Abstract
The main objective of this paper is to apply the onto-semiotic approach to analyze the mathematical concept of different coordinate systems, as well as some situations and university students’ actions related to these coordinate systems. The identification of objects that emerge from the mathematical activity and a first intent to describe an epistemic network that relates to this activity were carried out. Multivariate calculus students’ responses to questions involving single and multivariate functions in polar, cylindrical, and spherical coordinates were used to classify semiotic functions that relate the different mathematical objects.
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Notes
The duality personal-institutional will be defined in the Section 2. Informally, we are referring to the shared criteria within an institution (university, mathematics department, classroom, business) of what should be previously known, or what should be taught, learned, and understood in some didactical situation or evaluation process.
In single variable calculus, we treat real-variable and real-valued functions \(\left( {f:R \to R} \right)\), whose graph is in R 2 and we look at the “exponents” in terms of R n, in this case “1”, and point out that \(1 + 1 = 2\). When moving to multivariate calculus, we introduce R n. When our domain is two dimensional, we work with functions \(f:R^2 \to R\), whose graph can be represented in R 3 (2 + 1 = 3). Now, in terms of spatial dimensions, a function \(f:R^3 \to R\) would have to be graphed in \(R^4 \left( {3 + 1 = 4} \right)\), which is impossible. Triple integrals deal with functions of the type \(f\left( {x,y,z} \right) = w\). That is the reason for the relation with hyperspace.
As was mentioned, his actions as well as the properties and arguments used were impeccable, if the words function and region had been exchanged in the two cases. On the other hand, the pragmatic nature of the meanings of the terms ‘function’ and ‘region’ can themselves cause the mistake, and can be explained in terms of institutional use (including the professor’s explanations). The identification of a function with its graph and, consequently, the area (definite integral) with the region, shows a semantic and syntactic coherence, in spite of the mistaken use of the terms.
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Montiel, M., Wilhelmi, M.R., Vidakovic, D. et al. Using the onto-semiotic approach to identify and analyze mathematical meaning when transiting between different coordinate systems in a multivariate context. Educ Stud Math 72, 139–160 (2009). https://doi.org/10.1007/s10649-009-9184-2
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DOI: https://doi.org/10.1007/s10649-009-9184-2