Perceptual support promotes strategy generation: Evidence from equation solving

Abstract

Over time, children shift from using less optimal strategies for solving mathematics problems to using better ones. But why do children generate new strategies? We argue that they do so when they begin to encode problems more accurately; therefore, we hypothesized that perceptual support for correct encoding would foster strategy generation. Fourth-grade students solved mathematical equivalence problems (e.g., 3 + 4 + 5 = 3 + __) in a pre-test. They were then randomly assigned to one of three perceptual support conditions or to a Control condition. Participants in all conditions completed three mathematical equivalence problems with feedback about correctness. Participants in the experimental conditions received perceptual support (i.e., highlighting in red ink) for accurately encoding the equal sign, the right side of the equation, or the numbers that could be added to obtain the correct solution. Following this intervention, participants completed a problem-solving post-test. Among participants who solved the problems incorrectly at pre-test, those who received perceptual support for correctly encoding the equal sign were more likely to generate new, correct strategies for solving the problems than were those who received feedback only. Thus, perceptual support for accurate encoding of a key problem feature promoted generation of new, correct strategies. Statement of Contribution What is already known on this subject? With age and experience, children shift to using more effective strategies for solving math problems. Problem encoding also improves with age and experience. What the present study adds? Support for encoding the equal sign led children to generate correct strategies for solving equations. Improvements in problem encoding are one source of new strategies.

Relational equity and mathematics learning: Mutual construction during collaborative problem solving

“…Moreover, within each pair, one or both participants noticed the atypical location of the equal sign, and participants generated and discussed different strategies for solving the problems. Findings from previous literature suggest that under these conditions, students would be well positioned to generate new strategies and apply them to solve problems correctly (e.g., Alibali, Crooks, & McNeil, 2017). The students in the cases we present had different learning outcomes, as indicated by their post-assessment scores.…”

“…When elementary students are asked to recreate equivalence problems from memory, they often recall the structure of the original problems inaccurately, and they sometimes produce problems with the equal sign at the end, in "operations equal answer" format (e.g., they sometimes reconstruct 8 + 5 + 4 = 4 + ___ as 8 + 5 + 4 + 4 = ___, Alibali et al, 2009;McNeil & Alibali, 2004). Accurate encoding of the position of the equal sign is associated with using correct strategies to solve the problems (e.g., Alibali, Crooks, & McNeil, 2017;Crooks & Alibali, 2013;McNeil & Alibali, 2004). …”

Fraction Symbols and their Relation to Conceptual Knowledge

“…Similarly, narrowly spacing precedent operations within equations, for example, 3 + 5×4 versus 3 + 5 × 4, helped students attend to the order of operations and thus better discriminate between the correct (i.e., 23) and incorrect (i.e., 32) solutions (Landy & Goldstone, 2007). Finally, attending to the orthographic features of the equal sign was positively related to equation solving (Alibali et al, 2018;McNeil et al, 2011). Therefore, I predicted that students who are better able to attend to the orthographic features of fraction symbols, as evidenced by their knowledge of mathematical orthography, will be more successful fraction learners.…”

“…Thus, students may judge equations such as 3 + 5 = 2 + 6, as incorrect or meaningless (Li, 2008;Steinberg et al, 1991) and subsequently struggle to solve equations with missing numbers (Powell & Fuchs, 2010;Sherman & Bisanz, 2009). Interestingly, drawing attention to the orthographic features related to the equal sign -that is, identifying sides of an equation, noticing the location of the equal sign, and recognizing the order of the mathematical symbols in an equation -resulted in improved performance solving equations (Alibali et al, 2018;McNeil et al, 2011). These findings support the view that knowledge of mathematical orthography is related to mathematical skills.…”

Comparing Meta‐analysis and Individual Person Data Analysis Using Raw Data on Children's Understanding of Equivalence

“…Consistent with this view, encoding accuracy predicts solving accuracy across a wide range of problem types (Kotovsky, Hayes, & Simon, ; Rittle‐Johnson, Siegler, & Alibali, ; Siegler, ), including mathematical equivalence problems (McNeil & Alibali, ). Moreover, interventions designed to improve encoding accuracy improve solving accuracy (Alibali, Crooks, & McNeil, ; Alibali, McNeil, & Perrott, ; Siegler, ).…”