# An Analysis of Misunderstanding in Mathematics: The Case of Meanings of Equal Sign

## Introduction

Many mathematical symbols and notations are figured routinely by students in learning mathematics in the school contexts. Those symbols and notations are mostly faced by students, especially when they learn about arithmetic and algebra. In fact, many students are struggling to understand the meaning of those mathematical symbols and notations, and sometimes lead them to the misunderstandings (e.g. Kiran, 1981). Realizing this issue, many researchers and experts from many different fields and backgrounds have been trying to find out what kinds of misunderstandings that happen in the students’ thinking in learning mathematics and how to deal with them.

Furthermore, misunderstandings about mathematical symbols and notations are also happened in the case of equal sign (or more holistically, equality). Question about how students understand the equality symbol have largely been discussed by many researchers and experts from many different fields persist through elementary schools to high schools and colleges (e.g. Jones and Pratt, 2007; Hattikudur and Alibali, 2010). Based on those extensive researches focusing in this issue, many students do not interpret the equal sign, as an equivalence symbol. They misunderstand about the meaning of the equal sign. Moreover, as will be seen, understanding the equal sign as an equivalence relation does not seem to come easily to the students. The purpose of this essay is to analyze about misunderstandings of the equal sign among elementary and secondary school children.

## Children’s perceptions of the equal sign: literature review

According to the results of many researches and observations about children’ interpretation of equality, there are two different meanings of equal sign among students, namely, the operational and the relational meaning.  Students who interpret the equality as an operational meaning, symbol that has to be performed, tend to seeing the equal sign as a “do something signal” or “put the answer” when working with arithmetical notation. For example, the observation done by Behr and his colleagues (1976) interviewing children in grades 1 to 6 about their view of equality sentences, present that when students given the written statement like 3 = 3, a common response was they transformed it into addition or subtraction sentences such as 0 + 3 = 3 or 3 – 3 = 0. Moreover, Behr also found that

‘‘There is a strong tendency among all of the children to view the = symbol as being acceptable in a sentence only when one (or more) operation signs (+, –, etc.) precede it. Some children, in fact, tell us that the answer must come after the =.’’ (p. 15)

In contrast, students who view the equal sign as a relational meaning, expressing the relationship between quantities, tend to interpret the equal sign as meaning “the same as” (Hattikudur and Alibali, 2010). Although, the students’ view of the equal sign as an operational symbol may help them to solve the problem of standard equation such as 2 + 7 = …., they will face the difficulty when dealing with more sophisticated problems. For instance, many elementary school students were struggling and could not solve the problem like 4 + 3 + 5 = 4 +    . They applied their operational understanding of equal sign and added all the numbers before the equal sign to get a solution of 12 or added all of the numbers to get 16 (Hattikudur and Alibali, 2010).

Furthermore, based on the several researches, it is widely reported that most of children tend to hold an operational view of the equal sign, rather than relational view when working with arithmetical notation (e.g. Kieran, 1981). This report arises from analysing children’s responses when they presented various forms of equality statements. The form of this operational view of the equal sign is varied among the students. Besides seeing the equal sign as a “do something signal”, there are also students that seeing the equal sign as a “what it adds up to” (Behr et al, 1976). For example, when the students presented statement like 3 + 4 =    , they understand it as a calling for the answer to be placed on the box (Kieran, 1981).

Some researchers are also have different view in determining whether children’s view the statements as an operational meaning or relational meaning. For example, Kieran (1981) argued that the students who understand 4 + 3 = 6 + 1 have more of relational meaning rather than operational meaning:

“They justified them in terms of both sides being equal because they had the same value. The comparisons that subjects were eventually able to make between left and right sides of the equal sign suggest that the equality symbol was being seen at this stage more as a relational symbol than as a ‘do something signal’.” (p.321)

In contrast, Behr et al. argued that children who view 3 + 2 = 2 + 3 and read it as “3 plus 2 equals 5” and “2 plus 3 equals 5” respectively have an operational meaning. As Behr stated that

“They do not see these as indicating the sameness of sets of objects. Indeed, it appears that the children considered these as “do something” sentences”. (p. 14)

In general, Kieran interprets ‘the same value’ as more closely related to a relational meaning, whereas Behr and his colleagues interpret ‘the same value’ as more closely related to an operator meaning.

## Conclusion

There are many different interpretations of the equal sign among students. The concept of equivalence, as has been explained in this essay, is a complicated thing to be understood by the students. Many of them interpret the equal sign as a “do something signal” or “what it adds up to” rather than a relation symbol between quantities. Moreover, the experts are also had different views in determining ‘the same value’, whether it is more an operational or a relational meaning.

## Reference

Behr, M., Erlwanger, S., & Nichols, E. 1976. How children view equality sentences. PMDC Technical Report No. 3, Florida State University.

Jones, I., & Pratt, D. 2007. Connecting the equal signs. International Journal of Computers for Mathematical Learning, 11, 301-325.

Kieran, C. 1981. Concepts associated with the equality symbol. Educational Studies in Mathematics, 12, 317-326.

Hattikudur, S., & Alibali, M. W. 2010. Learning about the equal sign: does comparing with inequality symbols help. Journal of Experimental Child Psychology, 107, 15-30.