Mathematics

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    How Covid-19 changed our mathematics instruction
    (Association of Mathematics Teacher Educators in Texas, 2023) Wheeler, Ann; Harrington, Rachel; Driskell, Shannon; Rhine, Steve
    Initially, we struggled with what we were doing in the classroom during the start of COVID-19 (Driskell et al., 2023). While we were aware of the benefits of digital tools such as Desmos (Orr, 2017; Rosenburg et al., 2020), Flip (Angelone, 2020), Google Sheets (Rosenburg et al, 2020), and virtual manipulatives (Reiten, 2020), we had not utilized them in a remote teaching context. While actively aiming to meet AMTE’s Standards for Preparing Teachers of Mathematics Indicator C.1.6. (2017) to prepare teachers to be skilled in using technology in ways that promote standard-based learning, we grew to understand and appreciate different aspects of digital tools that aided our students’ learning in online, hybrid, and/or in-person instruction. The purpose of this paper is to detail the instructional positives that arose from spring 2020 instruction in our university mathematics education courses and how they continue to add value to our classes and promote deep learning of mathematics.
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    The Eliahou-Kervaire resolution over a skew polynomial ring
    (Taylor & Francis, 2023) Ferraro, Luigi; Hardesty, Alexis
    In a 1987 paper, Eliahou and Kervaire constructed a minimal resolution of a class of monomial ideals in a polynomial ring, called stable ideals. As a consequence of their construction they deduced several homological properties of stable ideals. Furthermore they showed that this resolution admits an associative, graded commutative product that satisfies the Leibniz rule. In this paper we show that their construction can be extended to stable ideals in skew polynomial rings. As a consequence we show that the homological properties of stable ideals proved by Eliahou and Kervaire hold also for stable ideals in skew polynomial rings.
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    The Tor algebra of trimmings of Gorenstein ideals
    (Springer, 2023) Ferraro, Luigi; Hardesty, Alexis
    Let (R,m,k) be a regular local ring of dimension 3. Buchsbaum and Eisenbud proved that there is a skew-symmetric matrix of odd size such that I is generated by the sub-maximal pfaffians of this matrix. Let J be the ideal obtained by multiplying some of the pfaffian generators of I by m; we say that J is a trimming of I. Building on a recent paper of Vandebogert, we construct an explicit free resolution of R=J and compute a partial DG algebra structure on this resolution. We provide the full DG algebra structure in the appendix. We use the products on this resolution to study the Tor algebra of such trimmed ideals and we use the information obtained to prove that recent conjectures of Christensen, Veliche and Weyman on ideals of class G hold true in our context. Furthermore, we address the realizability question for ideals of class G.
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    Bayesian estimation of in-game home team win probability for college basketball
    (2022) Maddox, Jason T.; Sides, Ryan; Harvill, Jane L.
    Two new Bayesian methods for estimating and predicting in-game home team win probabilities in Division I NCAA men’s college basketball are proposed. The first method has a prior that adjusts as a function of lead differential and time elapsed. The second is an adjusted version of the first, where the adjustment is a linear combination of the Bayesian estimator with a time-weighted pregame win probability. The proposed methods are compared to existing methods, showing the new methods are competitive with or outperform existing methods for both estimation and prediction. The utility is illustrated via an application to the 2012/2013 through the 2019/2020 NCAA Division I Men’s Basketball seasons.
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    Mathematics teacher educators’ migration to emergency remote teaching during the onset of COVID-19
    (University of Northern Colorado, 2023) Driskell, Shannon O.S.; Harrington, Rachel; Wheeler, Ann; Rhine, Steve
    The Mathematics Teacher Educators’ Migration to Online Teaching in Response to COVID-19 survey was designed and then administered to mathematics teacher educators (MTEs) regarding experiences with migrating in-person instruction to emergency remote teaching (ERT). MTEs reported change in fostering opportunities for students to teach and to learn from their own teaching and the teaching of others, engaging students, assessing students, and using both general and mathematical technology tools for students’ learning, just to name a few. Five themes emerged: Instruction, Tools, Affective Experience, Outside Influences, and Learner Development. Implications about Technological Pedagogical Content Knowledge (TPACK) are discussed.
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    Personalizing the Hispanic student experience through data disaggregation: Implications for fall semester preregistration and long-term success
    (The EvoLLLution, 2022) Stankey, Michael; Hamner, Mark
    Leveraging meaningful data can help inform conscious strategies to support the success of learners from underserved demographics, which is critical to meeting the mission of modern institutions.
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    Examining preservice teachers' opinions about using A Different Pond to teach math
    (Association of Mathematics Teacher Educators in Texas, 2019) Wheeler, Ann; Gutierrez, Beatriz
    Engaging activities that include high level problem-solving should be an integral part of math lessons (NCTM, 2000). To pique students’ interests in mathematical tasks, teachers may also introduce the idea of popular children’s books. Using children’s literature to enhance a math lesson is not a novel concept (Lamberg & Andrews, 2011; Wilburne & Napoli, 2007; Young & Marroquin, 2006), but connections that instructors might want to make can be presented in new and innovative ways that not only address math but people from potentially unfamiliar cultures.
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    Recording and/or writing? Weighing the benefits of reflective practices
    (Magna Publications, 2020) Wheeler, Ann; Waltje, Jörg
    Let’s come right out with it: Reflecting about learning is a commendable practice that should be embedded into any learning experience! For both teachers and students, reflecting is an important practice to make sense of what one has been doing and/or learning. The concept of reflection as an “educative process” dates back to the work of John Dewey (1933), who pointed out that experience alone does not constitute learning; instead, a conscious realization must occur so that an experience can truly become a source of learning. More explicitly, reflective assignments “require students to engage in critical reflection and higher order thinking; they force students to be more open-ended and less prescriptive; and they permit students to be creative and questioning” (Dyment & O’Connell, 2011, p. 92).
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    Examining the use of mathematics-themed children's books to help pre-service teachers write tasks
    (University of Northern Colorado, 2022) Mallam, Winifred; Cobbs, Georgia A.; Wheeler, Ann
    In this qualitative study, the researchers investigated the types of K-6th grade lessons 16 preservice elementary teachers created based on mathematics-themed children’s books. Lessons were analyzed using Stein et al.’s (2017) Task Analysis Guide (TAG), as well as grouped based on grade level and covered Common Core State Standards Mathematical Practices (CCSSO, 2010a) and Content Standards (CCSSOa, 2010). Through our analysis, we found that the most common CCSSM Standards used were Numbers and Operations as well as Expressions and Equations, while the most common CCSSMP were Reason Abstractly and Quantitatively and Model with Mathematics. Coding work showed the most common TAG classification was Procedures with Connections. Sample lessons and teacher implications were discussed.
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    Using literature to engage students mathematically
    (Ohio Council of Teachers of Mathematics, 2016) Wheeler, Ann; Champion, Joe; Dybvig, Holly
    In this article, the authors share two lessons that incorporate children’s literature with the Pythagorean theorem and area to engage preservice teachers (PSTs) mathematically. Sample responses, example texts, and future work are discussed.
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    Bayesian sample size determination in two-sample Poisson models
    (MedCrave, 2015) Sides, Ryan; Kahle, David; Stamey, James
    Sample size determination is a vital part of clinical studies where cost and safety concerns lead to greater importance of not using more subjects and resources than are required. The Bayesian approach to sample size determination has the advantages of being able to use prior data and expert opinion to possibly reduce the total sample size while also acknowledging all uncertainty at the design stage. We apply a Bayesian decision theoretic approach to the problem of comparing two Poisson rates and find the required sample size to obtain a desired power while controlling the Type I error rate.
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    A phase 3, long-term, open-label safety study of Galcanezumab in patients with migraine
    (BMC, 2018) Camporeale, Angelo; Kudrow, David; Sides, Ryan; Wang, Shufang; Van Dycke, Annelies; Selzler, Katherine J.; Stauffer, Virginia L.
    Background: Galcanezumab, a humanized monoclonal antibody that selectively binds to the calcitonin gene-related peptide, has demonstrated in previous Phase 2 and Phase 3 clinical studies (≤6-month of treatment) a reduction in the number of migraine headache days and improved patients’ functioning. This study evaluated the safety and tolerability, as well as the effectiveness of galcanezumab for up to 12 months of treatment in patients with migraine.
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    Comparison between prefilled syringe and autoinjector devices on patient-reported experiences and pharmacokinetics in Galcanezumab studies
    (Dove Press, 2018) Stauffer, Virginia L.; Sides, Ryan; Lanteri-Minet, Michel; Kielbasa, William; Jin, Yan; Selzler, Katherine J.; Tepper, Stewart J.
    Purpose: The aim of this study was to compare the usability and patient-rated experiences of an autoinjector with a prefilled syringe in patients with migraine, who self-administered galcanezumab, and to compare pharmacokinetic parameters between these devices.
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    Underreporting in mixed treatment comparisons meta-analysis for Poisson data
    (Pushpa Publishing, 2021) Sides, Ryan; Stamey, James
    Mixed treatment comparisons meta-analysis has become a popular methodology because of its ability to use separate trials to make comparisons about parameters, even when the parameters have not been directly compared. We consider a mixed treatment comparisons meta-analysis problem when analyzing Poisson data allowing for counts that are potentially underreported. Previously proposed methods do not allow for the presence of underreporting. Here, we illustrate how a constant underreporting rate for all treatments has no effect on relative risk comparisons; however, when the reporting rate changes with treatment, ignoring the underreporting can lead to considerable bias. We propose an approach that accounts for the underreporting and corrects for the bias.
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    Bayesian estimation of in-game home team win probability for National Basketball Association games
    (Cornell Tech, 2022) Maddox, Jason T.; Sides, Ryan; Harvill, Jane L.
    Maddox, et al. (2022) establish a new win probability estimation for college basketball and compared the results with previous methods of Stern (1994), Desphande and Jensen (2016) and Benz (2019). This paper proposes modifications to the approach of Maddox, et al. (2022) for the NBA game and investigates the performance of the model. Enhancements to the model are developed, and the resulting adjusted model is compared with existing methods and to the ESPN counterpart. To illustrate utility, all methods are applied to the November 23, 2019 game between the Chicago Bulls and Charlotte Hornets.
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    Using educational technology applications to enhance collegiate mathematics education
    (AMTE, 2020) Wheeler, Ann; Waltje, Jorg
    Classrooms in the United States today look very different than they did 10 or 20 years ago. Mobile technology has a pervasiveand too often disruptive presence in today’s learning environments, but it does not necessarily have to be an obstacle forstudents’ learning. In our own classrooms, we have embraced the idea that technology, such as cell phones and tablets, is hereto stay as a powerful force. Thus, we strive to find ways to harness our students’ obsessions with the technology they cannotseem to function without, by incorporating it into our weekly, if not daily, mathematics instruction at the collegiate level.
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    Examing type and quality of preservice teachers’ lessons based on children’s literature
    (International Journal on Teaching and Learning Mathematics, 2020) Wheeler, Ann; Mallam, Winifred
    This qualitiative research explores the types and levels of mathematical lesson tasks that 60 future elementary and middle school teachers created during an undergraduate mathematics content and pedagogy course. Data collection consisted of 51 children’s book inspired activities written by the preservice teachers. Using Stein et al.’s Task Analysis Guide as an assessment tool, the researchers coded the activities into 1 of 4 categories, as well as categorized each activity based on its mathematical content using the Common Core State Standards for Mathematics. Results showed that a majority of the PSTs wrote activities that were classified as Procedures with Connections. Also, a majority of the activities were geometry-based, specifically transformations and two-dimesional measurements. Implications for teaching include the fact that preservice elementary and middle school teachers can create mathematics lessons based on children’s literature, which often can include mathematical tasks that are making connections to procedural mathematics or even higher order thinking tasks.