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1. Cusps and commensurability classes ofhyperbolic 4–manifolds

   https://doi.org/10.2140/agt.2023.23.3805  

   Sell, Connor (January 2023, Algebraic & Geometric Topology)

   There are six orientable compact flat 3–manifolds that can occur as cusp cross-sections of hyperbolic 4–manifolds. We provide criteria for exactly when a given commensurability class of arithmetic hyperbolic 4–manifolds contains a representative with a given cusp type. In particular, for three of the six cusp types, we provide infinitely many examples of commensurability classes that contain no manifolds with cusps of the given type; no such examples were previously known for any cusp type. 

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2. Leveraging Prediction and Reflection in a Computational Setting to Enrich Undergraduate Students’ Combinatorial Thinking

   https://doi.org/10.1080/07370008.2021.2020793  

   Lockwood, Elise (January 2022, Cognition and Instruction)

   Enyedy, Noel (Ed.)

   In this paper, I discuss undergraduate students’ engagement in basic Python programming while solving combinatorial problems. Students solved tasks that were designed to involve programming, and they were encouraged to engage in activities of prediction and reflection. I provide data from two paired teaching experiments, and I outline how the task design and instructional interventions particularly supported students’ combinatorial reasoning. I argue that emergent computational representations and the prompts for prediction and reflection were especially useful in supporting students’ reasoning about fundamental combinatorial ideas. I argue that this particular mathematical example informs broader notions of disciplinary reflexivity and representational heterogeneity, providing insight into computational thinking practices in the domain of mathematics. Ultimately, I aim to explore the nature of computing and enumeration, shedding light on why the two disciplines are particularly well-suited to support each other. I conclude with implications and avenues for future research. 

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3. Integrative Generalized Convex Clustering Optimization and Feature Selection for Mixed Multi-View Data

   Wang, Minjie; Allen, Genevera I. (January 2021, Journal of Machine Learning Research)

   In mixed multi-view data, multiple sets of diverse features are measured on the same set of samples. By integrating all available data sources, we seek to discover common group structure among the samples that may be hidden in individualistic cluster analyses of a single data view. While several techniques for such integrative clustering have been explored, we propose and develop a convex formalization that enjoys strong empirical performance and inherits the mathematical properties of increasingly popular convex clustering methods. Specifically, our Integrative Generalized Convex Clustering Optimization (iGecco) method employs different convex distances, losses, or divergences for each of the different data views with a joint convex fusion penalty that leads to common groups. Additionally, integrating mixed multi-view data is often challenging when each data source is high-dimensional. To perform feature selection in such scenarios, we develop an adaptive shifted group-lasso penalty that selects features by shrinking them towards their loss-specific centers. Our so-called iGecco+ approach selects features from each data view that are best for determining the groups, often leading to improved integrative clustering. To solve our problem, we develop a new type of generalized multi-block ADMM algorithm using sub-problem approximations that more efficiently fits our model for big data sets. Through a series of numerical experiments and real data examples on text mining and genomics, we show that iGecco+ achieves superior empirical performance for high-dimensional mixed multi-view data. 

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4. Teaching Elementary Mathematics with Educational Robotics

   https://doi.org/10.30707/JSTE57.1.1664998343.900405  

   Yuling, Zhuang; Foster, Jonathan K.; Conner AnnaMarie; Crawford, Barbara A.; Foutz, Timothy; Hill, Roger B. (January 2022, Journal of STEM Teacher Education)

   Brown, Ryan; Antink-Meyer, Allison (Ed.)

   Current education reforms call for engaging students in learning science, technology, engineering, and mathematics (STEM) in an integrative way. This critical case study of one fourth grade teacher investigated the use of educational robots (ER) not only for teaching coding, but as an instructional support in teaching mathematical concepts. To support teachers in teaching coding in an integrative and logical manner, our team developed the Collective Argumentation Learning and Coding (CALC) approach. The CALC approach consists of three elements: choice of task, coding content, and teacher support for argumentation. After a cohort of elementary teachers completed a professional development course, we followed them into their classrooms to support and document implementation of the CALC approach. Data for this case consisted of video recordings of two lessons, a Pre-interview, and Post-interview after each lesson. Research questions included: How does an elementary teacher use the CALC approach (integrative STEM approach) to teach mathematics concepts with ER? What are the teacher’s perspectives towards teaching mathematics with ER using an integrative STEM approach? Results from this critical case provide evidence that teachers can successfully integrate ER into the mathematics curriculum without losing coherence of mathematics topics and while remaining sensitive to students’ needs. 

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5. Scoring Integrated Meaningful Play in Story-Driven Games with Q-Learning

   Bettis, Alan; Newberry, Danielle; Munoz-Avila, Hector (January 2020, CEUR workshop proceedings)

   Osborn, Joseph C. (Ed.)

   Integrated meaningful play is the idea that player’s choices should have a long-term effect on the game. In this paper we present I-score (for integrated), a scoring function for scoring integrative game play as a function of the game’s storylines. The I-scores are in the range [0,1]. In games with I-scores close to one, player’s early choices determine the game’s ending; choices made later in the game do not change the final ending of the game. In contrast, games with I-scores close to zero, players’s choices can change the ending until the very end. Games with scores closer to 0.5 provide a more balanced player choice whereby the game’s ending still can be changed despite early decisions, but not so much that the ending could be changed at any point. 

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