Representation Registers in the Solution of Calculus Problems

Abstract

This article is derived from the research project registered under number 20110343 (Ruiz, 2011), and developed in Escuela Superior de Cómputo del Instituto Politécnico Nacional (IPN) (School of Computer Sciences of the National Poly-technical Institute of Mexico). The article reports on the problems found among Engineering students with respect to their resistance to using different representation registers when solving optimization problems in the Calculus Learning Unit. Use of such registers could help the students to build mathematics know ledge and to solve calculus problems. As a didactic strategy, simulations are used in an electronic environment in order to support the students by fostering their use of tabular, graphical and algebraic representation registers. Interviews are undertaken of six of the professors who give the calculus courses, and a diagnostic questionnaire is applied to 68 students prior to and after working with the proposal. As for the theoretical framework, the work reported by Duval and Hitt is salient in this report, particularly their emphasis of the fact that working on activities by way of one single representation system is not sufficient. From the first responses provided by the students, one can conclude that the algebraic register is preferred by the majority of students. It is however used in a mechanical fashion without affording any meaning to the content of the problem and to the process of solving it. Another conclusion reported is that implementing tasks in the classroom in which the mathematics activity requires coherent use of different representations is necessary

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“…This study also supports the conclusion reached by other scholars (e.g., see Brijlall & Ndlovu, 2013;Ledesma, 2011) that the source of difficulties with optimization problems might not necessarily be rooted in the mathematization of the processes but in the difficulty of understanding the underlying mathematical principle. The ultimate extension of this study would be to investigate whether the lab experience helped students with formulating the solution processes to solve textbook problems on optimization.…”

“…They further suggested that "teachers needed to be aware of learners learning conflicts so as to reinforce the new concepts they encounter" (p. 17). Ledesma (2011) investigated how to identify a solid with a maximum volume built by cutting squares of various side lengths from a cardboard and folding the shape. She concluded that calculus students faced difficulties understanding the optimization process, which she considered a main obstacle in solving these problems.…”

Diagrammatic Language for Artificial Intelligence: Representation of Things that Flow

“…15. The belief sphere (circle 1) includes the belief itself (2), its truth value (3), the illness (4), and its truth value (5). Note that the belief (1) is in John Doe, and it includes two sub-spheres: itself (2) and its content illness (4).…”

“…In this paper, the term representation refers to "appearances" (public representation [4]), such as expressions in natural language, algebraic formulae, graphs, or geometric figures [5]. The focus is on a diagrammatic representation that depicts things (will be defined later) that "may exist in a possible or fictional world" [6].…”

“…Such a representation is utilized "to characterize the kind of phenomena that occur in any knowledge process or that constitute it" [8]. This type of representation is "the means through which a person can externalize his/her mental representations in order to make them visible or accessible to others" [5]. It plays "a central role in social exchange, sophisticated communicators such as humans have developed ways to talk and think directly about representational properties such as truth, accuracy, meaning, and entailment" [4].…”