Articulations among Computational Thinking, Mathematical Modeling, and Meaningful Learning

Kaminski, Márcia Regina; Boscarioli, Clodis; Klüber, Tiago Emanuel

doi:10.1590/1980-4415v40a250008

person Márcia Regina Kaminski

contributed substantially to the conception and planning of the study · to the acquisition, analysis, and interpretation of the data · to the writing and critical revision · and approved the final version to be published PhD in Science Education and Mathematics Education from the Western Paraná State University (Unioeste).

school Permanent faculty member in the Graduate Program in Science Education and Mathematics Education (PPGECEM) at Unioeste, Cascavel, Paraná, Brazil. E-mail: marciarkjf@gmail.com.

0000-0001-5705-0322

person Clodis Boscarioli

contributed substantially to the conception and planning of the study · to the acquisition, analysis, and interpretation of the data · to the writing and critical revision · and approved the final version to be published PhD in Electrical Engineering from the University of São Paulo (USP).

school Associate Professor at the State Western Paraná State University (Unioeste). Permanent faculty member in the Graduate Program in Science Education and Mathematics Education (PPGECEM) at Unioeste, Cascavel, Paraná, Brazil. E-mail: boscarioli@gmail.com.

0000-0002-7110-2026

person Tiago Emanuel Klüber

contributed substantially to the conception and planning of the study · to the acquisition, analysis, and interpretation of the data · to the writing and critical revision · and approved the final version to be published PhD in Scientific and Technological Education from the Federal University of Santa Catarina (UFSC).

school Associate Professor at the Western Paraná State University (Unioeste). Permanent faculty member in the Graduate Program in Science Education and Mathematics Education (PPGECEM) at Unioeste, Cascavel, Paraná, Brazil. E-mail: tiagokluber@gmail.com.

0000-0003-0971-6016

Editor-in-Chief:

Prof. Dr. Roger Miarka

Associate Editor:

Prof. Dr. Vicenç Font

.
Main premise	Each individual has a unique cognitive structure (Ausubel, 2003).
Cognitive structure	“The total content of ideas of a given individual, and its organization” (Ausubel, 1968 apud Moreira; Masini, 2001, p. 14), which results from “cognitive processes, that is, the processes through which knowledge is acquired and used” (Moreira; Masini, 2001, p. 14).
Learning	Expansion of the cognitive structure through the “organization and integration” of information within the cognitive structure (Moreira; Masini, 2001, p. 13).
Types of learning	Mechanical	When new information has little or no interaction with important concepts already available in the learner’s cognitive structure (called subsumers), being stored in an arbitrary way, that is, one that “does not result in the acquisition of new meanings” (Ausubel, 2003, p. 4).
Types of learning	Meaningful	Involves the “acquisition of new meanings” (Ausubel; Novak; Hanesian, 1980, p. 34). It occurs when new ideas relate in a non-arbitrary way to some relevant aspect of the cognitive structure, that is, it “results in a change either in the newly acquired information or in the specifically relevant aspect of the cognitive structure to which the new information is linked” (Ausubel, 2003, p. 3).

Form of interaction	Concept	Cognitive processes involved	Convergences with CT-related skills
Subordination	The new knowledge relates in a non-arbitrary way to a more general and inclusive idea already existing in the cognitive structure, establishing a subordinate relationship of the new idea in relation to the pre-existing idea. The new ideas are subsumed, becoming potential subsumers.	Progressive differentiation: when, through different interactions, a subsumer progressively acquires new meanings, becoming richer, more refined, more differentiated, and more capable of serving as an anchor for new Meaningful Learning (Moreira, 2011, p. 20). It involves noticing differences between meanings (Moreira, 2011).	Both Progressive Differentiation and Integrative Reconciliation require from the student an intense process of Abstraction in the sense of perceiving different meanings or similar and inclusive particularities among them. In addition, they involve a process of Pattern Recognition, Decomposition, and Composition of ideas, these being skills associated with CT.
Superordination	Superordinate interaction occurs when the “individual grasps a new inclusive proposition to which several preexisting, established, but less inclusive ideas can be subordinated” (Ausubel, 2003, p. 94).	Integrative reconciliation: prior knowledge may be reorganized through a movement of “recombination of elements previously existing in the cognitive structure” (Moreira, 2018, p. 168), acquiring new meanings.
Combinatorial	When new information cannot be subordinated nor superordinated to any prior idea. Instead, it relates to some broader and more general content.	Progressive Differentiation and Integrative Reconciliation.

Representational Meaningful Learning
Concept	Example	Convergences with CT-related skills
“[...] when the meaning of arbitrary symbols is equated with their referents (objects, events, concepts) and has for the learner whatever meaning the referents possess” (Ausubel, 2003, p. 1). It “involves the attribution of meanings to [...] words” (Moreira, 2018, p. 165).	When a child relates the word cat exclusively to their own pet, they have constructed Representational Meaningful Learning (Moreira, 2008).	Representational Meaningful Learning involves a process of Abstraction. Japiassú and Marcondes (2001, p. 7) understand Abstraction as “an element of representation” and emphasize that it may assume two perspectives: separating elements that should not be taken into account or isolating them in order to focus attention on them. Thus, we can say that the development of Abstraction may contribute to facilitating Representational Meaningful Learning, since in this type of learning, Abstraction is fundamental, as it requires identifying essential elements of new information and constructing a mental representation of its essence, as well as relating it to prior knowledge.
Conceptual Meaningful Learning
Concept	Example	Convergences with CT-related skills
Based on the idea that concepts are “objects, events, situations or properties that have common specific attributes and are designated by the same sign or symbol” (Ausubel, 2003, p. 2).	When the word cat comes to mean, for a child, not only their own pet but a group of animals with common characteristics and distinctions in relation to other classes, Conceptual Meaningful Learning has been constructed (Moreira, 2008).	Conceptual Meaningful Learning involves the formation of concepts which, according to Japiassú and Marcondes (2001, p. 39), “designates an abstract and general idea under which we can group various elements.” Based on these premises, it is possible to relate the construction of Conceptual Meaningful Learning to the skill of Pattern Recognition, associated with CT, since it involves identifying similarities or regularities (Brackmann, 2017), as well as to Abstraction, which is fundamental to the construction of a concept that is itself a representation (Japiassú; Marcondes, 2001, p. 7).
Propositional Meaningful Learning
Concept	Example	Convergences with CT-related skills
It involves understanding a “composite idea that is verbally expressed in a sentence with meanings of words,” in “syntactic functions and in the relationships between words” (Ausubel, 2003, p. 3), as well as learning the meanings of propositions constructed from concepts, going beyond their individual meanings (Moreira, 2018).	It requires understanding the meaning of propositions such as “the cat is a domestic animal,” or a mammal (Moreira, 2008). Language is fundamental in the reasoning process (Ausubel, 2003). Through it, concepts and propositions are learned and refined as development progresses (Ausubel; Novak; Hanesian, 1980).	Since Propositional Meaningful Learning is based on concepts, it can be related to the skills of Pattern Recognition and Abstraction. Furthermore, considering the strong relationship between all these types of learning and language, we observe the strong influence of Abstraction on the occurrence of any type of Meaningful Learning, given that Abstraction has a close connection with language (Abbagnano, 1998, p. 14).