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1. How do college students conceptualize algebraic properties?

   Wladis, C.; Sencindiver, B.; Offenholley, K. (July 2023, Proceedings for the 13th Congress of the European Society for Research in Mathematics Education (CERME13).)

   Here we explore how college students across different courses may conceptualize symbolic algebraic properties. This work draws on the theory of Grundvorstellungen (GVs) as a tool to analyse how learner conceptions do or do not align with some desired goals of instruction. In analysing interviews, several categories of conceptions, or descriptive GVs, emerged, which may be a helpful first step in understanding learners’ thinking and improving instruction on algebraic properties. 

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2. College Students’ Conceptualizations of Symbolic Algebraic Properties

   Wladis, C; Sencindiver, B; Offenholley, K; Aly, G (February 2024, Proceedings of the 26th Annual Conference on Research in Undergraduate Mathematics Education)

   Cook, S; Infante, N (Ed.)

   Here we explore how college students across a wide range of courses may conceptualize symbolic algebraic properties. We draw on the theory of Grundvorstellungen (GVs) to analyze how learner conceptions may or may not align with instructional goals. In analyzing interviews, several categories of conceptions (descriptive GVs) emerged that may help us to better understand how students conceptualize symbolic properties during instruction. 

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3. 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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4. When Wrong is Right: The Instructional Power of Multiple Conceptions

   https://doi.org/10.1145/3446871.3469750  

   Margulieux, Lauren; Denny, Paul; Cunningham, Kathryn; Deutsch, Michael; Shapiro, Benjamin R. (August 2021, Proceedings of the Sixteenth Annual Conference on International Computing Education Research)

   For many decades, educational communities, including computing education, have debated the value of telling students what they need to know (i.e., direct instruction) compared to guiding them to construct knowledge themselves (i.e., constructivism). Comparisons of these two instructional approaches have inconsistent results. Direct instruction can be more efficient for short-term performance but worse for retention and transfer. Constructivism can produce better retention and transfer, but this outcome is unreliable. To contribute to this debate, we propose a new theory to better explain these research results. Our theory, multiple conceptions theory, states that learners develop better conceptual knowledge when they are guided to compare multiple conceptions of a concept during instruction. To examine the validity of this theory, we used this lens to evaluate the literature for eight instructional techniques that guide learners to compare multiple conceptions, four from direct instruction (i.e., test-enhanced learning, erroneous examples, analogical reasoning, and refutation texts) and four from constructivism (i.e., productive failure, ambitious pedagogy, problem-based learning, and inquiry learning). We specifically searched for variations in the techniques that made them more or less successful, the mechanisms responsible, and how those mechanisms promote conceptual knowledge, which is critical for retention and transfer. To make the paper directly applicable to education, we propose instructional design principles based on the mechanisms that we identified. Moreover, we illustrate the theory by examining instructional techniques commonly used in computing education that compare multiple conceptions. Finally, we propose ways in which this theory can advance our instruction in computing and how computing education researchers can advance this general education theory. 

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5. The Algebra Concept Inventory: Creation and Validation with Students Across a Range of Math Courses in College

   Wladis, C; Offenholley, K; Sencindiver, B; Myszkowski, N; Aly, G (February 2024, Proceedings of the 26th Annual Conference on Research in Undergraduate Mathematics Education)

   Cook, S; Infante, N (Ed.)

   Even though algebraic conceptual understanding is recognized as a critical skill, existing larger scale validated algebra assessments consist mostly of computational tasks, or only assess a very narrow range of conceptions in a smaller focused domain. Further, few instruments have been validated for use with college students. In this paper, we describe the creation and validation of an algebra concept inventory for college students. We describe how items were administered, revised, and tested for validity and reliability. Results suggest that algebraic conceptual understanding is a measurable construct, and that the instrument has reasonable validity and reliability. Revision and validation is ongoing; however, lessons learned thus far provide information about what conceptual understanding in algebra might look like and how it might be assessed. 

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