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- Creators: Middleton, James
Description
This dissertation describes an investigation of four students' ways of thinking about functions of two variables and rate of change of those two-variable functions. Most secondary, introductory algebra, pre-calculus, and first and second semester calculus courses do not require students to think about functions of more than one variable. Yet vector calculus, calculus on manifolds, linear algebra, and differential equations all rest upon the idea of functions of two (or more) variables. This dissertation contributes to understanding productive ways of thinking that can support students in thinking about functions of two or more variables as they describe complex systems with multiple variables interacting. This dissertation focuses on modeling the way of thinking of four students who participated in a specific instructional sequence designed to explore the limits of their ways of thinking and in turn, develop a robust model that could explain, describe, and predict students' actions relative to specific tasks. The data was collected using a teaching experiment methodology, and the tasks within the teaching experiment leveraged quantitative reasoning and covariation as foundations of students developing a coherent understanding of two-variable functions and their rates of change. The findings of this study indicated that I could characterize students' ways of thinking about two-variable functions by focusing on their use of novice and/or expert shape thinking, and the students' ways of thinking about rate of change by focusing on their quantitative reasoning. The findings suggested that quantitative and covariational reasoning were foundational to a student's ability to generalize their understanding of a single-variable function to two or more variables, and their conception of rate of change to rate of change at a point in space. These results created a need to better understand how experts in the field, such as mathematicians and mathematics educators, thinking about multivariable functions and their rates of change.
ContributorsWeber, Eric David (Author) / Thompson, Patrick (Thesis advisor) / Middleton, James (Committee member) / Carlson, Marilyn (Committee member) / Saldanha, Luis (Committee member) / Milner, Fabio (Committee member) / Van de Sande, Carla (Committee member) / Arizona State University (Publisher)
Created2012
Description
The purpose of this project is to determine the feasibility of a water tunnel designed to meet certain constraints. The project goals are to tailor a design for a given location, and to produce a repeatable design sizing and shape process for specified constraints. The primary design goals include a 1 m/s flow velocity in a 30cm x 30cm test section for 300 seconds. Secondary parameters, such as system height, tank height, area contraction ratio, and roof loading limits, may change depending on preference, location, or environment. The final chosen configuration is a gravity fed design with six major components: the reservoir tank, the initial duct, the contraction nozzle, the test section, the exit duct, and the variable control exit nozzle. Important sizing results include a minimum water weight of 60,000 pounds, a system height of 7.65 meters, a system length of 6 meters (not including the reservoir tank), a large shallow reservoir tank width of 12.2 meters, and height of 0.22 meters, and a control nozzle exit radius range of 5.25 cm to 5.3 cm. Computational fluid dynamic simulation further supports adherence to the design constraints but points out some potential areas for improvement in dealing with flow irregularities. These areas include the bends in the ducts, and the contraction nozzle. Despite those areas recommended for improvement, it is reasonable to conclude that the design and process fulfill the project goals.
ContributorsZykan, Brandt Davis Healy (Author) / Wells, Valana (Thesis director) / Middleton, James (Committee member) / Barrett, The Honors College (Contributor) / Mechanical and Aerospace Engineering Program (Contributor)
Created2014-05
Description
The traditional early design phase of an aircraft involves a design approach in which the model's characteristics are defined before the CAD model is built. This thesis discusses an alternative to the early design process employing the use of a parametric model. A parametric model is one in which its characteristics are defined as functions of input parameters that a user will choose, as opposed to being pre-defined. This allows for faster iterations of the CAD design of an aircraft going through its first design phases. In order to demonstrate the feasibility and efficiency, a tool was developed in the form of a script written in Python that compiles into a plugin that a user can install into Rhino. With a full template of about 70 parameters that have significant effects on the performance characteristics of an aircraft, a user with the plugin can generate a full model. The overall design phase and development of the script into a publicly available installation file is discussed below. Results for the thesis took the form of insight gained into the field of parametric modeling. After development and implementation, emphasis points such as generation time, focus on parameters with large effect on aircraft performance, and interpolation of parameters dependent upon others were concluded.
ContributorsElliott, Steven Joseph (Author) / Takahashi, Tim (Thesis director) / Middleton, James (Committee member) / Mechanical and Aerospace Engineering Program (Contributor) / Barrett, The Honors College (Contributor)
Created2016-12
Description
This dissertation reports three studies about what it means for teachers and students to reason with frames of reference: to conceptualize a reference frame, to coordinate multiple frames of reference, and to combine multiple frames of reference. Each paper expands on the previous one to illustrate and utilize the construct of frame of reference. The first paper is a theory paper that introduces the mental actions involved in reasoning with frames of reference. The concept of frames of reference, though commonly used in mathematics and physics, is not described cognitively in any literature. The paper offers a theoretical model of mental actions involved in conceptualizing a frame of reference. Additionally, it posits mental actions that are necessary for a student to reason with multiple frames of reference. It also extends the theory of quantitative reasoning with the construct of a ‘framed quantity’. The second paper investigates how two introductory calculus students who participated in teaching experiments reasoned about changes (variations). The data was analyzed to see to what extent each student conceptualized the variations within a conceptualized frame of reference as described in the first paper. The study found that the extent to which each student conceptualized, coordinated, and combined reference frames significantly affected his ability to reason productively about variations and to make sense of his own answers. The paper ends by analyzing 123 calculus students’ written responses to one of the tasks to build hypotheses about how calculus students reason about variations within frames of reference. The third paper reports how U.S. and Korean secondary mathematics teachers reason with frame of reference on open-response items. An assessment with five frame of reference tasks was given to 539 teachers in the US and Korea, and the responses were coded with rubrics intended to categorize responses by the extent to which they demonstrated conceptualized and coordinated frames of reference. The results show that the theory in the first study is useful in analyzing teachers’ reasoning with frames of reference, and that the items and rubrics function as useful tools in investigating teachers’ meanings for quantities within a frame of reference.
ContributorsJoshua, Surani Ashanthi (Author) / Thompson, Patrick W (Thesis advisor) / Carlson, Marilyn (Committee member) / Roh, Kyeong Hah (Committee member) / Middleton, James (Committee member) / Culbertson, Robert (Committee member) / Arizona State University (Publisher)
Created2019