July 26, 2021
July 26, 2021
July 19, 2022
A Multi-level Diffusion Curriculum: Connecting Submicro- and Macro-levels with Computational, Graphical and Mathematical Representations
Diffusion is a central concept in materials science and engineering, but recent research has shown the difficulty undergraduate materials science students have in understanding even its introductory foundations . One difficulty lies in understanding the relationships represented in Fick’s laws, the differential equations describing the macro-level dynamics of diffusion. Another difficulty stems from the emergent nature of diffusion : the macro-level patterns described in Fick’s laws emerge from the random walk behavior of particles at the (sub)micro-level, but the connection between these levels is not obvious.
In the first iteration of this curriculum, a diffusion unit in a sophomore-level materials science course was augmented with lessons on random-walk behavior using computational representations. Analysis of learning outcomes showed that the computational representations helped students understand the random-walk behavior underlying diffusion, but students still struggled to connect this understanding to the macro-level relationships encoded in Fick’s laws. This paper describes the revised curriculum that was developed to fix this conceptual gap, as well as refinements to the theoretical framework guiding its design. Future work will analyze learning outcomes of students using the curriculum.
The theoretical framework guiding our curriculum builds on the “Crossing Levels and Representations” framework developed by Levy and Wilensky . They describe three spheres of knowledge: conceptual understanding, symbolic expressions, and physical experiences. To help students deeply understand a domain, a curriculum should ideally engage students in all three spheres and should connect between the spheres. Additionally, our curriculum is guided by a constructivist orientation. Thus, our learning activities aim to engage students in actively constructing their own understanding.
The curriculum starts with students building conceptual understanding of the submicro-level by exploring computational models of random-walk behavior. These activities culminate in students altering and extending the underlying code of the models, which requires them to connect their conceptual understanding with a symbolic expression (in the form of computer code) of random-walk behavior.
Next, the students interact with a model of interstitial atomic diffusion. In this model students can set the initial concentration profile of interstitial atoms as well as the boundary conditions. Students use this model both to explore how concentration profiles evolve over time and to connect this evolution with the random-walk behavior of individual atoms in the model.
Finally, using a modified version of concentration profiles from the interstitial diffusion model, students informally derive Fick’s 1st and 2nd laws by reasoning about the average number of atoms that will jump between each x-position and how this relates to the slope and curvature of the concentration profile. In this way, students construct the symbolic representation of Fick’s laws from their own conceptual understanding of random-walk behavior.
References  J. Z. Kelter, J. D. Emery, and U. Wilensky, “Learning About Diffusion at Two Levels: Micro-scale-computational to macro-scale-analytical,” presented at the American Society of Engineering Education, 2020.  M. T. H. Chi, R. D. Roscoe, J. D. Slotta, M. Roy, and C. C. Chase, “Misconceived Causal Explanations for Emergent Processes,” Cognitive Science, vol. 36, no. 1, pp. 1–61, 2012, doi: 10.1111/j.1551-6709.2011.01207.x.  S. T. Levy and U. Wilensky, “Crossing Levels and Representations: The Connected Chemistry (CC1) Curriculum,” Journal of Science Education and Technology, vol. 18, no. 3, pp. 224–242, Jun. 2009, doi: 10.1007/s10956-009-9152-8.
Kelter, J. Z., & Emery, J. D., & Wilensky, U. (2021, July), A Multi-level Diffusion Unit: Connecting Submicro- and Macro-levels with Computational, Graphical, and Mathematical Representations Paper presented at 2021 ASEE Virtual Annual Conference Content Access, Virtual Conference. 10.18260/1-2--36595
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