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1. Student Definitions of Equivalence: Structural vs. Operational Conceptions, and Extracted vs. Stipulated Definition Construction.

   Wladis, C.; Sencindiver, B.; Offenholley, K.; Jaffe, E.; & Taton, J. (January 2022, Proceedings for the 12th Congress of the European Society for Research in Mathematics Education (CERME12))

   Cho, Sung Je (Ed.)

   In this study, we describe a model of student thinking around equivalence (conceptualized as any type of equivalence relation), presenting vignettes from student conceptions from various college courses ranging from developmental to linear algebra. In this model, we conceptualize student definitions along a continuous plane with two-dimensions: the extent to which definitions are extracted vs. stipulated; and the extent to which conceptions of equivalence are operational or structural. We present examples to illustrate how this model may help us to recognize ill-defined or operational thinking on the part of students even when they appear to be able to provide “standard” definitions of equivalence, as well as to highlight cases in which students are providing mathematically valid, if non-standard, definitions of equivalence. We hope that this framework will serve as a useful tool for analyzing student work and exploring instructional and curricular handling of equivalence. 

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2. Modeling Student Definitions of Equivalence: Operational vs. Structural Views and Extracted vs. Stipulated Definitions.

   Wladis, C.; Sencindiver, B.; Offenholley, K.; E., & Taton (February 2022, Proceedings for the 24th Annual Conference on Research in Undergraduate Mathematics Education)

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

   This paper describes a model of student thinking around equivalence (conceptualized as any type of equivalence relation), presenting vignettes from student conceptions from various college courses ranging from developmental to linear algebra, and courses in between (e.g., calculus). In this model, we conceptualize student definitions along a continuous plane with two dimensions: the extent to which definitions are extracted vs. stipulated; and the extent to which conceptions of equivalence are operational or structural. We present examples to illustrate how this model may help us to recognize ill-defined or limited thinking on the part of students even when they appear to be able to provide “standard” definitions of equivalence, as well as to highlight cases in which students are providing mathematically valid, if non-standard, definitions of equivalence. We hope that this framework will serve as a useful tool for analyzing student work, as well as exploring instructional and curricular handling of equivalence. 

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3. Students’ conceptions of substitution

   Sencindiver, B.; Wladis, C.; Offenholley, K. (July 2021, Proceedings of the 44th Conference of the International Group for the Psychology of Mathematics Education)

   Inprasitha, Maitree; Changsri, Narumon; Boonsena, Nisakorn (Ed.)

   Substitution is a key idea that is woven throughout the mathematics curriculum. In secondary school, substitution is described as an interchangeability of equal numbers, and then as a method for finding solutions of systems of equations. In university, substitution is used as a means to recognize familiar structures in Integral Calculus. Despite its prevalence in mathematics, there is little research on substitution, especially on students’ understanding of substitution. This work aims to investigate students’ meanings for substitution, and how they use it. We draw on Tall and Vinner’s (1981) ideas of concept definition and concept image to explore students’ meanings of substitution through their personal definitions of substitution, what they identify as substitution, and how they perform substitution. In this presentation, we report on elementary algebra students’ responses to questions about substitution. Data comes includes written responses to multiple-choice and open-ended questions and transcripts from clinical interviews across multiple semesters at a community college. Through a combination of thematic and conceptual analysis, we categorized students’ thinking about substitution and what features appeared to impact how they enact it. We found that students often identify substitution as a process of replacement of one mathematical object for another but differ in the generality of the mathematical objects that they consider (e.g., strictly as the replacement of a number for a variable versus replacement of any expression for another expression). Students further differed in whether or not they thought that substitution entailed equivalence of the objects being replaced. When performing substitution (e.g., substituting x+1 for y in 2y^2), we found that students’ activity was heavily based on their understanding of the structure of the expression where the substitution is taking place (the unified ‘pieces’ of 2y^2). In addition to other findings, we elaborate on the mental processes that students engage in when performing substitution and synthesize our findings with the notion of substitution equivalence (Wladis et al., 2020). 

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4. An Exploration of How College Students Think About Parentheses in the Context of Algebraic Syntax.

   Wladis, C.; Sencindiver, B.; Offenholley, K. (November 2022, Proceedings of the 44th Annual Psychology of Mathematics Education-North America (PME-NA) Conference)

   Lischka; A. E., Dyer; Jones, R. S.; Strayer, J.; Drown, S. (Ed.)

   In this paper we explore how college students across different courses appeared to interpret the meaning of parentheses or brackets in the context of algebraic syntax. This work was influenced by theories of computational vs structural thinking, and also considered the extent to which students’ definitions, computational work, and explanations appeared to be consistent with specific normative definitions of parentheses. In analyzing student work, several categories of students’ conceptions emerged, which may be helpful in diagnosing which conceptions may be more productive or problematic as students progress through algebra. For students who appear to conceptualize parentheses as a cue to non-normative procedures, several categories of procedures were found, which could have implications for instruction. 

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5. Comparing study features is easy but identifying next steps is hard: Evaluating critical thinking through the Biology Lab Inventory of Critical Thinking in Ecology

   https://doi.org/10.1002/ece3.10071  

   Heim, Ashley B.; Esparza, David; Holmes, N. G.; Smith, Michelle K. (May 2023, Ecology and Evolution)

   Abstract Critical thinking, which can be defined as the evidence‐based ways in which people decide what to trust and what to do, is an important competency included in many undergraduate science, technology, engineering, and mathematics (STEM) courses. To help instructors effectively measure critical thinking, we developed the Biology Lab Inventory of Critical Thinking in Ecology (Eco‐BLIC), a freely available, closed‐response assessment of undergraduate students' critical thinking in ecology. The Eco‐BLIC includes ecology‐based experimental scenarios followed by questions that measure how students decide on what to trust and what to do next. Here, we present the development of the Eco‐BLIC using tests of validity and reliability. Using student responses to questions and think‐aloud interviews, we demonstrate the effectiveness of the Eco‐BLIC at measuring students' critical thinking skills. We find that while students generally think like experts while evaluating what to trust, students' responses are less expert‐like when deciding on what to do next. 

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