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The Equals Sign: The Challenges of Learning Arithmetic
most students perceive the meaning of the equals sign as an operational
symbol, calculating the number from the left-hand side to the right. The rea-
son for this understanding of the equals sign, or lack thereof, among students
arises from the manner in which this concept is learned in arithmetic. Stu-
dents first encounter this concept in arithmetic, usually in situations where
there is an expression on the left followed by an equals sign, and they are
expected to calculate, i.e. determine its value. Consequently, research shows
that students largely view the equals sign as the command ‘to calculate’ or
‘to compute’ (Baroody and Ginsburg 1983; Behr, Erlwanger, and Nichols 1980;
Cobb 1987; Ilič and Zeljič 2017; Kieran 1981; McNeil and Alibali 2005), i.e. they
interpret it as a command to perform arithmetic calculations, or ‘see “=” as an
instruction to complete an operation’ (Parslow-Williams and Cockburn 2008,
36). Kieran (1981) also draws attention to the fact that the use of the equals
sign in arithmetic classes steers students toward a wrong conceptualization
of this concept. It is crucial that students develop and understand the equals
sign both operationally in terms of performing certain operations, i.e. com-
mands ‘to calculate’ and ‘to compute’ and relationally, in terms of equivalence
of the left and the right side of the equality, at an early age (Kieran 1981; Knuth
et al. 2006; McNeil et al. 2006).
The difficulties reflected in mathematics education, and referring to a nar-
row understanding of the equals sign, i.e. in operational but not in relational
terms, stem from the fact that children are first introduced to arithmetic, at
the very outset of their mathematics education. The students’ understand-
ing of the equals sign in the operational sense is related to the habit that the
expression is always located on the left side of the equality, while its results
are located on the right. Let us take 5 + 6 = ___ as an example, because it
possesses a typical arithmetic interpretation of the equals sign. In the given
example, students naturally feel the need to calculate the sum of 5 and 6, and
write the number 11 in the blank. However, when they are presented with the
following problem: 5 + 6 = ___ + 2, they must understand the equivalence
between the left and the right side of the equality in order to successfully
solve the given problem. Research shows that using problems that deviate
from the standard form of equality in arithmetic, which implies that the ex-
pression is always on the left side of the equality (for example, ___ = 5 + 6),
increases the likelihood of developing relational understanding of the equals
sign (McNeil and Alibali 2005; McNeil et al. 2011). Solving problems with a non-
standard format helps students improve their understanding of the equals
sign, especially if that process involves the use of words that refer to the pri-
mary meaning of the equals sign, such as ‘equals to’ or ‘is the same as’ (Rittle-
473
most students perceive the meaning of the equals sign as an operational
symbol, calculating the number from the left-hand side to the right. The rea-
son for this understanding of the equals sign, or lack thereof, among students
arises from the manner in which this concept is learned in arithmetic. Stu-
dents first encounter this concept in arithmetic, usually in situations where
there is an expression on the left followed by an equals sign, and they are
expected to calculate, i.e. determine its value. Consequently, research shows
that students largely view the equals sign as the command ‘to calculate’ or
‘to compute’ (Baroody and Ginsburg 1983; Behr, Erlwanger, and Nichols 1980;
Cobb 1987; Ilič and Zeljič 2017; Kieran 1981; McNeil and Alibali 2005), i.e. they
interpret it as a command to perform arithmetic calculations, or ‘see “=” as an
instruction to complete an operation’ (Parslow-Williams and Cockburn 2008,
36). Kieran (1981) also draws attention to the fact that the use of the equals
sign in arithmetic classes steers students toward a wrong conceptualization
of this concept. It is crucial that students develop and understand the equals
sign both operationally in terms of performing certain operations, i.e. com-
mands ‘to calculate’ and ‘to compute’ and relationally, in terms of equivalence
of the left and the right side of the equality, at an early age (Kieran 1981; Knuth
et al. 2006; McNeil et al. 2006).
The difficulties reflected in mathematics education, and referring to a nar-
row understanding of the equals sign, i.e. in operational but not in relational
terms, stem from the fact that children are first introduced to arithmetic, at
the very outset of their mathematics education. The students’ understand-
ing of the equals sign in the operational sense is related to the habit that the
expression is always located on the left side of the equality, while its results
are located on the right. Let us take 5 + 6 = ___ as an example, because it
possesses a typical arithmetic interpretation of the equals sign. In the given
example, students naturally feel the need to calculate the sum of 5 and 6, and
write the number 11 in the blank. However, when they are presented with the
following problem: 5 + 6 = ___ + 2, they must understand the equivalence
between the left and the right side of the equality in order to successfully
solve the given problem. Research shows that using problems that deviate
from the standard form of equality in arithmetic, which implies that the ex-
pression is always on the left side of the equality (for example, ___ = 5 + 6),
increases the likelihood of developing relational understanding of the equals
sign (McNeil and Alibali 2005; McNeil et al. 2011). Solving problems with a non-
standard format helps students improve their understanding of the equals
sign, especially if that process involves the use of words that refer to the pri-
mary meaning of the equals sign, such as ‘equals to’ or ‘is the same as’ (Rittle-
473
