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The Anthropological Theory of Didactics describes mathematical activity in terms of mathematical organisations or praxeologies and considers the teacher as the director of the didactic process the students carry out, a process that is structured along six dimensions or didactic moments. This paper begins with an outline of this epistemological and didactic model, which appears as a useful tool for the analysis of mathematical and teaching practices. It is used to identify the main characteristics of the mathematical organisation around the limits of functions as it is proposed to be taught at high school level. The observation of an empirical didactic process will finally show how the internal dynamics of the didactic process is affected by certain mathematical and didactic constraints that significantly determine the teacher's practice and ultimately the mathematical organisation actually taught.

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Knowledge of functional groups provides students with a language for organic chemistry. However, students in a health science chemistry course do not plan to be synthetic organic chemists and, therefore, need examples of how functional group chemistry is relevant to their vocational goals. We have developed a lab to demonstrate how simple functional group chemistry is used in laboratory testing, namely, the "pee test", or dipstick urinalysis. Dipstick urinalysis is frequently used to screen for various conditions and is used weekly in the last month of pregnancy. Our lab models the prenatal clinical environment. The laboratory allows for testing of chemical species that support a medical diagnosis: albumin testing for preeclampsia; leukocytes, nitrites, and pH for urinary tract infection; glucose, ketones, and pH to test for gestational diabetes, alcoholism, or other serious metabolic diseases. The related lessons are designed to support students in understanding the reactions involved in testing, biochemistry related to diagnosis, and general understandings of how tests are interpreted. In developing this experiment, we were confronted with disparities in prenatal care, namely, the absence of culturally competent care, inequitable access to care, and toxic stress due to racism that contribute to dramatically increased maternal death rates for women of color in the US. We identified resources to educate students on culturally competent care. These resources include reference to the way in which the "pee test" is administered. We have incorporated this resource into our lab with appropriate reflection questions, based on our newly devised cultural competence and social justice framework for chemistry students. Our findings indicated that student understandings of the chemistry and testing methods were adequate; on average, 77% of samples were properly diagnosed. With respect to social justice and cultural competence, we found that students predominantly discussed two elements of the cultural competence and social justice framework: reflection on privilege in society and agency to promote equity in healthcare.

En este trabajo se presenta el diseño matemático de una pequeña parte de un modelo epistemológico de referencia que, una vez completado, deberá sustentar la organización didáctica de un proceso de estudio que incluya: el desarrollo de la modelización funcional con parámetros, la razón de ser del cálculo diferencial elemental en la última etapa de secundaria y los primeros desarrollos del cálculo en varias variables en la universidad. Aquí nos concentramos en el momento en que ya se dispone de la derivada como herramienta de trabajo para mostrar el desarrollo progresivo y la completación relativa de las praxeologías matemáticas que se construyen a medida que se avanza en los diferentes niveles de la modelización funcional. Esta propuesta se hace en el ámbito de la Teoría Antropológica de lo Didáctico y constituye una pequeña contribución al objetivo didáctico de situar la modelización matemática como un instrumento que permite articular y dar sentido a la matemática escolar.

An important role of theory in research is to provide new ways of conceptualizing practical questions, essentially by transforming them into scientific problems that can be more easily delimited, typified and approached. In mathematics education, theoretical developments around 'metacognition' initially appeared in the research domain of Problem Solving closely related to the practical question of how to learn (and teach) to solve non-routine problems. This paper presents a networking method to approach a notion as 'metacognition' within a different theoretical perspective, as the one provided by the Anthropological Theory of the Didactic. Instead of trying to directly 'translate' this notion from one perspective to another, the strategy used consists in going back to the practical question that is at the origin of 'metacognition' and show how the new perspective relates this initial question to a very different kind of phenomena. The analysis is supported by an empirical study focused on a teaching proposal in grade 10 concerning the problem of comparing mobile phone tariffs. The role of problems in the development of scientific theoriesAn important function of scientific theories is to transform problematic questions arising in different systems of the natural or social world into scientific problems formulated within a frame of notions, properties, statements and methodologies. A good formulation of a problem within a theoretical framework is always a first step to approach it and, thus, to explain, predict, control or modify the functioning of the system where the question or difficulty appeared. In this situation, theories represent an alternative to common sense. Instead of trying to directly handle problematic questions appearing in human activities with the means available at a given historical moment, theories propose to approach them by making a detour, apparently non-natural: considering a model that can sometimes be very far from the original system where difficulties came up and using the model to formulate a problem related to the initial question. 1 Moreover, a theory provides conceptual and procedural instruments not only to create the model but also to work with it in order to solve the initially formulated problems and eventually to come back to the system with appropriate solutions. An initial unexpected consequence of this work within the model-that also becomes a measure of the power and fecundity of the theory-is its capacity to formulate new questions about the system, which is inconceivable without the theory. The problems formulated in terms of the 'didactic contract' in the Theory of Didactic Situations (Brousseau, 1997) are a good example of this. 2 1 In mathematics, a well-known example of distance between a system and the model used to solve problems appearing in this system is Galois' theory (using groups and fields) and the problem of solving polynomial equations by radicals. In classical mechanics, the modelling of the solar system, where planets appear as points provided with mass...

The paper presents how two different theories-the APC-space and the ATD-can frame in a complementary way the semiotic (or ostensive) dimension of mathematical activity in the way they approach teaching and learning phenomena. The two perspectives coincide in the same subject: the importance given to ostensive objects (gestures, discourses, written symbols, etc.) not only as signs but also as essential tools of mathematical practices. On the one hand, APC-space starts from a general semiotic analysis in terms of ''semiotic bundles'' that is to be integrated into a more specific epistemological analysis of mathematical activity. On the other hand, ATD proposes a general model of mathematical knowledge and practice in terms of ''praxeologies'' that has to include a more specific analysis of the role of ostensive objects in the development of mathematical activities in the classroom. The articulation of both theoretical perspectives is proposed as a contribution to the development of suitable frames for Networking Theories in mathematics education.