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1. Reasoning about Uncertainties in Discrete-Time Dynamical Systems using Polynomial Forms.

   Sankaranarayanan, Sriram ; Chou, Yi ; Goubault, Eric ; Putot, Sylvie ( December 2020 , Part of Advances in Neural Information Processing Systems 33 (NeurIPS 2020))

   In this paper, we propose polynomial forms to represent distributions of state variables over time for discrete-time stochastic dynamical systems. This problem arises in a variety of applications in areas ranging from biology to robotics. Our approach allows us to rigorously represent the probability distribution of state variables over time, and provide guaranteed bounds on the expectations, moments and probabilities of tail events involving the state variables. First, we recall ideas from interval arithmetic, and use them to rigorously represent the state variables at time t as a function of the initial state variables and noise symbols that model the random exogenous inputs encountered before time t. Next, we show how concentration of measure inequalities can be employed to prove rigorous bounds on the tail probabilities of these state variables. We demonstrate interesting applications that demonstrate how our approach can be useful in some situations to establish mathematically guaranteed bounds that are of a different nature from those obtained through simulations with pseudo-random numbers. 

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2. Reasoning about Uncertainties in Discrete-Time Dynamical Systems using Polynomial Forms

   Sankaranarayanan, Sriram ; Chou, Yi ; Goubault, Eric ; and Putot, Sylvie ( December 2020 , Advances in Neural Information Processing System (NeurIPS))

   null (Ed.)

   In this paper, we propose polynomial forms to represent distributions of state variables over time for discrete-time stochastic dynamical systems. This problem arises in a variety of applications in areas ranging from biology to robotics. Our approach allows us to rigorously represent the probability distribution of state variables over time, and provide guaranteed bounds on the expectations, moments and probabilities of tail events involving the state variables. First we recall ideas from interval arithmetic, and use them to rigorously represent the state variables at time t as a function of the initial state variables and noise symbols that model the random exogenous inputs encountered before time t. Next we show how concentration of measure inequalities can be employed to prove rigorous bounds on the tail probabilities of these state variables. We demonstrate interesting applications that demonstrate how our approach can be useful in some situations to establish mathematically guaranteed bounds that are of a different nature from those obtained through simulations with pseudo-random numbers. 

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3. The Physics Inventory of Quantitative Reasoning: Assessing Student Reasoning About Sign

   Olsho, Alexis ; White Brahmia, Suzanne ; Smith, Trevor I. ; Boudreaux, Andrew ( May 2019 , Conference in Undergraduate Mathematics Education)
4. A Cross-Sectional Investigation of Students’ Reasoning About Integer Addition and Subtraction: Ways of Reasoning, Problem Types, and Flexibility

   Lamb, L.L. ; Bishop, J.P. ; Philipp, R. A. ; Whitacre, I. ; & Schappelle, B. P. ( November 2018 , Journal for research in mathematics education)

   In a cross-sectional study, 160 students in Grades 2, 4, 7, and 11 were interviewed about their reasoning when solving integer addition and subtraction open-number sentence problems. We applied our previously developed framework for 5 Ways of Reasoning (WoRs) to our data set to describe patterns within and across participant groups. Our analysis of the WoRs also led to the identification of 3 problem types: change-positive, all-negatives, and counterintuitive. We found that problem type influenced student performance and tended to evoke a different way of reasoning. We showed that those with more experience with negative numbers use WoRs more flexibly than those with less experience and that flexibility is correlated with accuracy. We provide 3 types of resources for educators: (a) WoRs and problem-types frameworks, (b) characterization of flexibility with integer addition and subtraction, and (c) development of a trajectory of learning about integers. 

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5. The Physics Inventory of Quantitative Reasoning: Assessing Student Reasoning About Sign

   Olsho, Alexis ; White Brahmia, Suzanne ; Boudreaux, Andrew ; Smith, Trevor I. ( January 2019 , Proceedings of the 22nd Annual Conference on Research in Undergraduate Mathematics Education)

   An increase in general quantitative literacy and discipline-specific Physics Quantitative Literacy (PQL) is a major course goal of most introductory-level physics sequences—yet there exist no instruments to assess how PQL changes with instruction in these types of courses. To address this need, we are developing the Physics Inventory of Quantitative Literacy (PIQL), a multiple-choice inventory to assess students’ sense-making about arithmetic and algebra concepts that underpin reasoning in introductory physics courses—proportional reasoning, covariational reasoning and reasoning about sign and signed quantities. The PIQL will be used to not only to assess students’ PQL at specific points in time, but also to track changes in and development of PQL that can be attributed to instruction. Data from early versions of the PIQL suggest that students experience difficulty reasoning about sign and signed quantities. 

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