Page 472 - 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
2022; Wahyuni and Herman, 2020; Carpenter et al. 2003; McNeil et al. 2006;
Stephens et al. 2013), and one of the very foundations for learning mathe-
matics, because there is no branch of mathematics that does not use it (Car-
penter et al. 2003; McNeil et al. 2006); especially, it is the essential concept for
algebra achievement (Van Amerom 2003; Chimoni, Pitta-Pantazi, and Chris-
tou 2018). It is particularly emphasized that the failure to grasp the meaning
of the equals sign in early grades can be a cause of learning difficulties later
in life, even at a higher level of education (Baiduri 2015; Knuth et al. 2006).
However, the authors also agree that, in addition to being an important
concept for understanding mathematics, understanding the equals sign is
very challenging for most students at all levels of mathematics education
(Alibali et al. 2007; Kieran 1981) and that there are numerous problems plagu-
ing its correct understanding (Carpenter and Levi 2000; Filloy and Rojano
1989; Freiman and Lee 2004; Kieran 1981; Knuth et al. 2006; McNeil and Al-
ibali 2005; Sfard and Linchevski 1994), which can later be reflected in the
learning of other mathematical content, especially algebra. ‘The difficulties
related to the misunderstanding of the equals sign in algebra stem from the
precarious foundations on which said concept is formed in arithmetic learn-
ing’ (Milinković, Maričić, and Ðokić 2022, 27). As pointed out by Ardiansari,
Suryadi, and Dasari, all the above shows that ‘understanding the equal sign
has become the focal point of reform and research efforts in mathematics
education as a proven strong and long-term “problem” that can be experi-
enced by elementary, middle school and even college students’ (Ardiansari,
Suryadi, and Dasari 2022, 95).
In the light of the considerations stated above, the paper will focus on ex-
amining students’ understanding of the equals sign.
Understanding the Concept of the Equals Sign
Mathematical equivalence is the relation between two quantities that are
equal and interchangeable (McNeil et al. 2015; Kieran 1981). The concept of
the equals sign implies the unity of symbolic notation and the meaning ex-
pressed through equality and equivalence. Numerous studies show that the
majority of students do not fully understand the equals sign and that there
are numerous problems that arise with regard to the equals sign and their
ability to understand it (Booth 1988; Kieran 1981; Sfard and Linchevski 1994,
Filloy and Rojano 1989; Carpenter and Levi 2000;). Research shows that stu-
dents view and understand the equals sign as a result of arithmetical oper-
ation rather than mathematical equality (Kieran 1981; Stephens et al. 2013;
Vermeulen and Meyer 2017). Falkner, Levi, and Carpenter (1999) showed that
472
2022; Wahyuni and Herman, 2020; Carpenter et al. 2003; McNeil et al. 2006;
Stephens et al. 2013), and one of the very foundations for learning mathe-
matics, because there is no branch of mathematics that does not use it (Car-
penter et al. 2003; McNeil et al. 2006); especially, it is the essential concept for
algebra achievement (Van Amerom 2003; Chimoni, Pitta-Pantazi, and Chris-
tou 2018). It is particularly emphasized that the failure to grasp the meaning
of the equals sign in early grades can be a cause of learning difficulties later
in life, even at a higher level of education (Baiduri 2015; Knuth et al. 2006).
However, the authors also agree that, in addition to being an important
concept for understanding mathematics, understanding the equals sign is
very challenging for most students at all levels of mathematics education
(Alibali et al. 2007; Kieran 1981) and that there are numerous problems plagu-
ing its correct understanding (Carpenter and Levi 2000; Filloy and Rojano
1989; Freiman and Lee 2004; Kieran 1981; Knuth et al. 2006; McNeil and Al-
ibali 2005; Sfard and Linchevski 1994), which can later be reflected in the
learning of other mathematical content, especially algebra. ‘The difficulties
related to the misunderstanding of the equals sign in algebra stem from the
precarious foundations on which said concept is formed in arithmetic learn-
ing’ (Milinković, Maričić, and Ðokić 2022, 27). As pointed out by Ardiansari,
Suryadi, and Dasari, all the above shows that ‘understanding the equal sign
has become the focal point of reform and research efforts in mathematics
education as a proven strong and long-term “problem” that can be experi-
enced by elementary, middle school and even college students’ (Ardiansari,
Suryadi, and Dasari 2022, 95).
In the light of the considerations stated above, the paper will focus on ex-
amining students’ understanding of the equals sign.
Understanding the Concept of the Equals Sign
Mathematical equivalence is the relation between two quantities that are
equal and interchangeable (McNeil et al. 2015; Kieran 1981). The concept of
the equals sign implies the unity of symbolic notation and the meaning ex-
pressed through equality and equivalence. Numerous studies show that the
majority of students do not fully understand the equals sign and that there
are numerous problems that arise with regard to the equals sign and their
ability to understand it (Booth 1988; Kieran 1981; Sfard and Linchevski 1994,
Filloy and Rojano 1989; Carpenter and Levi 2000;). Research shows that stu-
dents view and understand the equals sign as a result of arithmetical oper-
ation rather than mathematical equality (Kieran 1981; Stephens et al. 2013;
Vermeulen and Meyer 2017). Falkner, Levi, and Carpenter (1999) showed that
472
