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

By Molina, Marta; Ambrose, Rebecca | Focus on Learning Problems in Mathematics, Winter 2008 | Go to article overview

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

Molina, Marta, Ambrose, Rebecca, Focus on Learning Problems in Mathematics

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. …

• Questia's entire collection
• Automatic bibliography creation
• More helpful research tools like notes, citations, and highlights

If you are trying to select text to create highlights or citations, remember that you must now click or tap on the first word, and then click or tap on the last word.
One moment ...
Default project is now your active project.
Project items

Highlights (0)
Some of your highlights are legacy items.
Citations (0)
Some of your citations are legacy items.
Notes (0)
Bookmarks (0)

Project items include:
• Saved book/article
• Highlights
• Quotes/citations
• Notes
• Bookmarks
Notes

#### Cited article

Style
Citations are available only to our active members.
Sign up now to cite pages or passages in MLA, APA and Chicago citation styles.

(Einhorn, 1992, p. 25)

(Einhorn 25)

1

1. Lois J. Einhorn, Abraham Lincoln, the Orator: Penetrating the Lincoln Legend (Westport, CT: Greenwood Press, 1992), 25, http://www.questia.com/read/27419298.

#### Cited article

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

#### Settings

Typeface
Text size Reset View mode
Search within

Look up

#### Look up a word

• Dictionary
• Thesaurus
Please submit a word or phrase above.

Why can't I print more than one page at a time?

Full screen

## Cited passage

Style
Citations are available only to our active members.
Sign up now to cite pages or passages in MLA, APA and Chicago citation styles.

"Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn, 1992, p. 25).

"Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn 25)

"Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences."1

1. Lois J. Einhorn, Abraham Lincoln, the Orator: Penetrating the Lincoln Legend (Westport, CT: Greenwood Press, 1992), 25, http://www.questia.com/read/27419298.

## Welcome to the new Questia Reader

The Questia Reader has been updated to provide you with an even better online reading experience.  It is now 100% Responsive, which means you can read our books and articles on any sized device you wish.  All of your favorite tools like notes, highlights, and citations are still here, but the way you select text has been updated to be easier to use, especially on touchscreen devices.  Here's how:

1. Click or tap the first word you want to select.
2. Click or tap the last word you want to select.

## Thanks for trying Questia!

Please continue trying out our research tools, but please note, full functionality is available only to our active members.

Your work will be lost once you leave this Web page.