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1. Student reasoning about the least-squares problem in inquiry-oriented linear algebra

   Wawro, M.; Park, M.; Zandieh, M.; Bettersworth, Z.; & Lee, I. (January 2023, Proceedings of the Annual Conference on Research in Undergraduate Mathematics Education)

   Cook, S.; Katz, B.; Moore-Russo, D. (Ed.)

   The method of Least Square Approximation is an important topic in some linear algebra classes. Despite this, little is known about how students come to understand it, particularly in a Realistic Mathematics Education setting. Here, we report on how students used literal symbols and equations when solving a least squares problem in a travel scenario, as well as their reflections on the least squares equation in an open-ended written question. We found students used unknowns and parameters in a variety of ways. We highlight how their use of dot product equations can be helpful towards supporting their understanding of the least squares equation. 

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2. Student reasoning about the least-squares problem in inquiry-oriented linear algebra.

   Wawro, M. (January 2023, Proceedings of the Annual Conference on Research in Undergraduate Mathematics Education)

   Cook, S.; Katz, B.; Moore-Russo, D. (Ed.)

   The method of Least Square Approximation is an important topic in some linear algebra classes. Despite this, little is known about how students come to understand it, particularly in a Realistic Mathematics Education setting. Here, we report on how students used literal symbols and equations when solving a least squares problem in a travel scenario, as well as their reflections on the least squares equation in an open-ended written question. We found students used unknowns and parameters in a variety of ways. We highlight how their use of dot product equations can be helpful towards supporting their understanding of the least squares equation. 

   more » « less
3. CP decomposition for tensors via alternating least squares with QR decomposition

   https://doi.org/10.1002/nla.2511  

   Minster, Rachel; Viviano, Irina; Liu, Xiaotian; Ballard, Grey (June 2023, Numerical Linear Algebra with Applications)

   The CP tensor decomposition is used in applications such as machine learning and signal processing to discover latent low-rank structure in multidimensional data. Computing a CP decomposition via an alternating least squares (ALS) method reduces the problem to several linear least squares problems. The standard way to solve these linear least squares subproblems is to use the normal equations, which inherit special tensor structure that can be exploited for computational efficiency. However, the normal equations are sensitive to numerical ill-conditioning, which can compromise the results of the decomposition. In this paper, we develop versions of the CP-ALS algorithm using the QR decomposition and the singular value decomposition, which are more numerically stable than the normal equations, to solve the linear least squares problems. Our algorithms utilize the tensor structure of the CP-ALS subproblems efficiently, have the same complexity as the standard CP-ALS algorithm when the input is dense and the rank is small, and are shown via examples to produce more stable results when ill-conditioning is present. Our MATLAB implementation achieves the same running time as the standard algorithm for small ranks, and we show that the new methods can obtain lower approximation error. 

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4. What is a vector to students?

   Lee, I. (January 2022, Proceedings of the Annual Conference on Research in Undergraduate Mathematics Education)

   Karunakaran, S.; Higgins, A. (Ed.)

   This study presents linear algebra students’ vector conception found in the least-squares solution context through an IOLA (Inquiry-Oriented Linear Algebra) classroom teaching experiment. Students’ reflection writings after the classroom teaching experiment are the data source. Using the previously found student conception of vector in another study as a basic framing, the data have been analyzed to investigate how students used the word vector and what they referred to. A framework is developed as a tool to be useful in a wide range of describing student conception of a vector emphasizing their natural way of thinking of a vector and on their use of the vector. 

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5. What is a vector to students?

   Lee, I. (January 2022, Proceedings of the Annual Conference on Research in Undergraduate Mathematics Education)

   Karunakaran, S. S.; Higgins, A. (Ed.)

   This study presents linear algebra students’ vector conception found in the least-squares solution context through an IOLA (Inquiry-Oriented Linear Algebra) classroom teaching experiment. Students’ reflection writings after the classroom teaching experiment are the data source. Using the previously found student conception of vector in another study as a basic framing, the data have been analyzed to investigate how students used the word vector and what they referred to. A framework is developed as a tool to be useful in a wide range of describing student conception of a vector emphasizing their natural way of thinking of a vector and on their use of the vector. 

   more » « less