Question Design Affects Students' Sense-Making on Mathematics Word Problems.

Mathematics word problems provide students with an opportunity to apply what they are learning in their mathematics classes to the world around them. However, students often neglect their knowledge of the world and provide nonsensical responses (e.g., they may answer that a school needs 12.5 buses for a field trip). This study examined if the question design of word problems affects students' mindset in ways that affect subsequent sense-making. The hypothesis was that rewriting standard word problems to introduce inherent uncertainty about the result would be beneficial to student performance and sense-making because it requires students to reason explicitly about the context described in the problem. Middle school students (N = 229) were randomly assigned to one of three conditions. In the standard textbook condition, students solved a set of six word problems taken from current textbooks. In the modified yes/no condition, students solved the same six problems rewritten so the solution helped answer a "yes" or "no" question. In the disfluency control condition, students solved the standard problems each rewritten in a variety of fonts to make them look unusual. After solving the six problems in their assigned condition, all students solved the same three "problematic" problems designed to assess sense-making. Contrary to predictions, results showed that students in the modified yes/no condition solved the fewest problems correctly in their assigned condition problem set. However, consistent with predictions, they subsequently demonstrated more sense-making on the three problematic problems. Results suggest that standard textbook word problems may be able to be rewritten in ways that mitigate a "senseless" mindset.

Partial knowledge in the development of number word understanding.

A common measure of number word understanding is the give-N task. Traditionally, to receive credit for understanding a number, N, children must understand that N does not apply to other set sizes (e.g. a child who gives three when asked for 'three' but also when asked for 'four' would not be credited with knowing 'three'). However, it is possible that children who correctly provide the set size directly above their knower level but also provide that number for other number words ('N + 1 givers') may be in a partial, transitional knowledge state. In an integrative analysis including 191 preschoolers, subset knowers who correctly gave N + 1 at pretest performed better at posttest than did those who did not correctly give N + 1. This performance was not reflective of 'full' knowledge of N + 1, as N + 1 givers performed worse than traditionally coded knowers of that set size on separate measures of number word understanding within a given timepoint. Results support the idea of graded representations (Munakata, Trends in Cognitive Sciences, 5, 309-315, 2001.) in number word development and suggest traditional approaches to coding the give-N task may not completely capture children's knowledge.