Compared With What? The Effects of Different Comparisons on Conceptual Knowledge and Procedural Flexibility for Equation Solving

By Bethany Rittle-Johnson, Jon Star. 2009

324 citations recorded by the source corpus.

Researchers in both cognitive science and mathematics education emphasize the importance of comparison for learning and transfer. However, surprisingly little is known about the advantages and disadvantages of what types of things are being compared. In this experimental study, 162 seventh- and eighth-grade students learned to solve equations (a) by comparing equivalent problems solved with the same solution method, (b) by comparing different problem types solved with the same solution method, or (c) by comparing different solution methods to the same problem. Students' conceptual knowledge and procedural flexibility were best supported by comparing solution methods and to a lesser extent by comparing problem types. The benefits of comparison are augmented when examples differ on relevant features, and contrasting methods may be particularly useful in mathematics learning.

Subfields: Education, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/8780f408c42b52e4ba1baf0b813669f99b5ae0d9

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Citations in the corpus

Towards a Theory of When and How Problem Solving Followed by Instruction Supports Learning (2017, background): “Therefore, we would expect benefits on items requiring flexibility (including procedural flexibility items, cf. Rittle-Johnson and Star 2009) and transfer.”
Because Wisdom Can’t Be Told (2012, methodology): “Although comparison is a standard educational tool, Rittle-Johnson and Star (2009) noted that “surprisingly little is known about the advantages and disadvantages of what types of things are being compared” (p. 529).”
Effects of prompting multiple solutions for modelling problems on students’ performance (2015, methodology): “Students in one group learned to solve linear equations such as 3(x + 1) = 15 using the “traditional” solution method: “distribute, combine like terms, subtract constants and variables from both sides, and then divide both sides by the coefficient” (Rittle-Johnson and Star 2009).”
Practicing versus inventing with contrasting cases: The effects of telling first on learning and transfer. (2011, background): “Rittle-Johnson and Star (2007, 2009), for example, found that students developed more procedural flexibility in algebraic manipulations if they compared alternative solutions side by side than if they saw them sequentially.”
Teaching Mathematics by Comparison: Analog Visibility as a Double-Edged Sword. (2016, methodology): “These results could have been driven by the types of solution strategies that students used (Rittle-Johnson & Star, 2007, 2009), particularly subtraction (a common misconception).”
Understanding the Development of Flexibility in Struggling Algebra Students (2010, methodology): “However, research by Rittle-Johnson et al. (2009 ) suggests that some familiarity with a single approach is necessary before students can benefit from the comparison of multiple approaches, possibly because it eases the cognitive load necessary for comparison.”
Taking it to the classroom: Number board games as a small group learning activity (2012, background): “Collaboration with peers has been found to enhance learning under many circumstances (Azmitia & Montgomery, 1993; Howe et al., 2007; Rittle-Johnson & Star, 2009).”
A Is for Apple: Mnemonic Symbols Hinder the Interpretation of Algebraic Expressions (2010, methodology): “Encouraging students to compare different strategies is one method that has been found to improve students’ conceptual knowledge of algebra (Rittle-Johnson & Star, 2009), but benefits may depend on students’ level of prior knowledge (Rittle-Johnson, Star, & Durkin, 2009).”
Do Multiple Solutions Matter? Prompting Multiple Solutions, Interest, Competence, and Autonomy (2014, background): “Many studies that have investigated the use of multiple solutions have reported positive results with regard to students’ achievement (Große & Renkl, 2006; Rittle-Johnson & Star, 2007, 2009; Rittle-Johnson, Star, & Durkin, 2009; Star & Rittle-Johnson, 2009).”
Comparing Learning From Productive Failure and Vicarious Failure (2014, background): “…that although students did not have problems studying and understanding the RSMs, they did better evaluations when they were given a worked example that contrasted a mathematically valid evaluation from an invalid or trivial evaluation (Kapur & Bielaczyc, 2011; Rittle-Johnson & Star, 2009).”