**FEATURES** |

Nonstandard Student Conceptions About Infinitesimals
*Robert Ely* 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. |

Using Propensity Scores to Reduce Selection Bias in Mathematics Education Research
*Suzanne E. Graham* 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. |

Professional Noticing of Children’s Mathematical Thinking
*Victoria R. Jacobs, Lisa L. C. Lamb, and Randolph A. Philipp* 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. |