The Precalculus Concept Assessment: A Tool for Assessing Students’ Reasoning Abilities and Understandings

This article describes the development of the Precalculus Concept Assessment (PCA) instrument, a 25-item multiple-choice exam. The reasoning abilities and understandings central to precalculus and foundational for beginning calculus were identified and characterized in a series of research studies and are articulated in the PCA Taxonomy. These include a strong understanding of ideas of rate of change and function, a process view of function, and the ability to use covariational reasoning to examine and represent how two covarying quantities change together. This taxonomy guided the PCA development and now provides the theoretical basis for interpreting and reporting PCA results. A critical element of PCA's design was to identify the constructs essential for learning calculus and to employ methods to assure that PCA items are effective in assessing these constructs. We illustrate the role that cognitive research played during both the design and validation phases of the PCA instrument. We also describe our Four-Phase Instrument Development Framework that articulates the methods used to create and validate PCA. This framework should also be useful for others interested in developing similar instruments in other content areas. The uses of PCA are described and include (a) assessing student learning in college algebra and precalculus, (b) comparing the effectiveness of various curricular treatments, and (c) determining student readiness for calculus.     

Falling Man: encounters with catastrophic change

The assault on the twin towers thrust Americans into an encounter with catastrophic change. Previously protected by the illusion of security fed by our relative imperviousness to others' points of view, we are harshly awakened to our defensive blindness. This rupture helps us see the particular beyond the seeming universality, locating culture as a variable frame defining meanings through the narratives that hold complexities of human experience in conceptual space. Don DeLillo's Falling Man offers a reading of catastrophe as a forced encounter with fallibility, breaking apart illusions of sameness and difference, towards integration of what trauma excludes from awareness.     

Do production patterns influence the processing of speech in prelinguistic infants?

The headturn preference procedure was used to test 18 infants on their response to three different passages chosen to reflect their individual production patterns. The passages contained nonwords with consonants in one of three categories: (a) often produced by that infant (‘own’), (b) rarely produced by that infant but common at that age (‘other’), and (c) not generally produced by infants. Infants who had a single ‘own’ consonant showed no significant preference for either ‘own’ (a) or ‘other’ (b) passages. In contrast, infants’ with two ‘own’ consonants exhibited greater attention to ‘other’ passages (b). Both groups attended equally to the passage featuring consonants rarely produced by infants of that age (c). An analysis of a sample of the infant-directed speech ruled out the mothers’ speech as a source of the infant preferences. The production-based shift to a focus on the ‘other’ passage suggests that nascent production abilities combine with emergent perceptual experience to facilitate word learning.