Twenty-five years ago, a fast-food TV ad initiated a catchphrase, “Where’s the beef?” The phrase evolved into a way to question the amount or substance of an idea or product. It is now time for the phrase to make its way into discussions about mathematics education research.

This “Research Commentary” explores the roles of theoretical and empirical work in the development of productive lines of research by elaborating on three main points. First, the dialectic between theory and empirical practice is highly productive for both: Theory is enriched by close attention to data, and one’s understanding of empirical issues is deepened when one attends closely to theory. Second, an effective mechanism for making systematic progress is to focus on cases at the edge of one’s (and the field’s) collective theoretical understanding. Third, close observations of empirical phenomena often raise issues that have theoretical import and that give rise to new lines of work.

A case study of an undergraduate calculus student’s nonstandard conceptions of the real number line. Interviews with the student reveal robust conceptions of the real number line that include infinitesimal and infinite quantities and distances. Similarities between these conceptions and those of G. W. Leibniz are discussed and illuminated by the formalization of infinitesimals in A. Robinson’s nonstandard analysis. These similarities suggest that these student conceptions are not mere misconceptions, but are nonstandard conceptions, pieces of knowledge that could be built into a system of real numbers proven to be as mathematically consistent and powerful as the standard system.

This article describes general principles underlying propensity score methods and illustrates their application to mathematics education research using 2 examples investigating the impact of problem-solving emphasis in mathematics classrooms on students’ subsequent mathematics achievement and course taking. Selection bias is a problem for mathematics education researchers interested in using observational rather than experimental data to make causal inferences about the effects of different instructional methods in mathematics on student outcomes. Propensity score methods represent 1 approach to dealing with such selection bias. Limitations of the method are discussed.

The construct professional noticing of children's mathematical thinking is introduced as a way to begin to unpack the in-the-moment decision making that is foundational to the complex view of teaching endorsed in national reform documents. The authors define this expertise as a set of interrelated skills including (a) attending to children's strategies, (b) interpreting children's understandings, and (c) deciding how to respond on the basis of children's understandings. The findings help to characterize what this expertise entails; provide snapshots of those with varied levels of expertise; and document that, given time, this expertise can be learned.

Mathematics Educators Respond to Kaputs “Algebra Problem”: A Review of Algebra in the Early Grades. James Kaput, David Carraher, and Maria Blanton (Eds.) (2007). Mahwah, NJ: Lawrence Erlbaum Associates, 552 pp. ISBN 0-8058-5473-8 (pb) $65.95. Reviewed by Dan Chazan and Ann R. Edwards