Curricular approaches to algebra in Estonia, Finland and Sweden – a comparative study

Figures & data

Table 1. The subthemes within the big ideas

EEEI	GA	FT	PR	VAR
Equivalence, inequalities and equations	Relations between operations	Patterns	Proportional relationships	Unknown number
Handling of expressions	Arithmetic rules	Functional relations	Specific applications of proportional reasoning	Variable
Modeling and solving word problems		Tables and graphs	Percentage

Table 2. Distribution of the topics connected to the subtheme Equivalence, inequalities and equations across Grades 1–9

	Estonia	Finland	Sweden
1-2/3	Finding numerical value of a letter in equations on the basis of trying or on the basis of analogy.		The importance and the use of the equal sign in a relevant manner. Handling simple mathematical equalities.
3/4–6	The idea of equation. Solving simple equations.	Solving and examining equations by reasoning and experimentation.	Solving simple equations.
7-9	The concept of equation and its formal solving methods. Using linear equations and equation systems to model and solve word problems. Solving second-degree equations. Solving pairs of equations graphically and algebraically.	Forming and solving first-degree equations. Forming and solving incomplete second-degree equations. Solving pairs of equations graphically and algebraically. Solving first-degree inequalities.	Solving equations.

Table 3. Distribution of the topics connected to the subtheme Handling of expressions across Grades 1–9

	Estonia	Finland	Sweden
1-2/3	Determining correct order of operations in expressions (parentheses, mult/div, add/subtr).
3/4-6	Simplifying one-variable expressions. Calculating the value of letter expressions.		Simple algebraic expressions.
7-9	The concept of algebraic fraction. Operations with algebraic fractions. Formulas for difference of squares, squares of sums and differences . The concepts of monomial and multinomial. Add, subtract, multiply and divide monomials and multinomials. Simplify rational expressions.	The concept of variable. Calculating the value of a mathematical expression. Forming and reducing exponential expressions. The concept of polynomial. Addition, subtraction and multiplication of polynomials.	Algebraic expressions. Algebraic formulas.

Table 4. Distribution of the topics connected to the subtheme Modeling and solving word problems across Grades 1–9

	Estonia	Finland	Sweden
1-2/3	Representing internal quantitative relationships in word problems and expressing these relationships numerically. Solving word problems with one and two operations.
3/4-6	Solving and composing word problems with several operations, and checking and evaluating the result.		Equations in situations relevant to students. Equations in situations relevant to students.
7-9	Using linear equations and equation systems to model and solve word problems.	Modeling and problem solving. Forming and solving first-degree equations and incomplete second-degree equations. Using algorithmic thinking and skills in applying mathematics and programming in problem solving.

Table 5. Distribution of the topics connected to the subthemes of the big idea GA across Grades 1–9

		Estonia	Finland	Sweden
Relations between operations	1-2/3	Relation between addition and subtraction, subtraction defined as reverse of addition. Relations between multiplication and division, division defined as reverse of multiplication. Relation of multiplication to addition.	Principles and features of basic arithmetic operations. Relation between multiplication and division.	Relations between the four arithmetical operations.
	3/4-6	Relations between components in arithmetic operations and the results of operations.	Relations between operations.
	7-9
Arithmetic rules	1-2/3	Rules of mental and written arithmetic.	The commutative and the associative law in addition and multiplication.	Properties of arithmetic operations.
	3/4-6	Properties of operations. Formulating and applying divisibility properties (div by 2, 3, 5, 9, and 10).	Mental and written arithmetic using the properties of operations.
	7-9

Table 6. Distribution of the topics connected to the subthemes Patterns (highlighted in gray in the Table) and Functional relations across Grades 1–9

	Estonia	Finland	Sweden
1-2/3		Finding regularities.	Describing and expressing simple patterns in number sequences and simple geometrical forms. Constructing simple patterns in number sequences and simple geometrical forms.
3/4-6		Observing the regularities of number sequences. Continuing a number sequence following its rule.	Describing and expressing patterns in number sequences and simple geometrical forms. Constructing patterns in number sequences and simple geometrical forms.
7-9	Introduction of the function concept. Linear and quadratic functions with their graphs. Dependence of position and form of functions’ graphs on the coefficient in the function’s expression. The meaning of zeros of a function and finding zeros on graphs and formulas. The involution rules.	Deepening skills in examining and forming number sequences. The concept of function. Interpreting and producing a graph of first- and second-degree functions. Correlations depicted both graphically and algebraically. Direct and inverse proportionality. The concept of angular coefficient (slope) and constant term. Determining the zeros of a function.	Functions and linear equations. Using functions to investigate change, rate of change and other relations.

Table 7. The distribution of the topics connected to the subtheme Proportional relationships across Grades 1–9

	Estonia	Finland	Sweden
1-2/3			Different proportional relationships, including doubling and halving. Using and giving examples of simple proportional relationships in situations relevant to the students.
3/4-6			Graphs for expressing different types of proportional relations in simple investigations.
7-9	Proportional and inversely proportional relations. Proportional equations and decompositions. Meaning of proportional dependence based on real-life examples.	Using proportions in problem solving. Direct and inverse proportionality.

Table 8. Distribution of the topics connected to the subthemes of VAR across Grades 1–9

		Estonia	Finland	Sweden
Unknown number	1-2/3	The symbolic representation of the unknown.
	3/4-6		The concept of the unknown.	Unknown numbers and their properties. Situations in which there is a need to represent an unknown number with a symbol.
	7-9		The concept of the unknown.
Variable	1-2/3	The concepts of number and variable as its representation. Using letters to express the relationships between quantities.
	3/4-6	Simplifying expressions with one variable. Calculating the value of letter expressions.
	7-9	Variable quantity and function.	The concept of variable.	The meaning of the variable concept. The use of variable in algebraic expressions, formulas and equations.

Figure 1. Summarizing Tables 2–8

Figure 2. The number of different topics expressed within EEEI in each country

Figure 3. The number of the different topics connected to Patters and Functional relations (Table 6) and Proportional relationships (Table 7) across grades 1–9