Activation of Operational Thinking During Arithmetic Practice Hinders Learning And Transfer

Abstract

Many children in the U.S. initially come to understand the equal sign operationally, as a symbol meaning "add up the numbers" rather than relationally, as an indication that the two sides of an equation share a common value. According to a change-resistance account (McNeil & Alibali, 2005b), children's operational ways of thinking are never erased, and when activated, can interfere with mathematics learning and performance, even in educated adults. To test this theory, undergraduates practiced unfamiliar multiplication facts (e.g., 17-times table) in one of three conditions that differed in terms of how the equal sign was represented in the problems. In the operational words condition, the equal sign was replaced by operational words (e.g., "multiplies to"). In the relational words condition, the equal sign was replaced by relational words (e.g., "is equivalent to"). In the control condition, the equal sign was used in all problems. The hypothesis was that undergraduates' fluency with practiced facts and transfer problems would be hindered in the operational words condition compared to the other conditions. Results supported this hypothesis, indicating that the activation of operational thinking is indeed detrimental to learning and transfer, even in educated adults.

Persistent and Pernicious Errors in Algebraic Problem Solving

“…Consistent with this pattern, making equality/inequality errors during either the middle or end of the year-when they are becoming more prominent-is a troublesome sign for students' math achievement. Perhaps this is not surprising, given that the notion of balance across two sides of an equation is one of the most foundational concepts underlying algebra and that students have difficulty moving from an operational to a relational understanding of the equals sign (e.g., Chesney & McNeil, 2014). Perhaps having an operational understanding is sufficient for performance on simple equations, but the lack of a relational understanding becomes more problematic when students face more complex equations.…”

“…A variety of particularly problematic misconceptions typically plague beginning algebra students, including believing that the equals sign is an indicator of operations to be performed (Baroody & Ginsburg, 1983;Knuth, Stephens, McNeil, & Alibali, 2006); Chesney & McNeil, 2014), that negative signs represent only the subtraction operation and do not modify terms (Vlassis, 2004), and that variables cannot represent more than one value (Knuth, Alibali, Weinberg, & McNeil, 2005). Unfortunately, for many students these misconceptions persist even after typical classroom instruction (Vlassis, 2004).…”

Consequences of Individual Differences in Children's Formal Understanding of Mathematical Equivalence

“…This is unlikely to reflect a positive change in development because children in the older grades who rely on their knowledge of traditional arithmetic are less likely than those who are incorrect in other ways to benefit from instruction on mathematical equivalence (Byrd et al., ; McNeil & Alibali, ). This suggests that the knowledge structures that support arithmetic calculation skill do not automatically generalize to—and may sometimes even hinder—understanding of mathematical equivalence (McNeil & Alibali, ; see also Chesney & McNeil, ; McNeil, Rittle‐Johnson, Hattikudur, & Petersen, ).…”

“…Children's understanding does not improve much between the ages of 7 and 9, and some misconceptions even persist into middle school, high school, and beyond (Byers & Herscovics, ; Knuth et al., ). There is some evidence that activation of traditional arithmetic knowledge contributes to older students’ difficulties when working with symbolic expressions and equations (Knuth et al., ; McNeil et al., ), and other algebra problems (Chesney & McNeil, ; McNeil et al., ). This may be because of a competition between the operational and relational concepts, or because of students’ nonmeaningful learning of a relational concept.…”

Wait for it . . . Delaying Instruction Improves Mathematics Problem Solving: A Classroom Study

“…Further, they often solve mathematical equivalence problems incorrectly by adding up all of the numbers in the problem or only the numbers before the equal sign (e.g., answering 15 or 12 for 3 + 4 + 5 = 3 + ☐, rather than 9; McNeil & Alibali, 2005). Unfortunately, children's difficulties with mathematical equivalence are often robust, persisting into middle school, high school, and even adulthood (e.g., Chesney & McNeil, 2014;Knuth et al, 2006;McNeil & Alibali, 2005;McNeil, Rittle-Johnson, Hattikudur, & Petersen, 2010). In sum, evidence from past research stresses the importance of establishing understanding of mathematical equivalence early in elementary school.…”