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ed mathematical values as locations on the graph and drew horizontal lines on the set of axes to visualize the fixed values at positions zero and ten; this is element 8 and is categorized in the blended space because it graphically represents the boundary conditions and their physical meaning (element 4, implicit connection). Note that both element 1 and 8 are parts of the graph construction process and so both are placed in the blended space. Elements 7 and 8 are explicitly connected. In element 7, the “it” refers to the initial distribution (so at t 1⁄4 0), whereas the lines in element 8 deal with all values of t. Therefore, elements 2 and 8 are connected implicitly. 2. Question II The students continue and start discussing the evolution of the system over time. Figure 10 shows DBD 2 for the following piece of transcript. Evan 22:26 It (the graph) flattens out again, because that, that is just what things do under influence of the heat equation, (1) but how would it look like at the end? Something like eh, straight line or so maybe? (2) I don’t really know. David 22:40 Hm... Evan 22:40 What makes the most sense? David 22:43 (referring to the boundary condition at position 10 (3)) Wait, the end should be kept at 50 (4), so there should be some kind of heat source. (5) Evan 22:51 Yes. David 22:52 Yes, and if there is a heat source, heat is again flowing over there, to the beginning of the rod (6).(referring to the boundary condition at position 0 (7)) But the beginning of the rod is so cold that it stays zero (8) and so it cancels each other out a bit (9) and I think you will just get a straight line (10). Evan 23:13 Yes, I also have the feeling it will become a straight line, because ehm, if it is not going to be a straight line, what else... In element 1, Evan states that because of the effect of the heat equation, there is a tendency for the graph to “flatten out.” They seem to combine aspects of mathematics and physics in this element, so it is placed in the blended space. He ends the sentence with the thought that the graph will probably evolve to a straight line (2), which is placed in the mathematics space because it refers to the features of the graphical representation. Elements 1 and 2 are connected explicitly. Next, the students discuss the physical meaning of the boundary conditions. The placement of element 4 could be in the mathematics or the physics space. However, the continuation of the sentence suggests that the students are giving a physical interpretation here of the boundary condition at position 10, hence the placement in the physics space. Element 5 is then the physical tie to that; you have to know that it requires heat flow to keep a constant temperature. The boundary condition was given in the problem statement (3, gray box). This boundary condition is placed in the blended space because there is a physical interpretation (elements 4 and 5) tied to this equation. The connection between elements 3 and 4 is implicit because the students do not mention the boundary condition explicitly. However, this connection is important because it indicates blending: the students made a physical interpretation of a mathematical equation. Based on this, the students start a line of reasoning mainly in the physics space that is internally connected with solid lines. They state that heat has to flow to the beginning of the rod (6, physics space), where the DYNAMIC CONCEPTUAL BLENDING ANALYSIS ... PHYS. REV. PHYS. EDUC. RES. 16, 010114 (2020)

physical review. physics education research

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