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1. Student use of three registers for representing set relationships to reason about logic in undergraduate transition to proof courses

   Dawkins, PC; Roh, KH; Bruner, O; Gonzalez, M (November 2024, Psychology in Mathematics Education - North America)

   In transition to proof courses for undergraduates, we conducted teaching experiments supporting students to learn logic and proofs rooted in set-based meanings. We invited students to reason about sets using three representational systems: set notation (including symbolic expressions and set-builder notation), mathematical statements (largely in English), and Euler diagrams. In this report, we share evidence regarding how these three representations provided students with tools for reasoning and communicating about set relationships to explore the logic of statements. By analyzing student responses to tasks that asked them to translate between the representational systems, we gain insight into the accessibility and productivity of these tools for such instruction. 

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2. What Meaning Should We Attribute to the Colon in Set-builder Notation?

   Eckman, D; Roh, K_H (February 2024, Research in Undergraduate Mathematics Education Conference)

   In this study, we report one group of students’ efforts to create a community meaning for set- builder notation collectively. Students’ ability to develop and interpret set-builder notation is essential to transition-to-proof courses. Conventionally, a colon is used in set-builder notation to (1) separate the universe of discourse from the set’s defining property and (2) indicate an ordering to these components, with the universe to the left and the property to the right of the colon. We describe one normative and non-normative interpretation of this notation and how the students’ individual attribution of conventional meanings for the colon to different inscriptions within the notation helped (or inhibited) them from interpreting these expressions. We report how communicative discourse between the students affected their meanings and discussions. 

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3. Using multiple representations to improve student understanding of quantum states

   https://doi.org/10.1103/PhysRevPhysEducRes.20.020152  

   Marshman, Emily; Maries, Alexandru; Singh, Chandralekha (December 2024, Physical Review Physics Education Research)

   [This paper is part of the Focused Collection in Investigating and Improving Quantum Education through Research.] One hallmark of expertise in physics is the ability to translate between different representations of knowledge and use the representations that make the problem-solving process easier. In quantum mechanics, students learn about several ways to represent quantum states, e.g., as state vectors in Dirac notation and as wave functions in position and momentum representation. Many advanced students in upper-level undergraduate and graduate quantum mechanics courses have difficulty translating state vectors in Dirac notation to wave functions in the position or momentum representation and vice versa. They also struggle when translating the wave function between the position and momentum representations. The research presented here describes the difficulties that students have with these concepts and how the research was used as a guide in the development, validation, and evaluation of a Quantum Interactive Learning Tutorial (QuILT) to help students develop a functional understanding of these concepts. The QuILT strives to help students with different representations of quantum states as state vectors in Dirac notation and as wave functions in position and momentum representation and with translating between these representations. We discuss the effectiveness of the QuILT from in-class implementation and evaluation. Published by the American Physical Society2024 

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4. LINKING STRUCTURES ACROSS LOGIC AND SPACE: THE ROLE OF EULER DIAGRAMS

   Antonides, J; Norton, A; Arnold, R (July 2024, For the learning of mathematics)

   This theoretical article explores the affordances and challenges of Euler diagrams as tools for supporting undergraduate introduction-to-proof students to make sense of, and reason about, logical implications. To theoretically frame students’ meaning making with Euler diagrams, we introduce the notion of logico-spatial linked structuring (or LSLS). We argue that students’ use of Euler diagrams as representations of logical statements entails a conceptual linking between spatial and non-spatial representations, and the LSLS framework provides a tool for modeling this conceptual linking. Moreover, from our Piagetian epistemological perspective, reasoning with Euler diagrams entails engaging in spatial mental operations and making a logical conclusion from the result. We illustrate the utility of the LSLS framework through examples with two undergraduate students as they reasoned about the truth of the converse and contrapositive of a given logical implication, and we identify specific spatial operations that they used and coordinated in their problem solving. 

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5. Students’ Self-Regulated Use of Diagrams in a Choice-based Intelligent Tutoring System

   Nagashima, T.; Tseng, S.; Ling, E.; Bartel, A. N.; Vest, N. A.; Silla, E. M.; Alibali, M. W.; Aleven, V. (January 2022, Proceedings of the 16th International Conference of the Learning Sciences - ICLS 2022)

   Chinn, C.; Tan, E.; Chao, C.; Kali, Y. (Ed.)

   Learners’ choices as to whether and how to use visual representations during learning are an important yet understudied aspect of self-regulated learning. To gain insight, we developed a choice-based intelligent tutor in which students can choose whether and when to use diagrams to aid their problem solving in algebra. In an exploratory classroom study with 26 students, we investigated how learners choose diagrams and how their choice behaviors relate to learning outcomes. Students who proactively chose to use diagrams achieved higher learning outcomes than those who reactively used diagrams when they made incorrect attempts. This study contributes to understanding of self-regulated use of visual representations during problem solving. 

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