From an Operational to a Relational Conception of the Equal Sign: Third Graders' Developing Algebraic Thinking

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As algebra has taken a more prominent role in mathematics education, many are advocating introducing children to it in primary school (e.g., Carraher, Schliemann, Brizuela, & Earnest, 2006; Kaput, 2000; National Council of Teacher of Mathematics, 2000), and open number sentences can be a good context to address this goal. Students are frequently introduced to equations by considering number sentences with an unknown where a figure or a line is used instead of a variable, as in _ + 4 = 5 + 7 (Radford, 2000). Discussions about these equations and the properties they illustrate can help students to learn arithmetic with understanding and to develop a solid base for the later formal study of algebra by helping them to become aware of the structure underneath arithmetic (Carpenter, Franke, & Levi, 2003; Kieran, 1992; Resnick, 1992). Unfortunately students, conceptions about the equal sign interfere with their ability to successfully solve and analyze equations (Carpenter et al., 2003; Molina & Ambrose, 2006). Our aim in this paper is to describe a teaching experiment in which children developed an understanding of the equal sign and began to analyze expressions in sophisticated ways.

Difficulties in Solving Number Sentences

When elementary students encounter the equal sign in arithmetic sentences they tend to perceive it as an operational symbol, that is a "do something" signal, and tend to react negatively when number sentences challenge their conceptions about this symbol as in sentences of the form c = a + b. Behr, Erlwanger, and Nichols (1980) observed that most six-year old students thought that such sentences were "backwards" and tended to change them to c + a = b or a + b = c. Children also did not accept non-action sentences, that is sentences with no operational symbol (e.g., 3 = 3) or operational symbols on both sides (e.g., 3 + 5 = 7 + 1), and often changed them to action sentences, that is sentences with all the operations in one side of the equation. For example they changed 3 + 2 = 2 + 3 to 3 + 2 + 2 + 3 = 10 and 3 = 3 to 3 + 0 = 3 or 3 - 3 = 0.

In studies about elementary students' answers to open sentences of the forms a = a, c + a = b, a + b = c and a + b = c + d (Falkner, Levi, & Carpenter 1999; Freiman & Lee, 2004; Kieran, 1981), students provided a variety of responses: repeating one of the numbers in the sentence, the sum or differences of two numbers of the sentence, the sum of all the numbers in the sentence, and the correct answer. In sentences of the form a + b = _ + d, students tended to answer the sum of a + b or to write it as a string of operations.

These studies illustrate that children tend to read open number sentences from left to right and perform the computation as they go along. When children face unfamiliar number sentences, that is sentences different from the conventional form a [+ or -] b = c, they have trouble interpreting the equal sign as a symbol representing equivalence. To successfully solve a problem such as 8+ 4 = _ + 5, students have to read the whole sentence before computing and need to recognize that both sides of the equation need to have the same sum. The equal sign needs to be interpreted as a relational symbol expressing equivalence.

Considering students' difficulties with the equal sign, their understanding of the concept of equality can be questioned; however, Schliemann, Carraher, Brizuela, and Jones (1988) and Falkner et al. (1999) have observed that students show a correct understanding of equality when considering concrete physical contexts or verbal word problems. The children's misinterpretations are linked to the use of the equal symbol rather than an understanding of the concept of equality (Carpenter et al., 2003; Falkner et al., 1999, Schifter, Monk, Russell, & Bastable, in press). Most studies about the equal sign (Behr et al., 1980; Falkner et al., 1999; Saenz-Judlow & Walgamuth, 1998) have claimed that traditional curriculum does not promote a relational understanding of this symbol, mainly because of the repeated consideration of equations of the form a [+ or -] b = c throughout students' arithmetic learning. These misinterpretations may also be exacerbated by the linguistic convention of writing and reading from left to right (Rojano, 2002).

Recent research (Carpenter et al., 2003; Koehler, 2004; Saenz-Ludlow & Walgamuth, 1998) has shown that elementary students, even first graders, are capable of developing a relational understanding of the equal sign with suitable instruction, going against previous claims about the existence of cognitive limitations in developing a relational meaning of the equal sign in the elementary grades (Kieran, 1981). Our study contributes and extends these findings by describing how children developed broader conceptions about this symbol as the result of classroom activities and discussion.

Analyzing Expressions

Carpenter et al. (2003) have used the term relational thinking to describe an approach to solving open number sentences. They illustrate this with the example of the problem 27 - 48 + 48 = _ (p. 32). Children who recognize that 48 and 48 are the same number and that addition and subtraction are inverse operations will conclude that the answer to the problem is 27 without having done any computation. Koehler (2004) more broadly refers to relational thinking as "the many different relationships children recognize and construct between and within numbers, expressions, and operations" (p. 2). We have chosen to adopt the term analyzing expressions for this kind of thinking to better distinguish it from the relational meaning of the equal sign as a symbol representing equivalence. We say that students use analyzing expressions when they approach number sentences, by focusing on arithmetic relations instead of computing. Students engaged in analyzing expressions employ their number sense and what Slavit (1999) called, "operation sense" to consider arithmetic expressions from a structural perspective instead of a procedural one. Sentences have to be considered as wholes instead of as processes to do step by step. When students analyze expressions, they compare elements on one side of the equal sign to elements on the other side of the equal sign or they look for relations between elements on one side of the equation. For example, when considering the number sentence 8 + 4 = _ + 5 some students notice that both expressions include addition and that one of the addends, 4, on the left side is one less than the addend, 5, on the other side. Noticing this relation between these elements and having an implicit understanding of addition properties enables the student to solve this problem without having to perform the computations 8 plus 4 and 12 minus 5. Analyzing expressions by comparing elements on each side of the equal sign is the kind of thinking that students must do when solving algebraic equations in the form 3x + 7 = 2x + 18. As students analyze expressions, they employ and deepen their operation sense which is fundamental to algebraic thinking (Slavit, 1999).

Analyzing expressions may form a good foundation for the formal study of algebra, but little evidence is available to show that children are capable of doing so. Liebenberg, Sasman, and Olivier (1999) observed that most students were not able to solve open sentences without computing the answer due to a lack of knowledge about arithmetic operations and their properties. Kieran (1981) noted that "lack of closure" interfered with children's ability to solve sentences …