metric system, calculation of quantities, converting between units of measurement, perimeter of a polygon, formulas for the perimeter of a square and of a rectangle, circumference of a circle, volume of a rectangular parallelepiped
- Area—Comparing surfaces according to their area, regular units, conversions; formulas for area of a rectangle and of a triangle
- Angles—Comparing angles, using templates and set squares; acute, obtuse, and right angles
- Time—Reading clocks and calendars
- Duration—Units of measurement of duration, calculating time elapsed between two given moments
- Money
- Solving concrete problems helps to consolidate knowledge and skills related to quantities and their measurement and to reinforce conceptual understanding; in this context, measurement estimates may be provided and then validated
Organization and Management of Data
- Organization and data management skills are developed by solving problems in everyday life or in other subject domains; students gradually learn to sort data, to classify data, and to read, produce, and analyze tables and graphs
- Proportionality is taught in contexts involving percent, scale, unit conversion, and enlargement or reduction of figures, and using several different procedures (especially the rule of three)

In mathematics, the Common Base of Knowledge and Skills states that at the end of primary school Cycle 3, each student should be able to:

Write, name, compare, and use whole numbers, decimal numbers (up to hundredths), and some simple fractions
Reproduce addition and multiplication tables from 2 to 9
Perform the four operations on whole numbers and decimals (where the divisor is an integer in division)
Mentally calculate using the four operations
Assess the estimate of a result
Use a calculator
Recognize, describe, and name regular figures and solids
Use a ruler, a set square, and a compass to verify the properties of regular plane figures and construct figures with care and precision
Use regular units of measurement, use measuring instruments, and convert between units of measurement
Solve problems involving the four operations, proportionality, and different elements of mathematics (e.g., numbers, measurements, the rule of three, geometric figures, and diagrams)
Organize digital or geometric information, and justify and assess the likelihood of a result
Read, interpret, and construct simple representations (e.g., tables and graphs)