The goal of this research is to make a theoretical reflection about connections that contributes to the development of the model for mathematical connections originally proposed by Businskas, and extended with the contributions of other researchers. This model is used in the most relevant investigations in this field. As a consequence of the reflection, the introduction of a new type of connection is proposed. The context for the reflection was the transcription of several episodes of a lesson on derivative. We first analysed the connections given in this transcription grounded in a qualitative methodology based on a-priori categories given by the theoretical model. We then selected some paragraphs where the model proved to be ambiguous, and finally, argued the need to clarify or extend the categories in the model to deepen the study of the connections in the transcription. In particular, we propose metaphorical connections as a new type of connection.

This article presents a networking of two theories, the Extended Theory of Mathematical Connections (ETC) and the Onto-semiotic Approach (OSA). In particular, concordances and complementarities are identified in the respective conceptions of mathematical connections, as a result of applying these theoretical frameworks to a student’s response protocol when solving a task on the derivative. First the mathematical connections were identified from the perspective of the ETC and, second, an onto-semiotic analysis was carried out where the mathematical practices, the primary mathematical objects activated in them and the semiotic functions established between these objects were analysed. The main conclusion is that both theories complement each other to make a more detailed analysis of the mathematical connections. In particular, the more detailed analysis carried out with the OSA tools, visualizes a mathematical connection like the tip of an iceberg of a conglomerate of practices, processes, primary objects activated in these practices and semiotic functions that relate them, which enables a thorough analysis that details the structure and function of the connection.

The literature reports that students have difficulties connecting different meanings, multiple representations of the derivative, and performing reversibility processes between representations of f and f’. The research goal is to analyze the mathematical connections that university students establish when solving tasks that involve the graphs of f and f’ when the two functions do not have associated symbolic expressions. Seven students from the first year of undergraduate studies in mathematics from a university in southern Mexico participated. For data collection, two tasks involving the graphical context of the derivative were applied. An analysis of the mathematical activity was carried out by the participants with the analysis model proposed by the onto-semiotic approach, and thematic analysis with types of mathematical connections from the extended theory of connections was carried out to infer the connections made in that mathematical activity, which allowed us to consider the reversibility connection between the graphs of f and f’ as the encapsulation of a portion of the mathematical activity. Four students establish the reversibility relationship between the graph of f and the graph of f’. It has been concluded that some students can establish the reversibility connection between the graphs of f and f’, but the complexity of the mathematical activity that encapsulates the connection explains (by showing everything that the student must do) why some students are not able to establish it.

The quality of the intra-mathematical connections made by Mexican university students when solving tasks on the derivative was characterized. The typology of mathematical connections and the quality levels of mathematical connections were used. Interviews were conducted based on eight tasks applied to three case studies. The results showed that the quality of the intra-mathematical connections: different representations, procedural, implication, part-whole and meaning of the students, is found mostly at level 2 (consistent and argued mathematical connections) and only one student presented inconsistencies to find the correctness equation of the tangent line to the curve at a point (quality level 0 connection). Likewise, the metaphorical connection was identified when a student mentioned the metaphorical expression “the graph has no holes”, which suggests the conceptual metaphor “the graph is a path” referring to the continuity of a function. We believe that quality level 2 mathematical connections ensure understanding, and the quality level 0 mathematical connections are the reasons why students have difficulty understanding the derivative.

Se responde a la pregunta ¿Dada la expresión algebraica de , qué conexiones aseguran al estudiante una comprensión de que le permite esbozar la gráfica de y explicar la relación que tiene con la de ? Para ello, a un grupo de estudiantes se les aplicaron entrevistas y una tarea, en la que debían esbozar la gráfica de y explicar la relación que tiene con la de la derivada, y se analizó su actividad matemática usando como referente teórico la articulación TAC-EOS. Los estudiantes que evidenciaron una comprensión que les permitió resolver la tarea establecieron conexiones de: representaciones diferentes, significado, parte-todo, implicación y característica. Por otra parte, el no establecimiento de determinadas conexiones es una explicación plausible de por qué algunos estudiantes no resuelven la tarea.

The research goal is twofold: to articulate neuro-mathematics with the extended theory of mathematical connections that uses onto-semiotic approach tools and to explore the connections established by a teacher and his student when solving a problem about the volume of two boxes, one of toothpaste and the other of tomato. This research was developed in two stages: the theories were articulated assuming concordances and complementarities, highlighting the notion of connection, and a context of reflection was considered carried out in three phases where the participants were selected, participant observation was carried out in the classroom during solving a problem and then analyzing the data with the new tool to explore mathematical and neuro-mathematical connections. The findings present the mathematical connections established by the teacher and the student of meaning, feature, procedural, different representations (alternate, equivalent, and from a horizontal mathematization view), and part-whole, as well as neuro-mathematical connections of: recognition of terms and symbols; visual perception, spatial skills and motor coordination; association of mathematical concepts and formulas; intermediate calculations and unit conversion; solve operations step by step and understand the process; verification and conclusion, activated in the brain areas linked to each mathematical practice sequentially.