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1. Perceiving precedence: Order of operations errors are predicted by perception of equivalent expressions

   https://doi.org/10.5964/jnc.14103  

   Bye, Jeffrey Kramer; Chan, Jenny Yun-Chen; Closser, Avery H; Lee, Ji-Eun; Shaw, Stacy T; Ottmar, Erin R (January 2024, Journal of Numerical Cognition)

   Students often perform arithmetic using rigid problem-solving strategies that involve left-to-right-calculations. However, as students progress from arithmetic to algebra, entrenchment in rigid problem-solving strategies can negatively impact performance as students experience varied problem representations that sometimes conflict with the order of precedence (the order of operations). Research has shown that the syntactic structure of problems, and students’ perceptual processes, are involved in mathematics performance and developing fluency with precedence. We examined 837 U.S. middle schoolers’ propensity for precedence errors on six problems in an online mathematics game. We included an algebra knowledge assessment, math anxiety measure, and a perceptual math equivalence task measuring quick detection of equivalent expressions as predictors of students’ precedence errors. We found that students made more precedence errors when the leftmost operation was invalid (addition followed by multiplication). Individual difference analyses revealed that students varied in propensity for precedence errors, which was better predicted by students’ performance on the perceptual math equivalence task than by their algebra knowledge or math anxiety. Students’ performance on the perceptual task and interactive game provide rich insights into their real-time understanding of precedence and the role of perceptual processes in equation solving. 

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2. Using Learning Maps and Bloom’s Taxonomy to Develop a New Instrument to Assess Knowledge Transfer from Physics to Statics Courses.

   https://doi.org/10.18260/1-2--57307  

   Wijesinghe, Priyantha; Seneviratne, Varuni; Medsker, Larry; Giles, Courtney; Ottati, Marlee (June 2025, ASEE Conferences)

   This paper presents the design and analysis of a pilot problem set deployed to engineering students to assess their retention of physics knowledge at the start of a statics course. The problem set was developed using the NSF-IUSE (grant #2315492) Learning Map project (LMap) and piloted in the spring and fall of 2024. The LMap process is rooted in the Analysis, Design, Development, Implementation, and Evaluation (ADDIE) model [1] and Backward Design [2,3], extending these principles to course sequences to align learning outcomes, assessments, and instructional practices. The primary motivation for this problem set (Statics Knowledge Inventory, SKI) was to evaluate students' understanding and retention of physics concepts at the beginning of a statics course. The SKI includes a combination of multiple-choice questions (MCQ) and procedural problems, filling a gap in widely-used concept inventories for physics and statics, such as the Force Concept Inventory (FCI) and Statics Concept Inventory (SCI), which evaluate learning gains within a course, rather than knowledge retention across courses. Using the LMap analysis and instructor consultations, we identified overlapping concepts and topics between Physics and Statics courses, referred to here as “interdependent learning outcomes” (ILOs). The problem set includes 15 questions—eight MCQs and seven procedural problems. Unlike most concept inventories, procedural problems were added to provide insight into students’ problem-solving approach and conceptual understanding. These problems require students to perform calculations, demonstrate their work, and assess their conceptual understanding of key topics, and allow the instructors to assess essential prerequisite skills like drawing free-body diagrams (FBDs), computing forces and moments, and performing basic vector calculation and unit conversions. Problems were selected and adapted from physics and statics textbooks, supplemented by instructor-designed questions to ensure full coverage of the ILOs. We used the revised 2D Bloom’s Taxonomy [4] and a 3D representation of it [5] to classify each problem within a 6x4 matrix (six cognitive processes x four knowledge dimensions). This classification provided instructors and students with a clear understanding of the cognitive level required for each problem. Additionally, we measured students’ perceived confidence and difficulty in each problem using two questions on a 3-point Likert scale. The first iteration of the problem set was administered to 19 students in the spring 2024 statics course. After analyzing their performance, we identified areas for improvement and revised the problem set, removing repetitive MCQs and restructuring the procedural problems into scaffolded, multi-part questions with associated rubrics for evaluation. The revised version, consisting of five MCQs and six procedural problems, was deployed to 136 students in the fall 2024 statics course. A randomly selected subset of student answers from the second iteration was graded and analyzed to compare with the first. This analysis will inform future efforts to evaluate knowledge retention and transfer in key skills across sequential courses. In collaboration with research teams developing concept inventories for mechanics courses, we aim to integrate these procedural problems into future inventories. 

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3. Assessment of Metacognitive Skills in Design and Manufacturing

   https://doi.org/10.18260/1-2--34191  

   Elliott, L; Aqlan, F.; Zhao, R.; Janney, M. (June 2020, ASEE Annual Conference proceedings)

   Metacognition is the understanding of your own knowledge including what knowledge you do not have and what knowledge you do have. This includes knowledge of strategies and regulation of one’s own cognition. Studying metacognition is important because higher-order thinking is commonly used, and problem-solving skills are positively correlated with metacognition. A positive previous disposition to metacognition can improve problem-solving skills. Metacognition is a key skill in design and manufacturing, as teams of engineers must solve complex problems. Moreover, metacognition increases individual and team performance and can lead to more original ideas. This study discusses the assessment of metacognitive skills in engineering students by having the students participate in hands-on and virtual reality activities related to design and manufacturing. The study is guided by two research questions: (1) do the proposed activities affect students’ metacognition in terms of monitoring, awareness, planning, self-checking, or strategy selection, and (2) are there other components of metacognition that are affected by the design and manufacturing activities? The hypothesis is that the participation in the proposed activities will improve problem-solving skills and metacognitive awareness of the engineering students. A total of 34 undergraduate students participated in the study. Of these, 32 were male and 2 were female students. All students stated that they were interested in pursuing a career in engineering. The students were divided into two groups with the first group being the initial pilot run of the data. In this first group there were 24 students, in the second group there were 10 students. The groups’ demographics were nearly identical to each other. Analysis of the collected data indicated that problem-solving skills contribute to metacognitive skills and may develop first in students before larger metacognitive constructs of awareness, monitoring, planning, self-checking, and strategy selection. Based on this, we recommend that the problem-solving skills and expertise in solving engineering problems should be developed in students before other skills emerge or can be measured. While we are sure that the students who participated in our study have awareness as well as the other metacognitive skills in reading, writing, science, and math, they are still developing in relation to engineering problems. 

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4. The Findings and Developed Tools from the Exploration of Procedural Pathways to Bolster Algebraic Knowledge

   Pradhan, Siddhartha (April 2024, WPI eprojects)

   Algebra is essential for delving into advanced mathematical topics and STEM courses (Chen, 2013), requiring students to apply various problem-solving strategies to solve algebraic problems (Common Core, 2010). Yet, many students struggle with learning basic algebraic concepts (National Mathematics Advisory Panel (NMAP), 2008). Over the years, both researchers and developers have created a diverse set of educational technology tools and systems to support algebraic learning, especially in facilitating the acquisition of problem- solving strategies and procedural pathways. However, there are very few studies that examine the variable strategies, decisions, and procedural pathways during mathematical problem- solving that may provide further insight into a student’s algebraic knowledge and thinking. Such research has the potential to bolster algebraic knowledge and create a more adaptive and personalized learning environment. This multi-study project explores the effects of various problem-solving strategies on students’ future mathematics performance within the gamified algebraic learning platform From Here to There! (FH2T). Together, these four studies focus on classifying, visualizing, and predicting the procedural pathways students adopted, transitioning from a start state to a goal state in solving algebraic problems. By dissecting the nature of these pathways—optimal, sub-optimal, incomplete, and dead-end—we sought to develop tools and algorithms that could infer strategic thinking that correlated with post-test outcomes. A striking observation across studies was that students who frequently engaged in what we term ‘regular dead-ending behavior’, were more likely to demonstrate higher post-test performance, and conceptual and procedural knowledge. This finding underscores the potential of exploratory learner behavior within a low-stakes gamified framework in bolstering algebraic comprehension. The implications of these findings are twofold: they accentuate the significance of tailoring gamified platforms to student behaviors and highlight the potential benefits of fostering an environment that promotes exploration without retribution. Moreover, these insights hint at the notion that fostering exploratory behavior could be instrumental in cultivating mathematical flexibility. Additionally, the developed tools and findings from the studies, paired with other commonly used student performance metrics and visualizations are used to create a collaborative dashboard–with teachers, for teachers. 

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5. BOARD # 438: Research Initiation: Facilitating Knowledge Transfer within Engineering Curricula

   https://doi.org/10.18260/1-2--55817  

   De_Rosa, Alexander; Reed, Teri; Van_Horne, Samuel (June 2025, ASEE Conferences)

   It is well-established that students have difficulty transferring theory and skills between courses in their undergraduate curriculum. At the same time, many college-level courses only concern material relating to the course itself and do not cover how this material might be used elsewhere. It is unsurprising, then, that students are unable to transfer and integrate knowledge from multiple areas into new problems as part of capstone design courses, for example, or in their careers. More work is required to better enable students to transfer knowledge between their courses, learn skills and theory more deeply, and to form engineers who are better able to adapt to new situations and solve “systems-level” problems. Various authors in both the cognitive and disciplinary sciences have discussed these difficulties with the transfer of knowledge, and noted the need to develop tools and techniques for promoting knowledge transfer, as well as to help students develop cross-course connections. This work aimed to address these barriers to knowledge transfer, and crucially develop the needed activities and practices for promoting transfer by answering the following research questions: (1) What are the primary challenges experienced by students when tasked with transferring theory and skills from prior courses, specifically mathematics and physics? (2) What methods of prior knowledge activation are most effective in enabling students to apply this prior knowledge in new areas of study? In this paper we present a holistic summary of the work completed under this award. Initially, findings from a series of n=23 think aloud interviews, in which participants were asked to solve a typical engineering statics problem, is presented. These interviews evidenced multiple barriers to knowledge transfer (lack of prior knowledge, accuracy of prior knowledge, conceptual understanding, lack of teaching of applications, language of problem, curricular mapping) that hindered participant success in terms of using their mathematical skills to solve the problem. Findings also indicated the importance of reflective thinking on behalf of the participants to their problem solving success. Based on this initial work using think alouds, a further set of interviews (n=8) were conducted to more deeply examine student conceptions of important mathematical topics that are transferred into engineering such as integration and centroids. Findings indicated that participant knowledge and understanding of centroids in particular was generally based around more intuitive or geometrical conceptions rather than concrete physical or mathematical models. Following up on the initial study of problem solving, the importance of reflection on behalf of the problem solver was also examined in more detail. Comparison of expert (faculty) and novice (student) approaches to problem solving demonstrates how often experts reflect on their progress during the solving process and the manner in which they are able to connect problems in one context to similar problems they have encountered in the past in other areas of engineering. The ability of experts to “chunk” problems into smaller stages and reflect on individual elements of the problem at hand rather than the problem as a whole was also seen to be a differentiating factor in their approach as compared to novices. Similar to this paper, the associated poster presentation will cover a holistic representation of the findings of this study. 

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