The Mathematics Curriculum in Primary and Lower Secondary Grades
Both the primary and the lower secondary mathematics curricula in Israel have undergone intensive revision over the last decade. In 2002, a committee8 was appointed to examine the teaching of mathematics at all grade levels and to recommend improvements. The committee identified several deficiencies related to the teaching of mathematics, including a lack of sufficient instruction on solving word problems, a tendency to narrow the curriculum to topics that appear on matriculation examinations, and a lack of sufficient geometry instruction (compared to algebra) in lower secondary school. In addition, the committee identified the sectorial as well as socioeconomic achievement gaps between subpopulations in Israel as indicators of low achievement. At approximately the same time the committee’s recommendations were published, other committees9,10 appointed to examine the pedagogical approaches reflected in learning materials concluded that the most common instructional approach emphasized arithmetic skills and neglected mathematical reasoning and that the curricula for primary and secondary mathematics education were not connected in a learning progression. In addition, the committees found the conditions for teaching mathematics to be inadequate: classrooms were crowded and populated with students at disparate levels, teachers lacked sufficient subject knowledge, and there were not enough supervisors in the field.
In 2003, an intervention program, The Quantitative Reasoning Program, was implemented in approximately half of lower secondary schools in Israel. This intervention emphasized the study of six topics related to quantitative thinking, establishing a link between the mathematics taught at the primary and lower secondary levels, and improving student ability to solve mathematical problems that require the integration of skills learned in other school subjects. The intervention program went through several cycles of revision, and was the first step in the development of a new curriculum for primary schools introduced in 2006.
The goal of the new curriculum at the primary level is for students to learn basic concepts and structures in the numbers and geometry domains, as well as develop mathematical skills and abilities, such as number sense and geometric insight, computational skills, the ability to use mathematical tools to solve word problems, and conceptual understanding and knowledge of mathematical language.11 Attaining mathematical concepts is considered a cumulative process dependent on students’ ability to grasp mathematical concepts and link them to other school subjects and the real world. The primary mathematics school curriculum, up to fourth grade, includes the following topics:
- Numbers and Operation, and Data Investigation—Natural numbers, the four arithmetic operations (addition, subtraction, multiplication, and division), fractions (arithmetic operations, common denominators, and decimal equivalents), percentages, and proportion
- Geometry and Measurement—Geometric shapes (two-dimensional shapes, such as polygons, triangles, squares, and rectangles, and three-dimensional shapes, such as rectangular prisms); measurement of length, area, volume, angles, etc.; transformations and symmetry; properties of quadrilaterals; circles; classification of three-dimensional shapes; and computation of volume
In 2009, a new mathematics curriculum was developed at the lower secondary level. The curriculum integrates the mathematical knowledge learned in primary school with new and more advanced topics in lower secondary school, and uses a spiral approach to curricular planning to expand on topics previously taught. The spiral sequencing allows students to return to basic ideas as new subjects continually, and concepts are added over the course of a curriculum in order to solidify understanding over periodic intervals. The curriculum merges three domains—Numbers, Algebra, and Geometry—and cultivates student ability to use multidomain problem solving methods. The new curriculum is intended to include at least 150 instructional hours in each grade, and includes recommendations concerning the allocation of instructional hours to help teachers with planning. The allocation of instructional hours for mathematics content domains is presented in Exhibit 1.
Exhibit 1: Instructional Hours for Mathematics, Grades 7–9
| Grade | Content Domain | Instructional Hours |
| Grade 7 | Algebra | 70 |
| Numbers | 30 | |
| Geometry | 50 | |
| Grade 8 | Algebra | 70 |
| Numbers | 50 | |
| Geometry | 30 | |
| Grade 9 | Algebra | 65 |
| Numbers | 15 | |
| Geometry | 70 |
The main mathematics topics are distributed over the three grade levels of lower secondary school, as presented in Exhibit 2 (subjects for Grade 8 are described in greater detail).12
Exhibit 2: Mathematics Topics, Grades 7–9
| Grade | Content Domain | Percent of Curriculum | Topics |
| Grade 7 | Numbers | 20% | Negative numbers, fractions and decimals, and the Cartesian plane |
| Algebra | 45% | Patterns, algebraic expressions, and equations | |
| Geometry | 35% | Formulas for perimeter and area (rectangles, triangles, parallelograms, rhombuses, and trapezoids), volume and surface area (cubes and boxes), and measurements and concepts related to angles |
|
| Grade 8 | Numbers | 32% | Ratio and proportion, percentage, descriptive statistics, chance, real numbers, and square roots. Ratio is a major theme later in Grade 8, and includes a variety of topics (scale, proportion, similarity of triangles, the slope of a straight line, linear functions of the form y = mx, percent, relative frequency, and probability). |
| Algebra | 48% | Algebraic expressions; linear functions; solving equations, inequalities, and simultaneous equations; and using linear functions to solve problems. The concept of linear functions lays a foundation for studying methods for solving linear equations, systems of linear equations with two unknowns, inequalities, equations with absolute values, and word problems whose solutions involve these methods. | |
| Geometry | 20% | Geometric shapes and Euclidean geometry, including triangle congruence, properties of isosceles triangles, similar triangles, the Pythagorean theorem (including its use in the coordinate plane), theorems, and proofs. The key concepts taught in eighth grade geometry are isosceles triangles and similar triangles, which serve as a basis for further studies of deductive proof in geometry. Attention is directed mainly toward enhancing student awareness of the correctness and logic of statements of congruence and similarity, and why the conditions set forth in congruence and similarity theorems are necessary and sufficient. | |
| Grade 9 | Numbers | 44% | Fractions in algebraic expressions, multiplication, exponentiation, quadratic functions, and quadratic equations |
| Algebra | 10% | Chance and graphs | |
| Geometry | 46% | Euclidean geometry, including triangles (isosceles, equilateral, and right triangles), quadrilaterals, and circles |