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Expert and Novice Approaches to Reading Mathematical Proofs
*Matthew Inglis and Lara Alcock* A comparison of the proof validation behavior of beginning undergraduate students and research-active mathematicians is explored. Participants’ eye movements were recorded as they validated purported proofs. The main findings are that (a) contrary to previous suggestions, mathematicians sometimes appear to disagree about the validity of even short purported proofs; (b) compared with mathematicians, undergraduate students spend proportionately more time focusing on “surface features” of arguments, suggesting that they attend less to logical structure; and (c) compared with undergraduates, mathematicians are more inclined to shift their attention back and forth between consecutive lines of purported proofs, suggesting that they devote more effort to inferring implicit warrants. |

Measuring Mathematical Knowledge for Teaching Fractions With Drawn Quantities
*Andrew Izsák, Erik Jacobson, Zandra de Araujo, and Chandra Hawley Orrill* Researchers have recently used traditional item response theory (IRT) models to measure mathematical knowledge for teaching (MKT). Some studies (e.g., Hill, 2007; Izsák, Orrill, Cohen, & Brown, 2010), however, have reported subgroups when measuring middle-grades teachers’ MKT, and such groups violate a key assumption of IRT models. This study investigated the utility of an alternative called the mixture Rasch model that allows for subgroups. The model was applied to middle-grades teachers’ performance on pretests and posttests bracketing a 42-hour professional development course focused on drawn models for fraction arithmetic. |

A Proposed Instructional Theory for Integer Addition and Subtraction
*Michelle Stephan and Didem Akyuz* This article presents the results of a 7th-grade classroom teaching experiment that supported students’ understanding of integer addition and subtraction. The experiment was conducted to test and revise a hypothetical learning trajectory so as to propose a potential instructional theory for integer addition and subtraction. The instructional sequence, which was based on a financial context, was designed using the Realistic Mathematics Education theory. |

Coming to Understand the Formal Definition of Limit: Insights Gained From Engaging Students in Reinvention
*Craig Swinyard and Sean Larsen* The purpose of this article is to elaborate Cottrill et al.’s (1996) conceptual framework of limit, an explanatory model of how students might come to understand the limit concept. Drawing on a retrospective analysis of 2 teaching experiments, we propose 2 theoretical constructs to account for the students’ success in formulating and understanding a definition of limit. The 1st construct relates to the need for students to move away from their tendency to attend first to the input variable of the function. The 2nd construct relates to the need for students to overcome the practical impossibility of completing an infinite process. Together, these 2 theoretical constructs build on Cottrill et al.’s work, resulting in a revised conceptual framework of limit. |

Mathematics on the Inside: A Review of Loving + Hating Mathematics: Challenging the Myths of Mathematical Life
*David Kirshner* A review of Reuben Hersh and Vera John-Steiner’s book *Loving + Hating Mathematics: Challenging the Myths of Mathematical Life.* |

Announcement: Equity in Mathematics Education
A listing is given of current online offerings for *Equity in Mathematics Education*, the special issue of *JRME* that explores the subject of equity. |