This article presents the results from a study of 535 early undergraduate students at six universities that was designed to describe their views of the meaning of proof and how these views relate to their attitudes and beliefs towards proof and their classroom experiences with learning proof. Results show that early undergraduate students have difficulty with mathematical proof. In particular, the study showed that students' proof choices were strongly influenced by surface characteristics of the tasks. However, a large number of students appear to appreciate and acknowledge the rigor and central role of deductive proof in mathematics despite the difficulties they may face in producing proofs. Further, the study showed a strong positive relationship between students' beliefs about the role of proof and themselves as learners of proof, but weak relationship between proof ability and self-reported experiences with learning proof. Keywords Mathematical proof . Deduction . AttitudesArguments that the Bessence of mathematics lies in proofs^ (Ross 1998, p. 2) and that Bproof is not a thing separable from mathematics…. [but] is an essential component of doing, communicating, and recording mathematics^ (Schoenfeld 1994, p. 76) reinforce the centrality of proof in mathematical thinking. Moreover, not only does the act of proof Bdistinguish mathematical behavior from scientific behavior in other disciplinesÎ nt. J. Res. Undergrad. Math. Ed. (2015) (Dreyfus 1990, p. 126), it also serves as a tool for learning mathematics (Hanna 1990(Hanna , 1995Hersh 1993 MAA 2000), has emphasized this goal and has recommended that proof should be a part of students' mathematical experiences-especially of those students in mathematics-related programs. The report states that Bstudents should understand and appreciate the core of mathematical culture: the value and validity of careful reasoning, precise definition and close argument^(p. 6). Yet, research in mathematics education indicates that most students, in particular at the secondary school level, face substantial difficulties with proof (see Harel and Sowder 2007;Stylianou et al. 2009 for reviews). There also is evidence that college students face similar difficulties with proof as their high school counterparts. Most notably, Harel and Sowder (1998) showed that college students focus their attention on the format of the proof rather than the content. However, we know little about the conceptions of students as they are beginning to immerse themselves in college mathematics. Perhaps more importantly, we have not yet explored the relationship between students' understanding of proof and their beliefs towards proof. Indeed, we now know that mathematics performance in general, and the reading and writing of proofs in mathematics in particular, is a complex one that depends on a wide expanse of beliefs, knowledge, and cognitive skills and that is uniquely shaped by the realm in which learning occurs (Heinze et al. 2005). It is not at all clear, however, which of these beli...

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Traditionally, the fixed‐length linear paper‐and‐pencil (P&P) mode of administration has been the standard method of test delivery. With the advancement of technology, however, the popularity of administering tests using adaptive methods like computerized adaptive testing (CAT) and multistage testing (MST) has grown in the field of measurement in both theory and practice. In practice, several standardized tests have sections that include only set‐based items. To date, there is no study in the literature that compares these testing procedures when a test is completely set‐based under various item response theory (IRT) models. This study investigates the measurement precision of MST compared to CAT and compared to P&P tests for the one‐, two‐, and three‐parameter logistic (1‐, 2‐, and 3PL) models when the test is completely set‐based. Results showed that MST performed better for the 2‐ and 3PL models than an equivalent‐length P&P test in terms of reliability and conditional standard error of measurement. In addition, findings showed that MST performed better for the 1‐ and 2PL models than for an equivalent‐length CAT test. For the 3PL model, MST and CAT performed about the same.

This research report provides a description of the processes of evaluating the “deployability” of automated scoring (AS) systems from the perspective of large‐scale educational assessments in operational settings. It discusses a comprehensive psychometric evaluation that entails analyses that take into consideration the specific purpose of AS, the test design, the quality of human scores, the data collection design needed to train and evaluate the AS model, and the application of statistics and evaluation criteria. Finally, it notes that an effective evaluation of an AS system requires professional judgment coupled with statistical and psychometric knowledge and understanding of the risk assessment and business metrics.

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