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- Creators: Brahms, Johannes, 1833-1897
Description
Research on combinatorics education is sparse when compared with other fields in mathematics education. This research attempted to contribute to the dearth of literature by examining students' reasoning about enumerative combinatorics problems and how students conceptualize the set of elements being counted in such problems, called the solution set. In particular, the focus was on the stable patterns of reasoning, known as ways of thinking, which students applied in a variety of combinatorial situations and tasks. This study catalogued students' ways of thinking about solution sets as they progressed through an instructional sequence. In addition, the relationships between the catalogued ways of thinking were explored. Further, the study investigated the challenges students experienced as they interacted with the tasks and instructional interventions, and how students' ways of thinking evolved as these challenges were overcome. Finally, it examined the role of instruction in guiding students to develop and extend their ways of thinking. Two pairs of undergraduate students with no formal experience with combinatorics participated in one of the two consecutive teaching experiments conducted in Spring 2012. Many ways of thinking emerged through the grounded theory analysis of the data, but only eight were identified as robust. These robust ways of thinking were classified into three categories: Subsets, Odometer, and Problem Posing. The Subsets category encompasses two ways of thinking, both of which ultimately involve envisioning the solution set as the union of subsets. The three ways of thinking in Odometer category involve holding an item or a set of items constant and systematically varying the other items involved in the counting process. The ways of thinking belonging to Problem Posing category involve spontaneously posing new, related combinatorics problems and finding relationships between the solution sets of the original and the new problem. The evolution of students' ways of thinking in the Problem Posing category was analyzed. This entailed examining the perturbation experienced by students and the resulting accommodation of their thinking. It was found that such perturbation and its resolution was often the result of an instructional intervention. Implications for teaching practice are discussed.
ContributorsHalani, Aviva (Author) / Roh, Kyeong Hah (Thesis advisor) / Fishel, Susanna (Committee member) / Saldanha, Luis (Committee member) / Thompson, Patrick (Committee member) / Zandieh, Michelle (Committee member) / Arizona State University (Publisher)
Created2013
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
ContributorsBrahms, Johannes, 1833-1897 (Composer)
ContributorsBrahms, Johannes, 1833-1897 (Composer)
ContributorsBrahms, Johannes, 1833-1897 (Composer)
ContributorsBrahms, Johannes, 1833-1897 (Composer)
Description
There have been a number of studies that have examined students’ difficulties in understanding the idea of logarithm and the effectiveness of non-traditional interventions. However, there have been few studies that have examined the understandings students develop and need to develop when completing conceptually oriented logarithmic lessons. In this document, I present the three papers of my dissertation study. The first paper examines two students’ development of concepts foundational to the idea of logarithm. This paper discusses two essential understandings that were revealed to be problematic and essential for students’ development of productive meanings for exponents, logarithms and logarithmic properties. The findings of this study informed my later work to support students in understanding logarithms, their properties and logarithmic functions. The second paper examines two students’ development of the idea of logarithm. This paper describes the reasoning abilities two students exhibited as they engaged with tasks designed to foster their construction of more productive meanings for the idea of logarithm. The findings of this study provide novel insights for supporting students in understanding the idea of logarithm meaningfully. Finally, the third paper begins with an examination of the historical development of the idea of logarithm. I then leveraged the insights of this literature review and the first two papers to perform a conceptual analysis of what is involved in learning and understanding the idea of logarithm. The literature review and conceptual analysis contributes novel and useful information for curriculum developers, instructors, and other researchers studying student learning of this idea.
ContributorsKuper Flores, Emily Ginamarie (Author) / Carlson, Marilyn (Thesis advisor) / Thompson, Patrick (Committee member) / Milner, Fabio (Committee member) / Zazkis, Dov (Committee member) / Czocher, Jennifer (Committee member) / Arizona State University (Publisher)
Created2018
Description
The concept of distribution is one of the core ideas of probability theory and inferential statistics, if not the core idea. Many introductory statistics textbooks pay lip service to stochastic/random processes but how do students think about these processes? This study sought to explore what understandings of stochastic process students develop as they work through materials intended to support them in constructing the long-run behavior meaning for distribution.
I collected data in three phases. First, I conducted a set of task-based clinical interviews that allowed me to build initial models for the students’ meanings for randomness and probability. Second, I worked with Bonnie in an exploratory teaching setting through three sets of activities to see what meanings she would develop for randomness and stochastic process. The final phase consisted of me working with Danielle as she worked through the same activities as Bonnie but this time in teaching experiment setting where I used a series of interventions to test out how Danielle was thinking about stochastic processes.
My analysis shows that students can be aware that the word “random” lives in two worlds, thereby having conflicting meanings. Bonnie’s meaning for randomness evolved over the course of the study from an unproductive meaning centered on the emotions of the characters in the context to a meaning that randomness is the lack of a pattern. Bonnie’s lack of pattern meaning for randomness subsequently underpinned her image of stochastic/processes, leading her to engage in pattern-hunting behavior every time she needed to classify a process as stochastic or not. Danielle’s image of a stochastic process was grounded in whether she saw the repetition as being reproducible (process can be repeated, and outcomes are identical to prior time through the process) or replicable (process can be repeated but the outcomes aren’t in the same order as before). Danielle employed a strategy of carrying out several trials of the process, resetting the applet, and then carrying out the process again, making replicability central to her thinking.
I collected data in three phases. First, I conducted a set of task-based clinical interviews that allowed me to build initial models for the students’ meanings for randomness and probability. Second, I worked with Bonnie in an exploratory teaching setting through three sets of activities to see what meanings she would develop for randomness and stochastic process. The final phase consisted of me working with Danielle as she worked through the same activities as Bonnie but this time in teaching experiment setting where I used a series of interventions to test out how Danielle was thinking about stochastic processes.
My analysis shows that students can be aware that the word “random” lives in two worlds, thereby having conflicting meanings. Bonnie’s meaning for randomness evolved over the course of the study from an unproductive meaning centered on the emotions of the characters in the context to a meaning that randomness is the lack of a pattern. Bonnie’s lack of pattern meaning for randomness subsequently underpinned her image of stochastic/processes, leading her to engage in pattern-hunting behavior every time she needed to classify a process as stochastic or not. Danielle’s image of a stochastic process was grounded in whether she saw the repetition as being reproducible (process can be repeated, and outcomes are identical to prior time through the process) or replicable (process can be repeated but the outcomes aren’t in the same order as before). Danielle employed a strategy of carrying out several trials of the process, resetting the applet, and then carrying out the process again, making replicability central to her thinking.
ContributorsHatfield, Neil (Author) / Thompson, Patrick (Thesis advisor) / Carlson, Marilyn (Committee member) / Middleton, James (Committee member) / Lehrer, Richard (Committee member) / Reiser, Mark R. (Committee member) / Arizona State University (Publisher)
Created2019
Model Based Automatic and Robust Spike Sorting for Large Volumes of Multi-channel Extracellular Data
Description
Spike sorting is a critical step for single-unit-based analysis of neural activities extracellularly and simultaneously recorded using multi-channel electrodes. When dealing with recordings from very large numbers of neurons, existing methods, which are mostly semiautomatic in nature, become inadequate.
This dissertation aims at automating the spike sorting process. A high performance, automatic and computationally efficient spike detection and clustering system, namely, the M-Sorter2 is presented. The M-Sorter2 employs the modified multiscale correlation of wavelet coefficients (MCWC) for neural spike detection. At the center of the proposed M-Sorter2 are two automatic spike clustering methods. They share a common hierarchical agglomerative modeling (HAM) model search procedure to strategically form a sequence of mixture models, and a new model selection criterion called difference of model evidence (DoME) to automatically determine the number of clusters. The M-Sorter2 employs two methods differing by how they perform clustering to infer model parameters: one uses robust variational Bayes (RVB) and the other uses robust Expectation-Maximization (REM) for Student’s 𝑡-mixture modeling. The M-Sorter2 is thus a significantly improved approach to sorting as an automatic procedure.
M-Sorter2 was evaluated and benchmarked with popular algorithms using simulated, artificial and real data with truth that are openly available to researchers. Simulated datasets with known statistical distributions were first used to illustrate how the clustering algorithms, namely REMHAM and RVBHAM, provide robust clustering results under commonly experienced performance degrading conditions, such as random initialization of parameters, high dimensionality of data, low signal-to-noise ratio (SNR), ambiguous clusters, and asymmetry in cluster sizes. For the artificial dataset from single-channel recordings, the proposed sorter outperformed Wave_Clus, Plexon’s Offline Sorter and Klusta in most of the comparison cases. For the real dataset from multi-channel electrodes, tetrodes and polytrodes, the proposed sorter outperformed all comparison algorithms in terms of false positive and false negative rates. The software package presented in this dissertation is available for open access.
This dissertation aims at automating the spike sorting process. A high performance, automatic and computationally efficient spike detection and clustering system, namely, the M-Sorter2 is presented. The M-Sorter2 employs the modified multiscale correlation of wavelet coefficients (MCWC) for neural spike detection. At the center of the proposed M-Sorter2 are two automatic spike clustering methods. They share a common hierarchical agglomerative modeling (HAM) model search procedure to strategically form a sequence of mixture models, and a new model selection criterion called difference of model evidence (DoME) to automatically determine the number of clusters. The M-Sorter2 employs two methods differing by how they perform clustering to infer model parameters: one uses robust variational Bayes (RVB) and the other uses robust Expectation-Maximization (REM) for Student’s 𝑡-mixture modeling. The M-Sorter2 is thus a significantly improved approach to sorting as an automatic procedure.
M-Sorter2 was evaluated and benchmarked with popular algorithms using simulated, artificial and real data with truth that are openly available to researchers. Simulated datasets with known statistical distributions were first used to illustrate how the clustering algorithms, namely REMHAM and RVBHAM, provide robust clustering results under commonly experienced performance degrading conditions, such as random initialization of parameters, high dimensionality of data, low signal-to-noise ratio (SNR), ambiguous clusters, and asymmetry in cluster sizes. For the artificial dataset from single-channel recordings, the proposed sorter outperformed Wave_Clus, Plexon’s Offline Sorter and Klusta in most of the comparison cases. For the real dataset from multi-channel electrodes, tetrodes and polytrodes, the proposed sorter outperformed all comparison algorithms in terms of false positive and false negative rates. The software package presented in this dissertation is available for open access.
ContributorsMa, Weichao (Author) / Si, Jennie (Thesis advisor) / Papandreou-Suppappola, Antonia (Committee member) / He, Jingrui (Committee member) / Helms Tillery, Stephen (Committee member) / Arizona State University (Publisher)
Created2019