Mathematical connections promoted in multivariable calculus’ classes and in problems-solving about vectors, partial and directional derivatives, and applications

In a vector calculus course, the mathematical connections made by an in-service teacher and his engineering students in problems-solving involving vectors and partial and directional derivatives were explored. This study is relevant due to the difficulties in connecting multiple representations and meanings of ordinary and partial derivatives. Networking between the extended theory of connections and the onto-semiotic approach was used. The qualitative methodology included three stages: (1) selection of participants (in-service teacher and students), (2) data collection in four moments: design of the class on partial and directional derivatives and then, the development of this applying participant-observation and recording, design of a questionnaire and its application to the students, and (3) data analysis using theoretical tools. The results showed that the in-service teacher used various connections, starting with the instructional oriented and then others such as meaning, procedural and representations. Students defined and represented vector, partial and directional derivatives concepts, activating meaning connections and different representations. Also, they solved tasks using different connections (different representations, procedural, feature) to find partial and directional derivatives, gradient, curl and divergence. This analysis was carried out in terms of mathematical practices, processes, objects and semiotic functions. 72% of the students gave meaning, represented and appropriately used the concepts of vector calculus, while 28% had difficulties, especially in the procedural connection to find partial derivatives.