University students are often asked to learn abstract concepts. Abstract concepts are hard to learn. Giving specific examples can help learning abstract concepts. These examples might limit understanding to the similarities between the abstract domain and particular examples. The primary purpose of this study was to test whether exposure to…

This study investigated how students reason about the partial derivative and the directional derivative of a multivariable function at a given point, using different graphical representations for the function in the problem statement. Questions were formulated to be as isomorphic as possible in both mathematics and physics contexts and were given…

This full paper considers how collaborative discourse can reveal ways upper-class engineering students mechanically reason about engineering concepts. Argumentation and negotiation during collaborative, multimodal discourse using speech and gestures helps establish common ground between learners and fosters reflection on their conceptual…

Point-set topology is among the most abstract branches of mathematics in that it lacks tangible notions of distance, length, magnitude, order, and size. There is no shape, no geometry, no algebra, and no direction. Everything we are used to visualizing is gone. In the teaching and learning of mathematics, this can present a conundrum. Yet, this…

The authors outline some of the ways in which they teach student teachers to think about the challenges of abstract concepts in science. They ask the student teachers to think about abstract ideas in a non-science context first, and find they are enabled to discuss and engage (and then later apply) their thinking better. The authors think that…

Mass balance (MB) reasoning offers a rich topic for examination of students' scientific thinking and skills, as it requires students to account for multiple inputs and outputs within a system and apply covariational reasoning. Using previously validated constructed response prompts for MB, we examined 1,920 student-constructed responses (CRs)…

This article reports on a controlled study to investigate the efficacy of incorporating explicit discussions about the duality of human reasoning and its implications for learning in physics instruction. The central goal is to probe to what extent, if at all, such discussions improve student performance on tasks that tend to elicit intuitively…

This paper introduces two theoretical constructs, open-loop covariation and closed-loop covariation, that combine covariational reasoning and causality to characterize the way that three preservice mathematics teachers conceptualize a feedback loop relationship in a mathematical task related to climate change. The study's results suggest that the…

The Physiology Core Concept of flow down gradients is a major concept in physiology, as pressure gradients are the key driving force for the bulk flow of fluids in biology. However, students struggle to understand that this principle is foundational to the mechanisms governing bulk flow across diverse physiological systems (e.g., blood flow,…

Biological examples have been used to increase innovation in design activities and design education. And biologically inspired design has been studied in mechanical engineering using biological functions and principles as source domains. However, there were few studies regarding shape-based biological design in industrial design; furthermore, the…

[This paper is part of the Focused Collection on Instructional labs: Improving traditions and new directions.] Uncertainty is an important concept in physics laboratory instruction. However, little work has examined how students reason about uncertainty beyond the introductory (intro) level. In this work we aimed to compare intro and beyond-intro…

The algebraic structure is one of the axiomatic mathematical materials that consists of definitions and theorems. Learning algebraic structure will facilitate the development of logical reasoning, hence facilitating the study of other aspects of axiomatic mathematics. Even with this, several researchers say a lack of algebraic structure sense is a…

In this paper, we seek to evaluate the extent to which students can follow a deductive reasoning chain when it is presented to them. A great deal of instruction in introductory physics courses is centered on presenting students with a logical argument that starts from first principles and systematically leads to a particular conclusion. This…

Reasoning about organic chemistry reaction mechanisms requires engagement with multiple concepts and necessitates balancing the relative influence of different chemical properties. A goal of organic chemistry instruction is to support students with engaging in this type of reasoning. In this study, we describe our use of case comparison problems…

Isomorphism and homomorphism appear throughout abstract algebra, yet how algebraists characterize these concepts, especially homomorphism, remains understudied. Based on interviews with nine research-active mathematicians, we highlight new sameness-based conceptual metaphors and three new clusters of metaphors: sameness/formal definition, changing…