Page 474 - Istenič Andreja, Gačnik Mateja, Horvat Barbara, Kukanja Gabrijelčič Mojca, Kiswarday Vanja Riccarda, Lebeničnik Maja, Mezgec Maja, Volk Marina. Ur. 2023. Vzgoja in izobraževanje med preteklostjo in prihodnostjo. Koper: Založba Univerze na Primorskem
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ja Maričić, Nenad Milinković, and Marija Brković
Johnson and Alibali 1999), or the expressions ‘the left and the right side of
the equals sign have the same meaning’ and ‘same as’ (Knuth et al. 2006). In
addition to the concept of sameness in the relational sense, researchers also
propose using expressions that signify interchangeability instead of same-
ness, such as ‘the left and the right side are interchangeable’ (Jones et al. 2012;
Jones et al. 2013). The mathematical meaning of the equals sign referring to
interchangeability shows an analogy in the students’ cognitive understand-
ing, and a more sophisticated interpretation of this concept.
Although operational understanding may be sufficient in solving classical
elementary arithmetic problems, misconceptions about the meaning of the
equals sign can later lead to problems with solving equations (Alibali et al.
2007; Carpenter et al. 2003; Prediger 2009). Developing an understanding of
the symmetrical nature of equality requires one to grasp the fact that a math-
ematical symbol fundamentally expresses a relationship between quantities,
and that it should not be viewed merely as a signal for performing mathemat-
ical operations.
The ultimate goal that is being pursued is that the students’ understanding
of the concept of the equals sign ends at the relational level, i.e. the level that
expresses equivalence. According to Byrd et al. (2015), without relational un-
derstanding, the algebraic principle of maintaining equality is meaningless,
wherein children are left to struggle and memorize countless arbitrary rules
in equation transformations. In order for children to understand the equals
sign as ‘it is equivalent’ or ‘it is the same as,’ they first need to understand ex-
pressions and equalities as a whole, i.e. understand the equals sign relation-
ally. Thus, research by Molina and Ambrose (2008) confirms that the expres-
sion or equation must be observed as an entity of its own. Only through the
unity of the elements in the expression is it possible to influence the correct
notions about the equals sign, and thus, about the correct transformation of
the equation in its solution.
The solution to this problem requires comprehension ‘that the equals sign
lies in the deeper understanding of this concept and developing an under-
standing of this concept at the relational, instead of just the operational level’
(Milinković, Maričić, and Lazić 2022, 98). Accomplishing this task is a complex
process for teachers, so it must be systematic and gradual. A more detailed
operationalization of the levels of understanding of the equals sign proposed
by the following authors (Jones et al. 2013; Lee and Pang 2020; McAuliffe et
al. 2020; Milinković, Maričić, and Lazić 2022; Rittle-Johnson et al. 2011) can be
extremely helpful in this process. The levels of understanding of the equals
sign are shown in table 1.
474
Johnson and Alibali 1999), or the expressions ‘the left and the right side of
the equals sign have the same meaning’ and ‘same as’ (Knuth et al. 2006). In
addition to the concept of sameness in the relational sense, researchers also
propose using expressions that signify interchangeability instead of same-
ness, such as ‘the left and the right side are interchangeable’ (Jones et al. 2012;
Jones et al. 2013). The mathematical meaning of the equals sign referring to
interchangeability shows an analogy in the students’ cognitive understand-
ing, and a more sophisticated interpretation of this concept.
Although operational understanding may be sufficient in solving classical
elementary arithmetic problems, misconceptions about the meaning of the
equals sign can later lead to problems with solving equations (Alibali et al.
2007; Carpenter et al. 2003; Prediger 2009). Developing an understanding of
the symmetrical nature of equality requires one to grasp the fact that a math-
ematical symbol fundamentally expresses a relationship between quantities,
and that it should not be viewed merely as a signal for performing mathemat-
ical operations.
The ultimate goal that is being pursued is that the students’ understanding
of the concept of the equals sign ends at the relational level, i.e. the level that
expresses equivalence. According to Byrd et al. (2015), without relational un-
derstanding, the algebraic principle of maintaining equality is meaningless,
wherein children are left to struggle and memorize countless arbitrary rules
in equation transformations. In order for children to understand the equals
sign as ‘it is equivalent’ or ‘it is the same as,’ they first need to understand ex-
pressions and equalities as a whole, i.e. understand the equals sign relation-
ally. Thus, research by Molina and Ambrose (2008) confirms that the expres-
sion or equation must be observed as an entity of its own. Only through the
unity of the elements in the expression is it possible to influence the correct
notions about the equals sign, and thus, about the correct transformation of
the equation in its solution.
The solution to this problem requires comprehension ‘that the equals sign
lies in the deeper understanding of this concept and developing an under-
standing of this concept at the relational, instead of just the operational level’
(Milinković, Maričić, and Lazić 2022, 98). Accomplishing this task is a complex
process for teachers, so it must be systematic and gradual. A more detailed
operationalization of the levels of understanding of the equals sign proposed
by the following authors (Jones et al. 2013; Lee and Pang 2020; McAuliffe et
al. 2020; Milinković, Maričić, and Lazić 2022; Rittle-Johnson et al. 2011) can be
extremely helpful in this process. The levels of understanding of the equals
sign are shown in table 1.
474
