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1. The Findings and Developed Tools from the Exploration of Procedural Pathways to Bolster Algebraic Knowledge

   Pradhan, Siddhartha (April 2024, WPI eprojects)

   Algebra is essential for delving into advanced mathematical topics and STEM courses (Chen, 2013), requiring students to apply various problem-solving strategies to solve algebraic problems (Common Core, 2010). Yet, many students struggle with learning basic algebraic concepts (National Mathematics Advisory Panel (NMAP), 2008). Over the years, both researchers and developers have created a diverse set of educational technology tools and systems to support algebraic learning, especially in facilitating the acquisition of problem- solving strategies and procedural pathways. However, there are very few studies that examine the variable strategies, decisions, and procedural pathways during mathematical problem- solving that may provide further insight into a student’s algebraic knowledge and thinking. Such research has the potential to bolster algebraic knowledge and create a more adaptive and personalized learning environment. This multi-study project explores the effects of various problem-solving strategies on students’ future mathematics performance within the gamified algebraic learning platform From Here to There! (FH2T). Together, these four studies focus on classifying, visualizing, and predicting the procedural pathways students adopted, transitioning from a start state to a goal state in solving algebraic problems. By dissecting the nature of these pathways—optimal, sub-optimal, incomplete, and dead-end—we sought to develop tools and algorithms that could infer strategic thinking that correlated with post-test outcomes. A striking observation across studies was that students who frequently engaged in what we term ‘regular dead-ending behavior’, were more likely to demonstrate higher post-test performance, and conceptual and procedural knowledge. This finding underscores the potential of exploratory learner behavior within a low-stakes gamified framework in bolstering algebraic comprehension. The implications of these findings are twofold: they accentuate the significance of tailoring gamified platforms to student behaviors and highlight the potential benefits of fostering an environment that promotes exploration without retribution. Moreover, these insights hint at the notion that fostering exploratory behavior could be instrumental in cultivating mathematical flexibility. Additionally, the developed tools and findings from the studies, paired with other commonly used student performance metrics and visualizations are used to create a collaborative dashboard–with teachers, for teachers. 

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2. Assessing Bootstrap: Algebra Students on Scaffolded and Unscaffolded Word Problems

   https://doi.org/10.1145/3159450.3159498  

   Schanzer, Emmanuel; Fisler, Kathi; Krishnamurthi, Shriram (January 2018, Proceedings of the 49th ACM Technical Symposium on Computer Science Education)

   Bootstrap:Algebra is a curricular module designed to integrate introductory computing into an algebra class; the module aims to help students improve on various essential learning outcomes from state and national algebra standards. In prior work, we published initial findings about student performance gains on algebra problems after taking Bootstrap. While the results were promising, the dataset was not large, and had students working on algebra problems that had been scaffolded with Bootstrap's pedagogy. This paper reports on a more detailed study with (a) data from more than three times as many students, (b) analysis of performance changes in incorrect answers, (c) some problems in which the Bootstrap scaffolds have been removed, and (d) an IRT analysis across the elements of Bootstrap's program-design pedagogy. Our results confirm that students improve on algebraic word problems after completing the module, even on unscaffolded problems. The nature of incorrect answers to symbolic-form questions also appears to improve after Bootstrap. 

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3. Cyclic Shuffle-Compatibility Via Cyclic Shuffle Algebras

   https://doi.org/10.1007/s00026-023-00669-9  

   Liang, Jinting; Sagan, Bruce_E; Zhuang, Yan (October 2023, Annals of Combinatorics)

   Abstract A permutation statistic$${{\,\textrm{st}\,}}$$ $\text{st}$is said to be shuffle-compatible if the distribution of$${{\,\textrm{st}\,}}$$ $\text{st}$over the set of shuffles of two disjoint permutations$$\pi $$ $\pi$and$$\sigma $$ $\sigma$depends only on$${{\,\textrm{st}\,}}\pi $$ $\text{st}\pi$,$${{\,\textrm{st}\,}}\sigma $$ $\text{st}\sigma$, and the lengths of$$\pi $$ $\pi$and$$\sigma $$ $\sigma$. Shuffle-compatibility is implicit in Stanley’s early work onP-partitions, and was first explicitly studied by Gessel and Zhuang, who developed an algebraic framework for shuffle-compatibility centered around their notion of the shuffle algebra of a shuffle-compatible statistic. For a family of statistics called descent statistics, these shuffle algebras are isomorphic to quotients of the algebra of quasisymmetric functions. Recently, Domagalski, Liang, Minnich, Sagan, Schmidt, and Sietsema defined a version of shuffle-compatibility for statistics on cyclic permutations, and studied cyclic shuffle-compatibility through purely combinatorial means. In this paper, we define the cyclic shuffle algebra of a cyclic shuffle-compatible statistic, and develop an algebraic framework for cyclic shuffle-compatibility in which the role of quasisymmetric functions is replaced by the cyclic quasisymmetric functions recently introduced by Adin, Gessel, Reiner, and Roichman. We use our theory to provide explicit descriptions for the cyclic shuffle algebras of various cyclic permutation statistics, which in turn gives algebraic proofs for their cyclic shuffle-compatibility. 

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4. Worthwhile problems: How teachers evaluate the instructional suitability of contextual algebra tasks

   Patterson, Cody L; Bui, Mai; Guajardo, Lino; Acevedo, Carlos; Rygaard_Gaspard, Brandi; McGraw, Rebecca (October 2023, Proceedings of the forty-fifth annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education)

   Lamberg, Teruni; Moss, Diana L (Ed.)

   We investigate the beliefs that influence middle and high school algebra teachers’ appraisals of contextual problems having diverse mathematical and pedagogical features. We asked six teachers to analyze six contextual algebra tasks and indicate how they would apportion instructional time among the six tasks based on their structure, pedagogical features, and connections to the real world. We recorded small-group discussions in which teachers shared their responses to this activity, and qualitatively analyzed their discussions for evidence of beliefs that influenced their appraisals of the tasks. The teachers’ beliefs about contextual problems attended to task authenticity, opportunities for mathematical activity, obligations of tasks, and pedagogy and access. Our preliminary findings can inform future efforts to equip teachers with contextual tasks that develop students’ algebraic reasoning and problem solving. 

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5. CoLA: Exploiting Compositional Structure for Automatic and Efficient Numerical Linear Algebra

   Potapczynski, Andres; Finzi, Marc; Pleiss, Geoff; Wilson, Andrew (December 2023, Conference on Neural Information Processing Systems (NeurIPS) 2023)

   Many areas of machine learning and science involve large linear algebra problems, such as eigendecompositions, solving linear systems, computing matrix exponentials, and trace estimation. The matrices involved often have Kronecker, convolutional, block diagonal, sum, or product structure. In this paper, we propose a simple but general framework for large-scale linear algebra problems in machine learning, named CoLA (Compositional Linear Algebra). By combining a linear operator abstraction with compositional dispatch rules, CoLA automatically constructs memory and runtime efficient numerical algorithms. Moreover, CoLA provides memory efficient automatic differentiation, low precision computation, and GPU acceleration in both JAX and PyTorch, while also accommodating new objects, operations, and rules in downstream packages via multiple dispatch. CoLA can accelerate many algebraic operations, while making it easy to prototype matrix structures and algorithms, providing an appealing drop-in tool for virtually any computational effort that requires linear algebra. We showcase its efficacy across a broad range of applications, including partial differential equations, Gaussian processes, equivariant model construction, and unsupervised learning. 

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