This study investigated evidence of how students’ mental models of fundamental kinematic relations evolved (i.e., developed cognitively over time) as observed during an introductory course in calculus-based classical mechanics. The core of the curriculum is based on a claim known as Galileo’s principle of relativity. The course material comprised a standard sequence of topics in classical mechanics, reconfigured through a framework scaffolded from this principle. The research focused specifically on students’ mental models for the concept of velocity. Four instruments were developed and integrated into a suite of variegated tools for data collection. The suite probed indicators across diverse domains or modalities of mental processing: visual, quantitative, and verbal. Evidence of student mental models included student data derived from answers to multiple-choice questions, short written passages, symbolic computations, quantitative answers, pictorial sketches, and semi-structured interviews. A limiting model, or rubric, for approaching a comprehensive mental model for velocity coalesced after results from axial coding of sketches and interviews were considered in connection with the contingency tables made from short answer frequency counts and the Wilcoxon mean comparison of problem-solving tests. The rubric consisted of three identifiable tiers that ascended in cognitive sophistication. Statistically significant evidence was found for growth from the first to the second stage of this three-stage rubric for student velocity models. By the end of the semester, students showed increased capacity to treat velocity as a relation between two objects, rather than as a property of one object. Students typically developed a correct habit of demanding a second object when asserting velocity for a first. When provided with a second or reference object, many students demonstrated an acquired ability to adjust their conclusion for target velocity. Little to no statistically significant evidence was found, however, to suggest growth in student mental models from the second to the third stage of the three-stage rubric. In particular, the typical student mental model proved too fragile to manage problems involving a third moving object or a second dimension of space. Evidence was insufficient to indicate deliberate student distinction of the relational character of velocity from the relational quality of an interaction such as force. Across the visual and problem-solving domains, stage three difficulty was attributed to the mismanagement of arrows known as vectors. In the verbal domain, questions about three-object scenarios revealed conflation of velocity with force. In light of the data from all three domains, velocity vectors were considered in direct connection with force vectors. A cognitive connection between velocity vectors and force vectors was identified as a potential source of dissonance. The report of the study concludes by considering ways to scaffold the teaching of velocity vectors from the teaching of displacement vectors. The recommendation for improved physics pedagogy and future research involves increased visual, verbal, and quantitative emphases on velocity as a bearing, as distinct from an interaction.

This dissertation questions the dominant role that analogies to painting have played in the critical reception of the Italian madrigal—especially its flagship technique, the madrigalism—and argues for a more historically sensitive approach to sixteenth-century discussions of music–text relations in that genre. This approach centers rhetoric, understood broadly to encompass theories of style and subject matter, emotional response, and fictional representation, as the primary field of study to which musicians turned in their efforts to theorize musical expression. In its final chapter, the dissertation suggests that this rhetorical saturation of discourse around madrigals also to some degree influenced the composing of madrigals. Chapter 1 traces the outlines of what I call the “Galileian critical tradition” between the publication of Vincenzo Galilei’s 1581 "Dialogo della musica antica, et ella moderna" and the present. This tradition is characterized by a tendency among writers to dismiss the madrigalism on expressive grounds and, as time passes, increasingly in visual terms. Returning to the sixteenth century, chapter 2 argues that the phrase “imitating the words” (imitare le parole), which was adopted by Galilei, Gioseffo Zarlino, and others in the mid- to late Cinquecento to describe those novel techniques for the musical representation of text commonly seen in madrigals, acquires an affective connotation, in addition to its descriptive meaning, when placed in the context of contemporaneous literary studies of imitation (mimesis), especially those stemming from the recovery of Aristotle’s "Poetics." This affective dimension of the imitation principle, first theorized by Aristotle but creatively elaborated by his Renaissance commentators, highlights the cognitive pleasure that humans, because of a natural affinity for imitation, take from fiction—a species of pleasure, I suggest, relevant to the musical practice that “imitating the words” names. Turning from pleasure to passion, chapter 3 makes a case for the presence of a “two-stage model for emotional arousal” (as I call it) in sixteenth-century Italian music theory. Surveying this model’s foundation in physiological, rhetorical, and natural-philosophical texts, I show how Zarlino adapted its principles to musical purposes in his 1558 "Istitutioni harmoniche" and then trace the afterlife of Zarlino’s theory across a number of texts, including both music theory treatises and an epic poem. Chapter 4, finally, analyzes a late Cinquecento madrigal cycle—the three settings in Giaches de Wert’s "Ottavo libro de madrigali a cinque voci" (1586) from the “Armida” episode of Torquato Tasso’s "Gerusalemme liberata" (1581)—as a series of musical “stimuli” for listener responses patterned on the theoretical discussions studied in chapters 2 and 3. In its attention to the close-knit relationship between the expressive qualities of Wert’s music and the emotions they invite, this chapter follows the example of the ancient rhetorical tradition and its sixteenth-century inheritance, which emphasize the role of human psychology in determining the orator’s art.