Page 485 - 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
P. 485
The Equals Sign: The Challenges of Learning Arithmetic
– Students possess a developed understanding of the equals sign at op-
erational level, and are more successful in solving tasks (problems) in
which the expression is located on the left side of the equals sign than
those in which it is located on the right;
– Only a third of the examined first-grade students possess a developed
relational understanding of the equals sign, whereas almost two thirds
of the third-graders understand the equals sign relationally;
– Very few among the examined first-graders possess a developed un-
derstanding of the equals sign at the complex relational level in con-
trast to one third of the third-graders who understand the equals sign
in a relational sense.
In view of the fact that students form the concept of the equals sign from
the earliest days of their formal education, the obtained results point out that
students do not fully develop the meaning of this symbol in the process, and
that its operational meaning is still dominant. The intensive application of
the equals sign in elementary mathematics education for teaching calculus
is obviously the main cause of insufficient relational understanding. This may
indicate that teaching has long since turned into a routine, focused solely on
calculating the value of expressions, as well as that it largely relies on expres-
sions located on the left side of the equals sign. Students’ poor performance
at the complex relational level showcases the lack of essential understand-
ing of the meaning of this concept which signifies ‘sameness,’ ‘equivalence,’
and ‘transitivity.’ Such results can lead to serious problems in later stages of
mathematics education. Considering research findings by Rittle-Johnson et
al. (2011), it appears that the equals sign develops at all four levels of under-
standing at the same time. This does not mean that the understanding of the
equals sign exclusively pertains to any single level, or that its development at
the said level is finished, which is also confirmed in this research. In contrast,
Matthews et al. (2012) believe that operational and relational understanding
of the equals sign develop side by side and are co-dependent. In view of this
fact, we can say that there must be an obvious need for continuous practice
and development of the concept of the equals sign at the individual level
until it is complete at the highest relational level.
The obtained results suggest that maths teachers should form a deeper
understanding of the equals sign as a symbol of equivalence between two
sides of the equality. Falkner, Levi, and Carpenter point out that ‘teachers
should also be concerned about children’s conceptions of equality as soon as
symbols for representing number operations are introduced’ (Falkner, Levi,
485
– Students possess a developed understanding of the equals sign at op-
erational level, and are more successful in solving tasks (problems) in
which the expression is located on the left side of the equals sign than
those in which it is located on the right;
– Only a third of the examined first-grade students possess a developed
relational understanding of the equals sign, whereas almost two thirds
of the third-graders understand the equals sign relationally;
– Very few among the examined first-graders possess a developed un-
derstanding of the equals sign at the complex relational level in con-
trast to one third of the third-graders who understand the equals sign
in a relational sense.
In view of the fact that students form the concept of the equals sign from
the earliest days of their formal education, the obtained results point out that
students do not fully develop the meaning of this symbol in the process, and
that its operational meaning is still dominant. The intensive application of
the equals sign in elementary mathematics education for teaching calculus
is obviously the main cause of insufficient relational understanding. This may
indicate that teaching has long since turned into a routine, focused solely on
calculating the value of expressions, as well as that it largely relies on expres-
sions located on the left side of the equals sign. Students’ poor performance
at the complex relational level showcases the lack of essential understand-
ing of the meaning of this concept which signifies ‘sameness,’ ‘equivalence,’
and ‘transitivity.’ Such results can lead to serious problems in later stages of
mathematics education. Considering research findings by Rittle-Johnson et
al. (2011), it appears that the equals sign develops at all four levels of under-
standing at the same time. This does not mean that the understanding of the
equals sign exclusively pertains to any single level, or that its development at
the said level is finished, which is also confirmed in this research. In contrast,
Matthews et al. (2012) believe that operational and relational understanding
of the equals sign develop side by side and are co-dependent. In view of this
fact, we can say that there must be an obvious need for continuous practice
and development of the concept of the equals sign at the individual level
until it is complete at the highest relational level.
The obtained results suggest that maths teachers should form a deeper
understanding of the equals sign as a symbol of equivalence between two
sides of the equality. Falkner, Levi, and Carpenter point out that ‘teachers
should also be concerned about children’s conceptions of equality as soon as
symbols for representing number operations are introduced’ (Falkner, Levi,
485
