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1. The Algebra Concept Inventory for college students

   Wladis, C; Offenholley, K; Sencindiver, B; Myszkowski, N; Aly, G (November 2024, Proceedings of the 46th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Kent State University.)

   Kosko, K; Caniglia, J; Courtney, S; Zolfaghari, M; Morris, G (Ed.)

   There are currently no large-scale assessments to measure algebraic conceptual understanding, particularly among college students with no more than an elementary algebra, or Algebra I, background. Here we describe the creation and validation of the Algebra Concept Inventory (ACI), which was developed for use with college students enrolled in elementary algebra or above. We describe how items on the ACI were administered and tested for validity and reliability. Analysis suggests that the instrument has reasonable validity and reliability. These results could inform researchers and practitioners on what conceptual understanding in algebra might look like and how it might be assessed. 

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2. 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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3. The multiple usages of Thinking With Algebra

   Feikes, D; McCathey, N; Oppland-Cordell, S; Walker, WS; Sorge, BH (January 2024, Proceedings of the ninth annual Indiana STEM Education Conference. Purdue University: Purdue e-Pubs. https://docs.lib.purdue.edu/instemed/2024/briefs/4)

   Walker, WS; Bryan, LA; Guzey, E (Ed.)

   Thinking With Algebra (TWA) is a National Science Foundation Project (DUE 2021414) to develop a post-secondary curriculum for intermediate algebra. TWA focuses on six elements that align with building algebraic fluency with conceptual understanding, a mixed review approach, small-group work, and whole-class discussion (Feikes, et al., 2021). Using an equity lens (Oppland-Cordell et al., 2024), TWA is designed for students, including underrepresented students, who need additional mathematical supports at the college level. Seventeen college math instructors attended a workshop on the lessons and pedagogy in order to use TWA in their college courses. Feedback from instructors participating in the TWA first-year faculty workshop indicated that the curriculum was used in many different ways to help prepare students for college algebra and other STEM courses. 

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4. When Am I (N)ever Going to Use This? How Engineers Use Algebra

   Istas, Brooke; Walkington, Candace; Leyva, Elizabeth (July 2021, 2021 ASEE Virtual Annual Conference)

   Mathematics is an important tool in engineering practice, as mathematical rules govern many designed systems (e.g., Nathan et al., 2013; Nathan et al., 2017). Investigations of structural engineers suggest that mathematical modelling is ubiquitous in their work, but the nature of the tasks they confront is not well-represented in the K-12 classroom (e.g., Gainsburg, 2006). This follows a larger literature base suggesting that school mathematics is often inauthentic and does represent how mathematics is used in practice. At the same time, algebra is a persistent gatekeeper to careers in engineering (e.g., Harackiewicz et al., 2012; Olson & Riordan, 2012). In the present study, we interviewed 12 engineers, asking them a series of questions about how they use specific kinds of algebraic function (e.g., linear, exponential, quadratic) in their work. The purpose of these interviews was to use the responses to create mathematical scenarios for College Algebra activities that would be personalized to community college students’ career interests. This curriculum would represent how algebra is used in practice by STEM professionals. However, our results were not what we expected. In this paper, we discuss three major themes that arose from qualitative analyses of the interviews. First, we found that engineers resoundingly endorsed the importance of College Algebra concepts for their day-to-day work, and uniformly stated that math was vital to engineering. However, the second theme was that the engineers struggled to describe how they used functions more complex than linear (i.e., y=mx+b) in their work. Students typically learn about linear functions prior to College Algebra, and in College Algebra explore more complex functions like polynomial, logarithmic, and exponential. Third, we found that engineers rarely use the explicit algebraic form of an algebraic function (e.g., y=3x+5), and instead rely on tables, graphs, informal arithmetic, and computerized computation systems where the equation is invisible. This was surprising, given that the bulk of the College Algebra course involves learning how to use and manipulate these formal expressions, learning skills like factoring, simplifying, solving, and interpreting parameters. We also found that these trends for engineers followed trends we saw in our larger sample where we interviewed professionals from across STEM fields. This study calls into question the gatekeeping role of formal algebraic courses like College Algebra for STEM careers. If engineers don’t actually use 75% of the content in these courses, why are they required? One reason might be that the courses are simply outdated, or arguments might be made that learning mathematics builds more general modelling and problem-solving skills. However, research from educational psychology on the difficulty of transfer would strongly refute this point – people tend to learn things that are very specific. Another reason to consider is that formal mathematics courses like advanced algebra have emerged as a very convenient mechanism to filter people by race, gender, and socioeconomic background, and to promote the maintenance of the “status quo” inequality in STEM fields. This is a critical issue to investigate for the future of the field of engineering as a whole. 

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5. Self-Explanation of worked examples integrated in an intelligent tutoring system enhances problem solving and efficiency in algebra

   Vest, N. A.:; Bartel, A. N.; Nagashima, T.; Aleven, V.; Alibali, M. W. (July 2022, Proceedings of the 44th Annual Conference of the Cognitive Science Society)

   Culbertson, J.; Perfors, A.; Rabagliati, H.; Ramenzoni, V. (Ed.)

   One pedagogical technique that promotes conceptual understanding in mathematics learners is self-explanation integrated with worked examples (e.g.,Rittle-Johnson et al., 2017). In this work, we implemented self-explanations with worked examples (correct and erroneous) in a software-based Intelligent Tutoring System (ITS) for learning algebra. We developed an approach to eliciting self-explanations in which the ITS guided students to select explanations that were conceptually rich in nature. Students who used the ITS with self-explanations scored higher on a posttest that included items tapping both conceptual and procedural knowledge than did students who used a version of the ITS that included only traditional problem-solving practice. This study replicates previous findings that self-explanation and worked examples in an ITS can foster algebra learning (Booth et al., 2013). Further, this study extends prior work to show that guiding students towards conceptual explanations is beneficial. 

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