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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
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
Description
Many developing countries do not have health care systems that can afford technological biomedical devices or supplies to make such devices operational. To fill this void, nonprofit organizations, like Project C.U.R.E., recondition retired biomedical instrumentation so they can send medical supplies to help these developing countries. One of the issues with this is that sometimes the devices are unusable because components or expendable supplies are not available (Bhadelia). This issue has also been shown in the Impact Evaluations that Project C.U.R.E. receives from the clinics that explain the reasons why certain devices are no longer in use. That need underlies the idea on which this honors thesis has come into being. The purpose of this honors project was to create packing lists for biomedical instruments that Project C.U.R.E. recycles. This packing list would decrease the likelihood of important items being forgotten when sending devices. If an extra fuse, battery, light bulb, cuff or transducer is the difference between a functional or a nonfunctional medical device, such a list would be of benefit to Project C.U.R.E and these developing countries. In order to make this packing list, manuals for each device were used to determine what supplies were required, what was necessary for cleaning, and what supplies were desirable but functionally optional. This list was then added into a database that could be easily navigated and could help when packing up boxes for a shipment. The database also makes adding and editing the packing list simple and easy so that as Project C.U.R.E. gets more donated devices the packing list can grow.
ContributorsGraft, Kelsey Anne (Author) / Coursen, Jerry (Thesis director) / Walters, Danielle (Committee member) / Harrington Bioengineering Program (Contributor) / Barrett, The Honors College (Contributor)
Created2018-05
Description
The Hippo signaling pathway is responsible for regulating organ size through cell proliferation, stemness, and apoptosis. Through targeting proteins Yes-associated kinase 1(YAP) and transcriptional co-activator with a PDZ-binding domain(TAZ), YAP/TAZ are unable to enter the nucleus and bind with coactivators to express target genes. To understand YAP/TAZ dynamics and its role in tumorigenesis, tissue regeneration, and tissue degeneration, a regulatory network was modeled by ordinary differential equations. Using MATLAB, the deterministic behavior of the network was observed to determine YAP/TAZ activity in different states. Performing the bifurcation analysis of the system through Oscill8, three states were identified: tumorigenic/regenerative, degenerative, and homeostatic states. Further analysis through parameter modification allowed a better understanding of which proteins can be targeted for cancer and degenerative disease.
ContributorsBarra Avila, Diego Rodrigo (Author) / Tian, Xiaojun (Thesis director) / Wang, Xiao (Committee member) / Harrington Bioengineering Program (Contributor, Contributor) / Barrett, The Honors College (Contributor)
Created2020-05
Description
Based on James Marcia's theory, identity development in youth is the degree to which one has explored and committed to a vocation [1], [2]. During the path to an engineering identity, students will experience a crisis, when one's values and choices are examined and reevaluated, and a commitment, when the outcome of the crisis leads the student to commit to becoming an engineer. During the crisis phase, students are offered a multitude of experiences to shape their values and choices to influence commitment to becoming an engineering student. Student's identities in engineering are fostered through mentoring from industry, alumni, and peer coaching [3], [4]; experiences that emphasize awareness of the importance of professional interactions [5]; and experiences that show creativity, collaboration, and communication as crucial components to engineering. Further strategies to increase students' persistence include support in their transition to becoming an engineering student, education about professional engineers and the workplace [6], and engagement in engineering activities beyond the classroom. Though these strategies are applied to all students, there are challenges students face in confronting their current identity and beliefs before they can understand their value to society and achieve personal satisfaction. To understand student's progression in developing their engineering identity, first year engineering students were surveyed at the beginning and end of their first semester. Students were asked to rate their level of agreement with 22 statements about their engineering experience. Data included 840 cases. Items with factor loading less than 0.6 suggesting no sufficient explanation were removed in successive factor analysis to identify the four factors. Factor analysis indicated that 60.69% of the total variance was explained by the successive factors. Survey questions were categorized into three factors: engineering identity as defined by sense of belonging and self-efficacy, doubts about becoming an engineer, and exploring engineering. Statements in exploring engineering indicated student awareness, interest and enjoyment within engineering. Students were asked to think about whether they spent time learning what engineers do and participating in engineering activities. Statements about doubts about engineering to engineering indicated whether students had formed opinions about their engineering experience and had understanding about their environment. Engineering identity required thought in belonging and self-efficacy. Belonging statements called for thought about one's opinion in the importance of being an engineer, the meaning of engineering, an attachment to engineering, and self-identification as an engineer. Statements about self-efficacy required students to contemplate their personal judgement of whether they would be able to succeed and their ability to become an engineer. Effort in engineering indicated student willingness to invest time and effort and their choices and effort in their engineering discipline.
ContributorsNguyen, Amanda (Author) / Ganesh, Tirupalavanam (Thesis director) / Robinson, Carrie (Committee member) / Harrington Bioengineering Program (Contributor) / Barrett, The Honors College (Contributor)
Created2018-05
Description
Alzheimer’s Disease (AD) affects over 5 million individuals in the U.S. and has a direct cost estimated in excess of $200 billion per year. Broadly speaking, there are two forms of AD—early-onset, familial AD (FAD) and late-onset-sporadic AD (SAD). Animal models of AD, which rely on the overexpression of FAD-related mutations, have provided important insights into the disease. However, these models do not display important disease-related pathologies and have been limited in their ability to model the complex genetics associated with SAD.
Advances in cellular reprogramming, have enabled the generation of in vitro disease models that can be used to dissect disease mechanisms and evaluate potential therapeutics. To that end, efforts by many groups, including the Brafman laboratory, to generated patient-specific hiPSCs have demonstrated the promise of studying AD in a simplified and accessible system. However, neurons generated from these hiPSCs have shown some, but not all, of the early molecular and cellular hallmarks associated with the disease. Additionally, phenotypes and pathological hallmarks associated with later stages of the human disease have not been observed with current hiPSC-based systems. Further, disease relevant phenotypes in neurons generated from SAD hiPSCs have been highly variable or largely absent. Finally, the reprogramming process erases phenotypes associated with cellular aging and, as a result, iPSC-derived neurons more closely resemble fetal brain rather than adult brain.
It is well-established that in vivo cells reside within a complex 3-D microenvironment that plays a significant role in regulating cell behavior. Signaling and other cellular functions, such as gene expression and differentiation potential, differ in 3-D cultures compared with 2-D substrates. Nonetheless, previous studies using AD hiPSCs have relied on 2-D neuronal culture models that do not reflect the 3-D complexity of native brain tissue, and therefore, are unable to replicate all aspects of AD pathogenesis. Further, the reprogramming process erases cellular aging phenotypes. To address these limitations, this project aimed to develop bioengineering methods for the generation of 3-D organoid-based cultures that mimic in vivo cortical tissue, and to generate an inducible gene repression system to recapitulate cellular aging hallmarks.
Advances in cellular reprogramming, have enabled the generation of in vitro disease models that can be used to dissect disease mechanisms and evaluate potential therapeutics. To that end, efforts by many groups, including the Brafman laboratory, to generated patient-specific hiPSCs have demonstrated the promise of studying AD in a simplified and accessible system. However, neurons generated from these hiPSCs have shown some, but not all, of the early molecular and cellular hallmarks associated with the disease. Additionally, phenotypes and pathological hallmarks associated with later stages of the human disease have not been observed with current hiPSC-based systems. Further, disease relevant phenotypes in neurons generated from SAD hiPSCs have been highly variable or largely absent. Finally, the reprogramming process erases phenotypes associated with cellular aging and, as a result, iPSC-derived neurons more closely resemble fetal brain rather than adult brain.
It is well-established that in vivo cells reside within a complex 3-D microenvironment that plays a significant role in regulating cell behavior. Signaling and other cellular functions, such as gene expression and differentiation potential, differ in 3-D cultures compared with 2-D substrates. Nonetheless, previous studies using AD hiPSCs have relied on 2-D neuronal culture models that do not reflect the 3-D complexity of native brain tissue, and therefore, are unable to replicate all aspects of AD pathogenesis. Further, the reprogramming process erases cellular aging phenotypes. To address these limitations, this project aimed to develop bioengineering methods for the generation of 3-D organoid-based cultures that mimic in vivo cortical tissue, and to generate an inducible gene repression system to recapitulate cellular aging hallmarks.
ContributorsBounds, Lexi Rose (Author) / Brafman, David (Thesis director) / Wang, Xiao (Committee member) / Harrington Bioengineering Program (Contributor) / Barrett, The Honors College (Contributor)
Created2018-05
Description
The combination of immunohistochemical (IHC) stainings and optical microscopy has allowed for the visualization of specific microscopic structures within tissue; however, limitations in light and antibody penetration mitigate the scale on which these images can be taken (Alshammari et al, 2016; Marx, 2014). Tissue clearing, specifically the removal of lipids to improve sample transparency, solves the former weakness well, but does not improve antibody penetration significantly (Chung et al, 2013; Treweek et al, 2015). Therefore, there is a need to equalize the maximum depth that light can pass through a section with the depth at which there is recognizable fluorescence. This is particularly important when staining blood vessels as traditional size limitations exclusively allows for cross sectional visualization. Passive CLARITY Technique (PACT) has been at the forefront of tissue clearing protocols, utilizing an acrylamide hydrogel solution to maintain structure and sodium dodecyl sulfate to wash out lipids (Tomer et al, 2014). PACT is limited in its ability to clear larger sections and is not conducive to IHC antibody diffusion (Treweek et al, 2015). In order to circumvent these drawbacks, CUBIC was developed as an alternative passive protocol, aimed at being scalable to any tissue size (Richardson, 2015; Susaki et al, 2015). This study compared the effectiveness of both protocols in high and low lipid tissues in the context of blood vessel staining efficacy. Upon initial comparison, it became apparent that there was a statistically significant difference in mean DAPI intensity at all depths, up to 200 micrometers, between CUBIC and PACT \u2014 the former showcasing brighter stainings. Moreover, it was found that PACT does not remove erythrocytes from the tissue meaning that their auto-fluorescence is seen during imaging. Therefore, for blood vessel stainings, only CUBIC was optimized and quantitatively analyzed. In both tissue conditions as well as for two stainings, DAPI and fibronectin (FNCT), optimized CUBIC demonstrated a statistically significant difference from standard CUBIC with regards to mean fluorescent intensity.
ContributorsSidhu, Gurpaul Singh (Author) / VanAuker, Michael (Thesis director) / Kodibagkar, Vikram (Committee member) / Harrington Bioengineering Program (Contributor) / Barrett, The Honors College (Contributor)
Created2018-05