Compared with what? The effects of different comparisons on conceptual knowledge and procedural flexibility for equation solving.

Abstract

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.

Modeling Strategy Use in an Intelligent Tutoring System: Implications for Strategic Flexibility

“…The current study uses a method previously employed in a different domain [13] to identify groups of students in a math ITS that differ in how they apply strategies to solving equations. First, our findings from the behavioral pre-test replicated results from RittleJohnson and Star [1,7], showing a correlation between strategic flexibility and procedural knowledge. Understanding how strategic flexibility is represented during tutoring is less clear.…”

“…In the current study, we use Hidden Markov Modeling (HMM) to cluster participants into strategically distinct groups. We present evidence supporting the hypothesis that these groups differ on a measure of strategic flexibility collected using the pencil and paper test developed by Rittle-Johnson and Star [7]. Furthermore, we use this HMM method to explore how flexibility presents itself in a math ITS as differences in tutor behavior.…”

“…Assessment We used a modified version of the Rittle-Johnson and Star [7] assessment, which assesses mathematical knowledge (both conceptual and procedural) and strategic flexiblity for one-and two-step algebra equation problem solving. We implemented only the procedural skill and strategic flexibility portion of the assessment.…”

“…Measures of strategic flexibility correlate with both the student's procedural and conceptual knowledge [1,7,4]. Schneider et al [2] found a bidirectional relationship between procedural and conceptual knowledge and hypothesized that these two types of knowledge improve strategic flexibility in an iterative fashion.…”

Learning to Think: Cognitive Mechanisms of Knowledge Transfer

“…Gick and Hoyoak found that the best transfer was achieved by instruction that asked students to compare two examples (or analogs) of a general solution schema in the context of a symbolic abstraction (a diagram) representing the general solution schema. Other studies have also demonstrated the eff ectiveness of prompting for example comparisons (Gentner et al, 2009;Rittle-Johnson & Star, 2009), or for providing an abstraction of a general rule, pattern, or theory behind solutions (e.g., Judd, 1908;Holland, Holyoak, Nisbett, & agard, 1986). How might we best map laboratory results like these onto educational practice?…”