This article reports on an exploratory study of the way that eight mathematics and secondary education mathematics majors read and reflected on four student-generated arguments purported to be proofs of a single theorem. The results suggest that such undergraduates tend to focus on surface features of arguments and that their ability to determine whether arguments are proofs is very limited-perhaps more so than either they or their instructors recognize. The article begins by discussing arguments (purported proofs) regarded as texts and validations of those arguments, that is, reflections of individuals checking whether such arguments really are proofs of theorems. It relates the mathematics research community's views of proofs and their validations to ideas from reading comprehension and literary theory. Then, a detailed analysis of the four student-generated arguments is given and the eight students' validations of them are analyzed.

It is widely reported that partisanship in the United States Congress is at an historic high. Given that individuals are persuaded to follow party lines while having the opportunity and incentives to collaborate with members of the opposite party, our goal is to measure the extent to which legislators tend to form ideological relationships with members of the opposite party. We quantify the level of cooperation, or lack thereof, between Democrat and Republican Party members in the U.S. House of Representatives from 1949–2012. We define a network of over 5 million pairs of representatives, and compare the mutual agreement rates on legislative decisions between two distinct types of pairs: those from the same party and those formed of members from different parties. We find that despite short-term fluctuations, partisanship or non-cooperation in the U.S. Congress has been increasing exponentially for over 60 years with no sign of abating or reversing. Yet, a group of representatives continue to cooperate across party lines despite growing partisanship.

ABSTRACT. This study focuses on undergraduate students' ability to unpack informally written mathematical statements into the language of predicate calculus. Data were collected between 1989 and 1993 from 61 students in six small sections of a "bridge" course designed to introduce proofs and mathematical reasoning. We discuss this data from a perspective that extends the notion of concept image to that of statement image and introduces the notion of proof framework to indicate that part of a theorem's image which corresponds to the top-level logical structure of a proof. For simplified informal calculus statements, just 8.5% of unpacking attempts were successful; for actual statements from calculus texts, this dropped to 5%. We infer that these students would be unable to reliably relate informally stated theorems with the top-level logical structure of their proofs and hence could not be expected to construct proofs or validate them, i.e., determine their correctness.

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