This study investigates the evolution in conceptual understanding of cloud physics among learners from diverse academic backgrounds, using the mathematical framework of graph theory.

Cloud physics is an inherently multidisciplinary area of research, and therefore also of teaching in higher education. Challenges related to the understanding and modeling of clouds influence one of the main uncertainties in climate models [1], as well as a range of other areas, like affect aircraft operations and remote sensing technologies. Cloud physics education therefore represents a key aspect in atmospheric science education and, more widely, in geoscience education [2]. Recent academic efforts have addressed the difficulties encountered by learners in the discipline [3, 4, 5, 6], yet more is to be done to connect these with the conceptual structure of cloud physics.

Graph theory is an established field of mathematics, but the use of graph structures in education is relatively new [7, 8, 9, 10, 11], offering new perspectives to discipline-based educational research. Graph structures are networks of nodes connected with edges, and in our case networks of concepts from cloud physics connected with directed arrows by the participants of our study. The algorithmic power of graph theory affords characterization of both the mathematical graph structure and the role of the nodes that compose it. In this study, participants hand-drew concept maps depicting the life-cycle of a cloud, reflecting their understanding of cloud physics. We coded the maps according to thematic analysis and transformed them into graph structures in Python.

A "map of cloud physics" is created, depicting the joint graph representation of all participants. Studying this representation presents a novel way to look at the field and inspires a series of follow-up investigations to inform the disciplinary teaching and learning practices. We present sub-graphs based on the participants' academic experience. While Novice represents the group with no formal academic exposure to cloud physics, a comparison of the Adept and Advanced groups highlights the main changes induced by an increasing experience in the discipline. We represent the core knowledge of each group, corresponding to the nodes and edges of highest consensus, using a hierarchical structure. We also compute the groups' agreement with regard to the predecessors and successors of the used concepts, and define a new node-level metric measuring these quantities.

The evolution of the computed metrics through the experience-gradient provides a diagnosis of both the changes occurring along a learner's journey in cloud physics, and of the structure of the discipline and its inherent conceptual complexities. Overall, our results both qualify and quantify the epistemological shift in the description of the life-cycle of a cloud, from the general physics of the water cycle to detailed description of cloud microphysical processes, as learners mature in their understanding of the discipline.

Our findings can be used by lecturers to tailor their teaching towards the identified expert-like concepts, and by students to anticipate the main complexities in the field during their learning process.

(As our work in this study is very graphical, for both visualisation and analysis purposes, the above explanations would undoubtedly profit from a few visual inputs, which we would be happy to provide to the reader upon request.)

We collected concept maps from 117 participants from five different academic teaching and research institutions in Norway between Nov 2022 and Sep 2023. The participants were asked to graphically depict the life-cycle of a cloud, from the early conditions for formation to their dissipation, using around ten minutes for the exercise. The instructions were to draw and label nodes representing scientific concepts, and connect them with unlabeled directed arrows wherever they seemed appropriate. The information collected from the participants informed about their disciplinary field (six disciplines of STEM), academic level (bachelor, master, PhD, researcher), and experience with cloud physics (Novice, Adept, Proficient, Expert). The concept maps were coded according to thematic analysis (with thematic saturation reached at about 110 concepts) and converted to graph structures via the creation of adjacency matrices in Python. The joint weighted graph of all the collected data presents a “map of cloud physics” reflecting the collective understanding of all the participants. Setting threshold levels of consensus on edges reveals valuable substructures on this map. A 3D web-visualization allows to navigate the map and highlight specific areas according to criteria set by the user. We computed graph-level metrics such as density, diameter and intertwinement for each participant, and created box-plots of these metrics according to the participants’ disciplinary field, academic level and cloud physics experience. Grouping the participants according to their experience with cloud physics led to the largest variance of graph metrics, motivating clustering the data into Novice, Adept, Proficient and Expert groups. A further grouping of Proficient and Expert into Advanced was also introduced. We identified for each group a “layer-structure” in their collective graph according to consensus threshold values set on edges. The layers of highest consensus correspond to the core knowledge of each group, which we represent using a hierarchical structure that indicates the optimized way of navigating their sub-graphs. For the Advanced group, the core knowledge sub-graph can directly be used to inform teaching and learning. Node-level metrics were then computed for each group, in particular right/left-eigenvector, betweenness, and out/in-degree centralities. Expanding on the degree centrality measures, we created a new metric that quantifies the agreement of a group on the successors and predecessors of a node. A study of the rate of change of these node-level metrics across groups highlights the concepts becoming central, and thus important, in the conceptual understanding of these groups as their disciplinary experience increases.

Our analysis shows that the agreement on the origins and effects of the concepts Adiabatic Cooling and Heterogeneous Nucleation increases with experience, indicating an increasingly precise understanding and knowledge. This agreement decreases with experience for Evaporation, Rain and Shortwave Radiation, making us suggest that these concepts have an inherently more complex role within the storyline of a cloud. We also show that the importance of specific concepts such as Droplet Growth and Convection increases with experience in explanations of more advanced learners, whereas that of more general concepts such as Water Mass and Condensation decreases. Convection, Droplet Growth and Maturation also gain importance as bridges enabling the flow of information in the graphs of more experienced groups of learners. The hierarchical graph of the Advanced-group reveals a three-part structure of cloud physics: 1) the atmospheric physics and thermodynamics, from an ascending mass of moist air to droplet nucleation; 2) the aerosol physics behind cloud formation; and 3) the mechanisms behind droplet growth and ice crystal nucleation during the maturation phase of the cloud. Such a result can be used as a recommendation to introduce the topic sequentially in a teaching and learning setting. Using concept mapping narratives as a proxy and the theoretical framework of graph theory, differences in understanding of cloud physics in groups of varying experience have been quality-tested and quantified. We believe that the methodology developed within this study has the potential to be applied to other disciplines of the STEM curriculum, and could thus inform their teaching and learning practices. The visual representation of a discipline through a large and dense network could, in particular, form a helpful tool for both teachers and learners. The applied methodology makes structures emerge from large "maps", and reveals how increasing experience in a discipline changes how learners navigate them.

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