Blending of mathematics and physics:undergraduate students' reasoning in the context of the heat equation

The role of mathematics in physics is multifaceted. Moreover, mathematics has not only been essential to the development of physics, but conversely, many mathematical concepts arose from a desire to describe nature. The intertwining of mathematics and physics is so strong that it is sometimes difficult to separate the two. Yet this strong interconnectedness is not always reflected in education. This dichotomy makes it difficult for pupils and students to integrate their knowledge of both disciplines. The question of how best to bring both together to support learners' learning remains largely unanswered. In our study, for a specific topic (partial differential equations), we examined how students bring mathematics and physics together in their reasoning, what the specific difficulties they face in doing so are, and ultimately how we can respond to them in designing educational learning activities. We did this through three qualitative studies, in which we conducted interviews with small numbers of students and which we analyzed in detail. In the first study, we identified the specific difficulties that students face with blending mathematics and physics in this context. Second, we focused on the ways in which students combine mathematical and physical ideas in their reasoning, and how graphic reasoning can help them foster that combination. As a final step, we then designed a tutorial that draws on our previous findings and aims to support students in combining mathematics and physics

Dynamic conceptual blending analysis to model student reasoning processes while integrating mathematics and physics:A case study in the context of the heat equation

In recent years, there has been an increased interest in conceptual blending in physics and mathematics education research as a theoretical framework to study student reasoning. In this paper, we adapt the conceptual blending framework to construct a blending diagram that not only captures the product but also the process of student reasoning when they interpret a mathematical description of a physical system. We describe how to construct a dynamic blending diagram (DBD) and illustrate this using two cases from an interview study. In the interview, we asked pairs of undergraduate physics and mathematics students about the physical meaning of boundary conditions for the heat equation. The selected examples show different aspects of the DBD as an analysis method. We show that by using a DBD, we can judge the degree to which students integrate their understandings of mathematics and physics. The DBD also enables the reader to follow the line of reasoning of the students. Moreover, a DBD can be used to diagnose difficulties in student reasoning

Undergraduate students' difficulties with boundary conditions for the diffusion equation

Combining mathematical and physical understanding in reasoning is difficult, and a growing body of research shows that students experience problems with the combination of physics and mathematics in reasoning beyond the introductory level. We investigated students' reasoning about boundary conditions (BCs) for the diffusion equation by conducting exploratory task-based, think-aloud interviews with twelve undergraduate students majoring in physics or mathematics. We identified several difficulties students experienced while solving the interview task and categorized them using the conceptual blending framework. This framework states that in reasoning, people draw from separate input spaces, in this case the mathematics and the physics input space, to form a blended space, where they make connections between elements from these spaces. To identify difficulties, we used open coding techniques. We observed few difficulties in the physics space. In the mathematics space, we identified several difficulties that we clustered in two main groups: findings about the mathematical meaning of BCs, and findings about reasoning with functions of two variables. Lastly, we identified four ways in which blending failed. Starting from our findings, we formulate recommendations for teaching and future research

C-ITS road-side unit deployment on highways with ITS road-side systems : a techno-economic approach

Connectivity and cooperation are considered important prerequisites to automated driving, as they are crucial elements in increasing the safety of future automated vehicles and their full integration in the overall transport system. Although many European Member States, as part of the C-Roads Platform, have implemented and are still implementing Road-side Units (RSUs) for Cooperative Intelligent Transportation Systems (C-ITS) within pilot deployment projects, the platform aspires a wide extension of deployments in the coming years. Therefore, this paper investigates techno-economic aspects of C-ITS RSU deployments from a road authority viewpoint. A two-phased approach is used, in which firstly the optimal RSU locations are determined, taking into account existing road-side infrastructure. Secondly, a cost model translates the amount of RSUs into financial results. It was found that traffic density has a significant impact on required RSU density, hence impacting costs. Furthermore, major cost saving can be obtained by leveraging existing road-side infrastructure. The proposed methodology is valuable for other member states, and in general, to any other country aspiring to roll out C-ITS road infrastructure. Results can be used to estimate required investment costs based on legacy infrastructure, as well as to benchmark with the envisioned benefits from the deployed C-ITS services

Translating between graphs and equations: The influence of context, direction of translation, and function type

We report on a study investigating the influence of context, direction of translation, and function type on undergraduate students’ ability to translate between graphical and symbolic representations of mathematical relations. Students from an algebra-based and a calculus-based physics course were asked to solve multiple-choice items in which they had to link a graph to an equation or vice versa and explain their answer. The first part of the study focuses on the accuracy of the chosen alternative. Using a generalized estimating of equations (GEE) analysis we find that mathematics items are solved better than physics or kinematics items; that items starting from a graph are solved better than those starting from an equation; and that items on inversely proportional functions are the hardest for students. Quantitatively we see big differences in the number of correct answers between the algebra-based and calculus-based cohorts, but qualitatively the effects of the item variables on the accuracy are the same for both groups. The second part of the study focuses on students’ argumentations for the chosen alternative. We observe that students use their physical knowledge 2 to even 3 times as much in kinematics items than in other physics items. When there is an effect of context on argument use, we observe that the use of an argument in a mathematics context differs significantly from the use of that argument in the physics and/or kinematics context. With regard to function type, students explain their choice of answer in items on inversely proportional functions with different arguments and fail more often to answer correctly. In general, the data also show that students from the calculus-based course use more mathematical arguments and score better on the items.status: Published onlin

Translating between graphs and equations: The influence of context, direction of translation, and function type

We report on a study investigating the influence of context, direction of translation, and function type on undergraduate students’ ability to translate between graphical and symbolic representations of mathematical relations. Students from an algebra-based and a calculus-based physics course were asked to solve multiple-choice items in which they had to link a graph to an equation or vice versa and explain their answer. The first part of the study focuses on the accuracy of the chosen alternative. Using a generalized estimating of equations (GEE) analysis we find that mathematics items are solved better than physics or kinematics items; that items starting from a graph are solved better than those starting from an equation; and that items on inversely proportional functions are the hardest for students. Quantitatively we see big differences in the number of correct answers between the algebra-based and calculus-based cohorts, but qualitatively the effects of the item variables on the accuracy are the same for both groups. The second part of the study focuses on students’ argumentations for the chosen alternative. We observe that students use their physical knowledge 2 to even 3 times as much in kinematics items than in other physics items. When there is an effect of context on argument use, we observe that the use of an argument in a mathematics context differs significantly from the use of that argument in the physics and/or kinematics context. With regard to function type, students explain their choice of answer in items on inversely proportional functions with different arguments and fail more often to answer correctly. In general, the data also show that students from the calculus-based course use more mathematical arguments and score better on the items