Sarah Theule Lubienski, Rochelle Gutierrez
In this rejoinder, the authors further detail their positions on the role of gaps analyses in mathematics education research as outlined in the previous 2 articles. They clarify areas of agreement and probe areas of disagreement, focusing on the benefits and dangers they see in either emphasizing educational disparities between groups or shifting the focus to the advancement of particular groups.
Sarah Theule Lubienski
There are three articles in this month's "Research Commentary" section. Analyses of disparities in students' mathematics experiences and outcomes are an essential part of efforts to promote equity. Scholars concerned about equity should not write off such analyses as mere "gap gazing." Research on gaps between underserved groups and their more advantaged peers are important for shaping public opinion and informing education policy. Analyses of gaps also inform mathematics education research and practice, illuminating which groups and curricular areas are most in need of intervention and additional study.
Rochelle Gutierrez
A substantial amount of research in mathematics education seeks to document disparities in achievement between middle-class White students and students who are Black, Latina/Latino, First Nations, English language learners, or working class. I outline the dangers in maintaining an achievement-gap focus. These dangers include offering little more than a static picture of inequities, supporting deficit thinking and negative narratives about students of color and working-class students, perpetuating the myth that the problem (and therefore solution) is a technical one, and promoting a narrow definition of learning and equity
Heather C. Hill, Deborah Loewenberg Ball, Steven G. Schilling
There is widespread agreement that effective teachers have unique knowledge of students' mathematical ideas and thinking. However, few scholars have focused on conceptualizing this domain, and even fewer have focused on measuring this knowledge. In this article, we describe an effort to conceptualize and develop measures of teachers' combined knowledge of content and students by writing, piloting, and analyzing results from multiple-choice items.
Anderson Norton
This article reports on students' learning through conjecturing, by drawing on a semester-long teaching experiment with 6 sixth-grade students. It focuses on 1 of the students, Josh, who developed especially powerful ways of operating over the course of the teaching experiment. Through a fine-grained analysis of Josh's actions, this article integrates Piaget's scheme theory (1950/2001) and Peirce's logic of abduction (1998) into a new theory about conjecturing that explains Josh's learning.
Keith Weber
The purpose of this article is to investigate the mathematical practice of proof validation-that is, the act of determining whether an argument constitutes a valid proof. The results of a study with 8 mathematicians are reported. The mathematicians were observed as they read purported mathematical proofs and made judgments about their validity; they were then asked reflective interview questions about their validation processes and their views on proving. The results suggest that mathematicians use several different modes of reasoning in proof validation, including formal reasoning and the construction of rigorous proofs, informal deductive reasoning, and example-based reasoning.