More Like this

1. Conceptual Knowledge, Procedural Knowledge, and Metacognition in Routine and Nonroutine Problem Solving

   https://doi.org/10.1111/cogs.13048  

   Braithwaite, David W.; Sprague, Lauren (October 2021, Cognitive Science)

   Abstract When, how, and why students use conceptual knowledge during math problem solving is not well understood. We propose that when solving routine problems, students are more likely to recruit conceptual knowledge if their procedural knowledge is weak than if it is strong, and that in this context, metacognitive processes, specifically feelings of doubt, mediate interactions between procedural and conceptual knowledge. To test these hypotheses, in two studies (Ns = 64 and 138), university students solved fraction and decimal arithmetic problems while thinking aloud; verbal protocols and written work were coded for overt uses of conceptual knowledge and displays of doubt. Consistent with the hypotheses, use of conceptual knowledge during calculation was not significantly positively associated with accuracy, but was positively associated with displays of doubt, which were negatively associated with accuracy. In Study 1, participants also explained solutions to rational arithmetic problems; using conceptual knowledge in this context was positively correlated with calculation accuracy, but only among participants who did not use conceptual knowledge during calculation, suggesting that the correlation did not reflect “online” effects of using conceptual knowledge. In Study 2, participants also completed a nonroutine problem‐solving task; displays of doubt on this task were positively associated with accuracy, suggesting that metacognitive processes play different roles when solving routine and nonroutine problems. We discuss implications of the results regarding interactions between procedural knowledge, conceptual knowledge, and metacognitive processes in math problem solving. 

   more » « less
2. Applying a mathematical sense-making framework to student work and its potential for curriculum design

   https://doi.org/17010138  

   Gifford, J.D. and (May 2021, Physical review)

   null (Ed.)

   This paper extends prior work establishing an operationalized framework of mathematical sense making (MSM) in physics. The framework differentiates between the object being understood (either physical or mathematical) and various tools (physical or mathematical) used to mediate the sense-making process. This results in four modes of MSM that can be coordinated and linked in various ways. Here, the framework is applied to novel modalities of student written work (both short answer and multiple choice). In detailed studies of student reasoning about the photoelectric effect, we associate these MSM modes with particular multiple choice answers, and substantiate this association by linking both the MSM modes and multiple choice answers with finer-grained reasoning elements that students use in solving a specific problem. Through the multiple associations between MSM mode, distributions of reasoning elements, and multiple- choice answers, we confirm the applicability of this framework to analyzing these sparser modalities of student work and its utility for analyzing larger-scale (N > 100) datasets. The association between individual reasoning elements and both MSM modes and MC answers suggest that it is possible to cue particular modes of student reasoning and answer selection. Such findings suggest potential for this framework to be applicable to the analysis and design of curriculum. 

   more » « less
3. Board 414: Understanding and Scaffolding the Productive Beginnings of Engineering Judgment in Undergraduate Students

   Caserto, Melissa J; Swenson, Jessica ES; Johnson, Aaron W (June 2024, American Society for Engineering Education)

   This work presents the first year of work on a project addressing the productive beginnings of engineering judgment in undergraduate engineering students. In particular, we discuss a new research question about how open-ended modeling problems (OEMPs), which engage students in engineering judgment, foster the growth of conceptual knowledge. Because OEMPs are open-ended with multiple answers, they are different from the typical well-defined “textbook” problems given in engineering science courses where students learn canonical mathematical models and apply relevant formulas to find a single correct answer. By looking at the conceptual gains that result from assigning an OEMP, we aim to convince other instructors to create and assign open-ended questions. More practice using engineering judgment will give students experience with engineering judgment before receiving their engineering degree. Ideally, this will increase the number of graduates prepared for real-world engineering application. 

   more » « less
4. Sharing scenarios facilitate division performance in preschoolers.

   https://doi.org/https://doi.org/10.1016/j.cogdev.2020.100954  

   Hamamouche, K.; Chernyak, N.; Cordes, S. (January 2020, Cognitive development)

   null (Ed.)

   Understanding division is critical for later mathematical achievement. Yet division concepts are difficult to grasp and are often not explicitly taught until middle childhood. Given the structural similarity between sharing and division, we investigated whether contextualizing division problems as sharing scenarios improved preschool-aged children’s abilities to solve them, as compared with other arithmetic problems which do not share structural similarities with sharing. Preschoolers (N = 113) completed an addition, subtraction, and division problem in either a sharing context that presented arithmetic via contextualized sharing scenarios, or a comparable, linguistically-matched non-social context (randomly assigned). Children were assessed on their formal, verbal responses and their informal, non-verbal, action-based responses (abilities to solve the problems using manipulatives) to these arithmetic problems. Most critically, context predicted children’s performance on the division, but not the addition or subtraction trial, supporting a structural link between sharing and division. Results also revealed that children’s action-based responses to the arithmetic problems were much more accurate than their verbal ones. Results are discussed in terms of the conceptual link between division and sharing. 

   more » « less
5. 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. 

   more » « less