Don’t teach me 2 + 2 equals 4: Knowledge of arithmetic operations hinders equation learning
Year: 2004
Type: article
Abstract: Don’t teach me 2 + 2 equals 4: Knowledge of arithmetic operations hinders equation learning Nicole M. McNeil ([email protected]) University of Wisconsin-Madison Department of Psychology, 1202 W. Johnson Street Madison, WI 53706 USA that the two sides of an equation represent the same quantity. Elementary school children (ages 7-11) have significant difficulties with equations and the principle of mathematical equivalence (Carpenter & Levi, 2000; Kieran, 1981; Baroody & Ginsburg, 1983). Their difficulties are most apparent when they are presented with equations that have operations on both sides of the equal sign (e.g., 3 + 4 + 5 = 3 + __). In the absence of instruction, approximately 80% of second- through fifth-grade children solve these types of equations incorrectly (Alibali & Goldin-Meadow, 1993; Alibali, 1999; McNeil & Alibali, 2000; NCISLA, 2000; Perry, Church, & Goldin-Meadow, 1988; Rittle-Johnson & Alibali, 1999). Although there are many possible accounts of children’s difficulties, including immature working memory function (Adams & Hitch, 1997; Gathercole & Pickering, 2000) or insufficient knowledge of necessary prerequisite skills (Haverty, 1999), the c h a n g e - resistance account suggests that children’s equation- learning difficulties are due, at least in part, to children’s existing knowledge (McNeil, 2004). More specifically, the account posits that children construct knowledge on the basis of their early experiences with arithmetic operations and that this knowledge contributes to children’s difficulties with more complex equations. There are at least three knowledge structures that children learn from their early experiences with arithmetic operations that may ultimately hinder the ability to learn complex equations (see McNeil & Alibali, 2002). First, children may learn an operational strategy for solving math problems—perform all the given operations on all the given numbers. For example, in a typical addition problem like 3 + 4 + 5 + 3 = __, a problem solver simply needs to add up all the numbers and put the total in the blank. Second, children may learn an operational perceptual pattern related to the structure of math problems—the traditional “operations = answer” problem structure. For example, in the typical addition problem above, all of the numbers and operations are on the left hand side of the equation, and the answer blank is on the right side of the equation (directly following the equal sign). Third, children may learn an operational concept of the equal sign—the equal sign means “the total.” Although these three operational patterns facilitate fast and accurate Abstract This study investigated whether children’s knowledge of arithmetic operations hinders their ability to solve novel equations after instruction. Second- and third-grade children completed a timed arithmetic pretest as a means for assessing their proficiency with arithmetic operations. Next, they received lessons on the principle of mathematical equivalence either in a context designed to activate their knowledge of arithmetic operations (e.g., 15 + 13 = 28), or in a context designed to not activate their knowledge of arithmetic operations (e.g., 28 = 28). Then, children completed an equation-solving posttest (e.g., 3 + 9 + 5 = 6 + __). After the posttest, children switched lesson contexts and completed the posttest again. Children solved more equations incorrectly after receiving lessons in the operational context. Additionally, the operational context led children who were most proficient with arithmetic operations to solve more equations using the typical addition strategy of adding up all the numbers. Results highlight that the activation of existing knowledge can interfere with the acquisition of new information. Some domains of knowledge are particularly difficult for people to learn, even after significant amounts of training or instruction. There are many examples of this in our formal education system, including reading, mathematics, science, and foreign language. Over the past several years, a number of scientists (e.g., Flege, Yeni Komshian, & Liu, 1999; Kuhl, 2000; McNeil & Alibali, 2002; Schauble, 1990; Zevin & Seidenberg, 2002) have begun to consider how existing knowledge may contribute to these difficulties. The general theoretical view is that later learning is strongly constrained by early learning (cf. Tolman, 1948). If this is true, it obviously has implications in domains, like second language learning, where people learn one thing for many years (native language) before switching gears and learning something new, but closely related (second language). The domain of mathematics is another domain in which people learn one topic for many years before switching gears and learning a new, but closely related, topic. Specifically, in most American mathematics classrooms, children learn arithmetic operations for many years (i.e., grades K-6) before eventually reaching algebra and being introduced formally to equations and the principle of mathematical equivalence. Mathematical equivalence is the principle
Author: Nicole M. McNeil
Language: English
Subfield: Developmental and Educational Psychology
Field: Psychology
Domain: Social Sciences
SDG: Quality education
Open Access status: green
OpenAlex ID: W2768395944