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.

Ethnomathematics, STEAM Education, and the Globalized Approach are articulated to analyze mathematical connections in daily practices. For this, the networking strategies were considered (understand the theoretical approaches; contrast-compare; coordinate-combine; synthesize-local integration). In the coordination and comparison, the complementarities between the approaches were evidenced and based on these, an empirical phenomenon was analyzed on the connections in daily practices carried out by five artisans who worked in the elaboration of kites, cabinets, masks, and drawers, agriculture, and masonry. From the integration strategy, three results were identified: (i) intra-disciplinary connections, which make it possible to present mathematics as an integrated whole, rather than as a set of isolated knowledge; (ii) interdisciplinary connections, between knowledge from different disciplines that feed into each other; and (iii) connections between the mathematics practiced by cultural groups with institutionalised mathematics that are, at the same time, globalized because they relate mathematics to the socio-cultural context.

En el artículo, se analizan las conexiones etnomatemáticas entre conceptos geométricos evidenciados en la elaboración de la tortilla por un comerciante de Chilpancingo, México. La investigación se fundamentó teóricamente en la Etnomatemática y la metodología fue cualitativa-etnográfica. Los datos se recolectaron a través de una entrevista semiestructurada, aplicada a un comerciante de tortillas de la ciudad de Chilpancingo, México. En los resultados se presentan las conexiones entre los conceptos (circunferencia-círculo-cilindro), mediciones, conteos y sucesiones, que surgen en el proceso de elaboración, forma y comercialización de la tortilla. El análisis de la práctica de elaboración de tortillas, permitió identificar conceptos geométricos que aportan herramientas a la enseñanza de las matemáticas escolares, desde situaciones o tareas basadas en la cotidianidad. Se muestra a los profesores la existencia de otras formas de abordar los conceptos geométricos asociados, usando: papel, regla, lápiz y compás, junto con el software GeoGebra que permite visualizar los objetos matemáticos, y evidencia permanentemente las conexiones establecidas por una persona en las vistas gráfica y algebraica-simbólica.

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.