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The Equals Sign: The Challenges of Learning Arithmetic
Table 1 Levels of Understanding of the Equals Sign
Level of understanding Expected outcomes
Level : Real relational Student understands equivalence in real-world context prob-
Level : Complex relational lems.
Level : Basic relational
Student understands equivalence in complex equalities that
Level : Operational feature multiple equals signs.
Student understands the equals sign as a symbol of equiv-
alence in equalities that feature expressions on both sides
of the equality. Student uses relational thinking and under-
stands equivalence in simple equalities.
Student understands the equals sign as a command ‘to calcu-
late.’ Student understands simple equalities that feature ex-
pressions on both sides of the equals sign.
Notes Adapted from Milinković, Maričić, and Lazić (2022, 98).
Level 1 implies operational understanding of equality where the student
views the equals sign solely in the function of performing a calculation, i.e.
as the command ‘to calculate.’ Level 2 implies a somewhat more advanced
understanding of the equals sign. The development of this level reflects the
student’s ability to understand equivalence, and not understand the equals
sign solely in an operational sense. At this level, the student views the equals
sign as a mathematical equivalence, in other words, as a sign that signifies the
equivalence of the left and the right side of the equation. Level 3 involves stu-
dents’ understanding of complex equalities in which the equals sign occurs
repeatedly (Lee and Pang 2020; Milinković, Maričić, and Lazić 2022). Under-
standing the equals sign at a complex relational level involves examples of
equations where the equals sign occurs repeatedly as a link between expres-
sions (e.g. 2 + 3 = ___ – 1 = 9 – ___ = ___ ). Level 4 involves understanding of
the equals sign in the context of real world problem solving. ‘This involves
situations in which students are expected to solve specific problems using
the balance method, i.e. jumping from one side of the equation to the other’
(Milinković, Maričić, and Lazić 2022, 99).
All individual levels of understanding are interconnected, but it should be
considered ‘that the continuous nature of the model means that the levels
should not be interpreted as discrete stages’ (Rittle-Johnson et al. 2011, 3).
The development of one level of understanding does not exclude the devel-
opment of a different level, nor does the development of one level cancel
another. The concept of the equals sign formed at an operational level can
be gradually transformed into a relational meaning, which is what arithmetic
classes aim to accomplish, and which will later lay the foundations for learn-
475
Table 1 Levels of Understanding of the Equals Sign
Level of understanding Expected outcomes
Level : Real relational Student understands equivalence in real-world context prob-
Level : Complex relational lems.
Level : Basic relational
Student understands equivalence in complex equalities that
Level : Operational feature multiple equals signs.
Student understands the equals sign as a symbol of equiv-
alence in equalities that feature expressions on both sides
of the equality. Student uses relational thinking and under-
stands equivalence in simple equalities.
Student understands the equals sign as a command ‘to calcu-
late.’ Student understands simple equalities that feature ex-
pressions on both sides of the equals sign.
Notes Adapted from Milinković, Maričić, and Lazić (2022, 98).
Level 1 implies operational understanding of equality where the student
views the equals sign solely in the function of performing a calculation, i.e.
as the command ‘to calculate.’ Level 2 implies a somewhat more advanced
understanding of the equals sign. The development of this level reflects the
student’s ability to understand equivalence, and not understand the equals
sign solely in an operational sense. At this level, the student views the equals
sign as a mathematical equivalence, in other words, as a sign that signifies the
equivalence of the left and the right side of the equation. Level 3 involves stu-
dents’ understanding of complex equalities in which the equals sign occurs
repeatedly (Lee and Pang 2020; Milinković, Maričić, and Lazić 2022). Under-
standing the equals sign at a complex relational level involves examples of
equations where the equals sign occurs repeatedly as a link between expres-
sions (e.g. 2 + 3 = ___ – 1 = 9 – ___ = ___ ). Level 4 involves understanding of
the equals sign in the context of real world problem solving. ‘This involves
situations in which students are expected to solve specific problems using
the balance method, i.e. jumping from one side of the equation to the other’
(Milinković, Maričić, and Lazić 2022, 99).
All individual levels of understanding are interconnected, but it should be
considered ‘that the continuous nature of the model means that the levels
should not be interpreted as discrete stages’ (Rittle-Johnson et al. 2011, 3).
The development of one level of understanding does not exclude the devel-
opment of a different level, nor does the development of one level cancel
another. The concept of the equals sign formed at an operational level can
be gradually transformed into a relational meaning, which is what arithmetic
classes aim to accomplish, and which will later lay the foundations for learn-
475
