Understanding the Meaning of the Equal SignĪcross elementary, middle, and high school, students often misunderstand the equal sign ( Kieran, 1981 Knuth, Stephens, McNeil, & Alibali, 2006). A mathematical equation is an equation with zero or one variables (e.g., 9 = 6 + 3 9 = x + 3), whereas an algebraic equation is an equation with two or more variables (e.g., x − 3 = y). Nonstandard equations occur in any form other than standard (e.g., 6 + 4 = _ + 8 6 = 2 + 4) and can also be open or closed. Standard equations can be open (i.e., incorporating a blank or variable to solve) or closed (without any missing information). As the equation is read left to right, the equal sign is always in the second-to-last position, and the answer is after the equal sign. Standard equations are in the form of operations equal an answer (e.g., 2 + 4 = _ 2 + 4 = 6 2 + 2 + 2 = 6). The dividing point between the sides is the equal sign. Every equation has two sides (i.e., left and right). An equation is a mathematical statement where the equal sign is used to show equivalence between a number or expression on one side of the equal sign to the number or expression on the other side of the equal sign. An expression is a combination of numbers and operations without an equal sign (e.g., 9 ÷ 3 1 + 1 + 4 y × 6). To date, an evaluation of the types of equations presented in elementary mathematics textbooks has not been conducted.īefore proceeding, I comment briefly on equation terminology.
These nonstandard equations are generally believed to be necessary to promote a relational understanding of the equal sign ( McNeil et al., 2006). The purpose of this study was to evaluate eight elementary curricula across grades K–5 to determine the degree to which students receive exposure on nonstandard equation types and to understand how teachers are encouraged to define the equal sign and provide instruction on nonstandard equation types. One possible reason for misinterpretation of the equal sign is a lack of exposure to a variety of equation types. Research has shown, however, that ongoing classroom dialogue (e.g., Blanton & Kaput, 2005 Saenz-Ludlow & Walgamuth, 1998) or explicit instruction ( McNeil & Alibali, 2005b Powell & Fuchs, 2010 Rittle-Johnson & Alibali, 1999) can change students’ incorrect interpretations of the equal sign. If the equal sign is interpreted in an operational manner, this typically leads to mistakes in solving equations with missing numbers (e.g., 5 − _ = 1) and difficulties with algebraic thinking (e.g., x − 2 = 2 y + 4 Lindvall & Ibarra, 1980 McNeil & Alibali, 2005b). The number or expression on one side of the equal sign should have the same value as the number or expression on the other side of the equal sign. Students should understand the equal sign as relational, indicating that a relationship exists between the numbers or expressions on each side of the equal sign ( Jacobs, Franke, Carpenter, Levi, & Battey, 2007). Students in elementary school often misinterpret the equal sign (=) as an operational (i.e., do something or write an answer) symbol even though the equal sign should be viewed as a relational symbol ( Sherman & Bisanz, 2009).
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