Comparison of lower secondary school mathematics standards

One of the project’s aims is to show that, in Europe, it is possible to design a curriculum for lower secondary school mathematics teacher training that, despite the differences between teacher training systems throughout Europe, includes a relevant common set of topics to be introduced to trainees.

Before trying to show this fact, it is, however, necessary to investigate which are the standards for mathematics education at lower secondary school level in the different European countries. If they turn out to be quite different, any attempt to outline a European curriculum will be hard or even impossible. In this chapter standards accepted in the project’s partner countries are compared.

The analysis of the tables below actually show that, despite some differences between the input standards (that is the output primary schools standards), there is little difference, as expected, between the output lower secondary school mathematics standards accepted in the project’s partner countries.

TOPICS	Czech Republic	Denmark	France	Italy	Slovak Republic
Number systems (operations incl.)
Rational numbers	+	+	+	+	+
Fractions	+	+	+	+	+
Decimal numbers	+	+	+	+	+
Real numbers	+	+	+	+	+
Powers	+	+	+	+	+
Roots	+	+	+	+	+
Proportionality
Percent	+	+	+	+	+
Ratio	+	+	+	+	+
Proportionality, rule of three	+	+	+	+	+
Divisibility
Multiple and divisor	+	+	+	+	+
Prime numbers	+	+	-	+	+
GCD	+	-	+	+	+
LCM	+	-	-	+	+
Factorization	+	+	-	+	+
Expressions
Numerical and algebraic expressions	+	+	+	+	+
Polynomials	+	+	-	+	+
Rational expressions	+	+	-	+	+
Equations, Inequalities
Expressions	+	+	+	+	+
Linear equation	+	+	+	+	+
Quadratic equations	+	-	-	+	+
Linear inequalities		+	+	+	+
Systems of linear equations	+	+	+	-	+
Functions
Coordinate system	+	(primary)	+	+	+
Properties of functions	+	+	-	-	+
Direct proportion	+	+	+	+	+
Indirect proportion	+	+	+	+	+
Linear function	+	+	+	+	+
Quadratic function	+	-	-	+	+
Trigonometric functions	+	-	-	-	+
Basic 2D notions
Point, straight line, plane	+	+	+	+	+
Half-line, segment, half-plane, angle	+	+	+	+	+
Circle, circumference	+	+	+	+	+
Triangle, quadrilateral, polygon	+	+	+	+	+
Sets of points of given property	+	+	+	+	+
Trigonometry in rectangular triangle	+	-	+	+	+
Basic solids
Polyhedron	+	-	+	+	+
Cube, cuboid, prism	+	+	+	+	+
Pyramid	+	+	+	+	+
Sphere, cylinder, cone	+	+	+	+	+
Geometric mappings
Congruence of geometric figures	+	+	-	+	+
Similarity of geometric figures	+	+	-	+	+
Point symmetry, line reflection, translation	+	+	+	+	+
Constructions	+	+	+	+	+
Measurement	+	+	+	+	+

Output lower secondary school mathematics standards

Detailed descriptions for each project’s partner country can be found by following the links below:

Further topics included in standards

The following topics are not explicitly mentioned in all standards, but they are implicitly present in all mathematical educational systems at the metacognitive^[1]level. Except Statistical probability they belong to both content knowledge and metacognitive domain. The metacognitive level is not explicitly stated.

TOPICS	Czech Republic	Denmark	France	Italy	Slovak Republic
Data representation and organization	-	+	+	+	+
Estimates	-	+	+	+	-
Possibilities and limitations in using mathematics as a description and a basis for decisions	-	+	+	-	-
Statistical probability	-	+	-	-	+
Communication and problem solving	-	+	+	-	-

Further topics included in standards

Metacognition combines various attended thinking and reflective processes. It can be divided into five primary components:

preparing and planning for learning,
selecting and using learning strategies,
monitoring strategy use,
orchestrating various strategies,
evaluating strategy use and learning.

Teachers should model strategies for learners to follow in all five areas. To be effective, metacognitive instruction should explicitly teach students a variety of learning strategies and also when to use them^[2].

Some theories underpinning learning strategy research:

O’Malley and Chamot (1990)^[3] classify strategies in the following way:
- cognitive strategies;
- metacognitive strategies;
- social strategies;
- affective strategies.

Rebecca Oxford (1990)^[4] distinguishes:
- Direct Strategies (memorizing, cognitive processing, compensation);
- Indirect Strategies (metacognitive, social and affective).

¹ In education, the term metacognition can be defined as "awareness of one's own knowledge or problem-solving abilities".
²Anderson, J. (2002). The Role of Metacognition in Second Language Teaching and Learning. Available at [http://www.cal.org/resources/digest/0110anderson.html].
³ O’Malley, J.M. & Chamot, A.U. (1990). Learning Strategies in Second Language Acquisition. Cambridge University Press.
⁴ Oxford, R.L. (1990). Language Learning Strategies: What Every Teacher Should Know. Newbury House.