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