Problem Solving as a Means toward Mathematics for All: An Exploratory Look through a Class Lens

ion From Contexts Because the open problems could be explored in various ways, students could take unexpected approaches. Although this openness is not necessarily problematic, in this study I raise questions about complications that can arise if, as Bernstein (1975) and Holland (1981) argued, lower SES students have a contextualized orientation toward meanings. Many of the CMP problems were set in realistic situations, and these contexts were often effective motivators, prompting students to delve into the problems. Whereas the higher SES students seemed able and inclined to pull back from the context and analyze the intended mathematical ideas involved, the lower SES students seemed to focus more on the real-world constraints involved with the contexts as they solved the problems, thereby missing some of the intended mathematical ideas. Usually the lower SES students' thinking about these real-world constraints was very sensible, and I (and the other CMP authors) did want students 6Additionally, Kohr et al. (1988) found that SES correlates with students' general self-esteem in school (more so than race or gender). 7Again, in this study I say nothing about the published versions of the CMP curriculum. The CMP problems gained clarity with each revision, and, therefore, all students' struggles with the open nature of the problems would probably have been lessened if the published CMP materials had been used. The issue raised in this study is SES-related differences in students' reactions to open, contextualized curricular problems, in general. This content downloaded from 157.55. 39.163 on Wed, 21 Sep 2016 05:58:33 UTC All use subject to http://about.jstor. org/terms Sarah Theule Lubienski 477 to be able to think about mathematical ideas in the messy, real-world contexts in which they are often embedded. But these contextualized problems were primarily intended to be a means of learning more general mathematical ideas that could then be applied in other contexts. The tendency for the lower SES students to focus on the immediate context, as well as their desire for more specific teacher direction for their mathematical thinking and less intrinsic interest in understanding the mathematics for its own sake, seemed to contribute to the lower SES students' tendencies to become stuck8 in the contexts. Hence, although students with more of a contextualized orientation toward meaning might be expected to benefit from contextualized mathematics problems, in this study I raise questions about possible drawbacks of contextualized problems in the curriculum. Other researchers recently raised similar questions about the fairness of contextualized mathematics assessment items. In comparing British workingand middle-class students' approaches to standardized test items, Cooper, Dunne, and Rodgers (1997) found that class differences were greater on questions involving realistic situations. In examining these differences, they discovered that the contexts were obstacles for working-class students, who approached the problems in heavily context-laden ways unintended by test writers. Ball (1995) raised additional equity issues surrounding the use of realistic contexts. She suggested that teaching mathematics through real-world problems can pose difficulties because of (a) differences in students' interpretations and approaches and (b) uneven access to relevant contextual knowledge. In her classroom experience, Ball found that abstract mathematical contexts often seemed more inclusive, giving students a sense of common understanding and purpose. When her students explored real-world problems, they "were distracted, or confused, or the differences among them were accentuated in ways that diminished the sense of collective purpose and joint work" (p. 672). Although students need to become able and inclined to critically analyze realworld problems, particularly those involving inequities, I question whether these problems are equitable means of building the mathematical understandings necessary for such critical analyses.