Academic journal article Memory & Cognition

# An Eye for Relations: Eye-Tracking Indicates Long-Term Negative Effects of Operational Thinking on Understanding of Math Equivalence

Academic journal article Memory & Cognition

# An Eye for Relations: Eye-Tracking Indicates Long-Term Negative Effects of Operational Thinking on Understanding of Math Equivalence

## Article excerpt

Abstract Prior knowledge in the domain of mathematics can sometimes interfere with learning and performance in that domain. One of the best examples of this phenomenon is in students' difficulties solving equations with operations on both sides of the equal sign. Elementary school children in the U.S. typically acquire incorrect, operational schemata rather than correct, relational schemata for interpreting equations. Researchers have argued that these operational schemata are never unlearned and can continue to affect performance for years to come, even after relational schemata are learned. In the present study, we investigated whether and how operational schemata negatively affect undergraduates' performance on equations. We monitored the eye movements of 64 undergraduate students while they solved a set of equations that are typically used to assess children's adherence to operational schemata (e.g., 3 + 4 + 5 = 3 + __). Participants did not perform at ceiling on these equations, particularly when under time pressure. Converging evidence from performance and eye movements showed that operational schemata are sometimes activated instead of relational schemata. Eye movement patterns reflective ofthe activation ofrelational schemata were specifically lacking when participants solved equations by adding up all the numbers or adding the numbers before the equal sign, but not when they used other types of incorrect strategies. These findings demonstrate that the negative effects of acquiring operational schemata extend far beyond elementary school.

Keywords Einstellung * Mathematical equivalence * Eye-tracking * Mental set * Problem solving

Conventional wisdom suggests that the process of learning is a progressive march forward. As individuals gain domain knowledge, they become more proficient at operating within that domain. However, several decades of research indicate that this is not always the case. Prior domain knowledge sometimes interferes with the ability to operate successfully within that domain, particularly when task demands change (Bilalic, McLeod, & Gobet, 2008a, Bilalic et al. 2008b; Croskerry, 2003; Lippman, 1994; Lovett, & Anderson, 1996; Luchins, 1942; Wiley, 1998). One reason is that learners tend to rely on prior knowledge instead of encoding new information or generating new strategies. This change resistance has been shown in several areas of psychology (e.g., Allport, 1954; Diamond & Kirkham, 2005; Luchins, 1942; Munakata, 1998; Rescorla, 1996; Schäuble, 1990; Wiley, 1998; Zelazo, Frye, & Rapus, 1996). In the present study, we focused on change resistance involving adherence to schemata that form through repeated experience in a given domain. We will refer to this type of change resistance as the Einstellung effect (Luchins, 1942), although it historically has been described using a variety of terms, including habit (James, 1890), fixation (Duncker, 1945; see also Maier, 1931), mental set (Wiley, 1998), and rigidity (Schultz & Searleman, 2002), among others. Our goal was to use eye-tracking methods to detect evidence of long-term, pernicious Einstellung effects in the domain of mathematics.

Einstellung effects

The Einstellung effect was classically demonstrated by Luchins's (1942) water jar problems. In these problems, individuals used water jars of known volumes (e.g., 18, 43, and 10 quarts) to construct a third volume (e.g., 5 quarts). After solving several problems that required a particular multistep solution strategy (e.g., 43 - 10 - 10 - 18 = 5), problem solvers continued to employ this set strategy even when they later encountered problems for which that strategy was inefficient (e.g., finding 20 quarts given 23-, 49-, and 3-quart jars by 49 - 3 - 3 - 23, rather than 23 - 3) or even ineffective (e.g., finding 25 quarts given 28-, 76-, and 3-quart jars; see also Lovett & Anderson, 1996).

Einstellung effects have been linked to what is dubbed the insight problem'. …

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