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Advances in Social Science, Education and Humanities Research, volume 574 Proceedings of the 6th International Conference on Science, Education and Technology (ISET 2020) Developing Mathematical Conceptual Understanding through Problem-Solving: The Role of Abstraction Reflective Lulu Choirun Nisa St. Budi Waluya Universitas Negeri Semarang, Universitas Negeri Semarang, Indonesia Indonesia lulu.choirunnisa@gmail.com s.b.waluya.math.unnes@gmail.com Kartono Scholastika Mariani Universitas Negeri Semarang, Universitas Negeri Semarang, Indonesia Indonesia scmariani.unnes@gmail.com Abstract---Reflective abstraction is a mechanism that lower level to a higher level) and reflexion moves individuals from one level to a higher level of (rearranging a higher structure). 1]. This may be the knowledge. Reflective abstraction is a mechanism that first study of reflective abstraction and is a very builds novelty. Therefore, the study of reflective important part of how mathematical knowledge is abstraction is dominant in the process of how reflective formed. Beth & Piaget explicitly states that reflective abstraction forms new knowledge or understanding. abstraction is very important for the development of For example, Piaget, Dubinsky, David Tall, advanced mathematical concepts because Mitchelmore, are some researchers who focus on the mathematical constructs are processed through process of reflective abstraction in concept formation. reflective abstraction [2]. Dubinsky also stated that The resulting mastery of the students' concepts played a lot in the problem-solving process. A good mathematics is a product of reflective abstraction [3]. understanding of concepts, students will be able to Simon et.al stated that reflective abstraction is also a reason, comprehend, operate, and connect the method that supports and animates large buildings of mathematics idea that will play a role in problem- mathematical logic construction [4]. Arnon et. al. solving. However, when students have to solve emphasized that reflective abstraction is concerned assignments or problems that are not routine, this with the extraction of basic structures by considering problem-solving process also contributes to the the relationship between actions or actions, and is a development of understanding mathematical concepts. mental mechanism where all mathematical logic The problem-solving process will result in structures are developed in the thinking of an understanding a new concept if there is a reflective abrasion in it. This paper is the result of a literature individual [5]. review that will describe the role of reflective Piaget's study of reflective abstraction was abstraction in problem-solving so that students can get continued by Dubinsky who explained the mental new concepts. mechanism as a reflective abstraction in the formation of mental structures [5]. Also, Dreyfus describes the Keywords: reflective abstraction, conceptual processes of representation, generalization, and understanding, problem-solving. synthesis required in reflective abstraction [6]. Meanwhile, according to Hershkowitz, the I. INTRODUCTION abstraction process occurs through recognition, building-with, and construction [7]. The three studies One of Piaget's phenomenal works is Genetic form a new family in the study of reflective Psychology which talks about what knowledge abstraction. consists of and how knowledge develops. However, reflective abstraction as a means of Assimilation and accommodation are the keywords in developing cognition does not occur only in the the process of cognitive development. Piaget believed formation or understanding of concepts. Conversely, that assimilation and accommodation occur naturally with a proper reflective abstraction concept and that the development of cognition is driven by a understanding can develop in the problem-solving tilted process towards equilibration between process. assimilation and accommodation. How a person constructs a new cognitive structure from a pre-existing structure is described in reflective abstraction which consists of two phases, namely reflechissement (projecting a structure at a Copyright © 2021 The Authors. Published by Atlantis Press SARL. This is an open access article distributed under the CC BY-NC 4.0 license -http://creativecommons.org/licenses/by-nc/4.0/. 38 Advances in Social Science, Education and Humanities Research, volume 574 II. THEORITICAL BACKGROUND CONCEPTUAL UNDERSTANDING Skemp stated that understanding something means assimilating it into a suitable schema [12]. REFLECTIVE ABSTRACTION Harel & Sowder stated that understanding Reflective abstraction is one of the three types of mathematical activities refers to (1) certain abstraction mentioned by Piaget. The other two are interpretations or meanings of concepts, relationships empirical abstraction and pseudo empirical between concepts, statements, or problems; (2) a abstraction. Compared to the other two types, particular solution offered by an individual to a reflective abstraction is a type of abstraction that is problem; and (3) certain evidence offered by an closely related to mathematical knowledge. individual to build or reject a mathematical statement According to Piaget (1980) reflective abstraction is a [13]. general coordination of actions, and reflective As for concepts, Gray & Tall argues that there are abstraction takes place entirely internally [8]. This at least three types of mathematical concepts, namely type of abstraction leads to constructive (1) concepts based on perceptions of objects, (2) generalizations and results in a new synthesis which concepts based on processes that are symbolized and Damerow calls a feature by which the level of understood as processes and objects (procept), and (3) intelligence has increased [9]. Thus the result of a concept based on a set of properties acting as a reflective abstraction - in Piaget's paradigm - is the concept definition for constructing axiomatic systems logical structure of mathematics that specifically in advanced mathematical thinking [14]. Each of distinguishes human thought from previous forms of these concepts, according to Gray & Tall, is an intelligence. abstraction, namely a mental image of an object The reflective abstraction process involves two received (for example a triangle), a mental process inseparable elements, namely refechissement and that becomes a concept (such as counting into reflexion. Reflechissement is a projection of numbers), and a formal system (such as a permutation something borrowed from a previous level to a higher group). which is based on its properties with a concept level, and reflexion is an awareness of cognitive built through deductive logic [15]. reconstruction or reorganization of what has been The need for conceptual understanding in transferred. This two-component abstraction mathematics learning is emphasized by the National reflection can be observed at all stages, from sensory Mathematics Advisory Panel which states that motor [10]. learning mathematics requires three types of The process that is characterized by reflective knowledge, namely factual, procedural and abstraction is the process of constructing the structure. conceptual knowledge. NCTM also states that Thus, the emergence of reflective abstraction can be conceptual understanding is one of the five indicators identified in the form of developmental psychology, of math proficiency. The other four indicators are in which reflective abstraction evokes a transition problem-solving, reasoning, connection, period from the sensory-motor intelligence stage to representation and communication [16]. the concrete operation stage, or in all subsequent Operationally, indicators of understanding the transitions in the development of intelligence. concept are described in various versions. According to Piaget, the process of reflective Engelbrecht, Harding & Potgier also stated that abstraction takes place during cognitive development understanding operations and relationships is part of and does not have an absolute beginning, and has understanding concepts [17]. Concept understanding appeared since the earliest stages in motor sensory [2]. consists of relationships that are built internally and This process lasted until mathematics advanced and relate to pre-existing ideas; and it will be necessary formed a history of the development of mathematics when an individual identifies and applies principles, [8]. knows and applies facts and definitions, and Piaget distinguished various types of constructs compares and contrasts concepts. in reflective abstraction, namely interiorization, The existence of a connection in conceptual coordination, encapsulation, and generalization [1]. understanding is also emphasized by Hiebert and Meanwhile, Dreyfus (2002) states that abstraction Lefevre [18]. They describe conceptual requires a process of representation, generalization understanding as knowledge that is rich in and synthesis [11]. Meanwhile, Hershkowitz et.al connectedness, so that all pieces of information are (2001) stated that the abstraction process occurs linked into some information. Hiebert and Lefevre through the process of recognition, building-with, and also made a distinction between what is called the construction [7]. This model is hereinafter known as ground-level conceptual understanding relationship the RBC model. and what they call the reflective level. Basic level refers to pieces of knowledge that are at the same level of abstraction. The reflective level refers to the higher level of abstraction of two pieces of knowledge that were originally conceived as separate pieces of 39 Advances in Social Science, Education and Humanities Research, volume 574 knowledge. The National Assessment of Educational students in tasks whose solving methods were not Progress shows that there is a slice in the definition of previously known [23]. conceptual understanding between those used by In general, when researchers use the term NCTM and those used by the National Research problem-solving they refer to tasks that provide Council (NRC), namely that students have intellectual challenges that can encourage students' demonstrated understanding of mathematical mathematical development. This task, which is a concepts when they are proven to be able to (1) problem, can encourage conceptual understanding, recognize, label, and generate examples of concepts, reasoning and communication skills and capture their (2) using and interpreting various models, diagrams, mathematical interest and curiosity [23] [16] [18]. manipulations and representations of concepts, (3) Even according to badger, problem-solving is a identifying and applying principles, (4) knowing and student skill that will be most useful if they graduate applying facts and definitions, (5) comparing, [20]. contrasting and integrating related concepts and principles, and (6) recognizing, interpreting the signs, symbols, and forms used to represent concepts. REFLECTIVE ABSTRACTION IN PROBLEM- Meanwhile, the Mathematics Core Curriculum SOLVING document issued by New York Education The reflective abstraction that occurs in problem- Development (NYED) states that conceptual solving, Piaget hinted at when Piaget stated that when understanding consists of relationships that are built a problem is raised or confronted, the individual can internally and are connected to existing ideas. to this go beyond the things that can be observed and put indicator slice used in this study. The indicators of them into relationships, producing logico- conceptual understanding set out by NYED are mathematical knowledge or endogenous knowledge. identifying and applying principles, knowing and That reflective abstraction occurs when there is a applying facts and definitions, and comparing and confrontation, this problem is related to the idea of contrasting related concepts [19]. equilibration from Piaget's constructivism theory. PROBLEM-SOLVING Equilibration itself is defined as a process where the In general, the mathematics curriculum subject tries to understand a concept by placing the differentiates assignments or questions given to concept in the context of the cognitive system as a students into the form of excercises and problems. whole [24]. Exercise is a question whose solution requires a Reflective abstraction is a linking mechanism in routine procedure. Meanwhile, the problem is a equilibration that moves the individual to a higher question or assignment that is not an exercise. In other level, and is a mechanism that builds novelty [25]. words, a problem is a question whose resolution This novelty is what distinguishes problem-solving process is not clear. However, a question cannot be from ordinary math practice questions. The novelty separated into exercise or problem categories, possessed by problem-solving problems includes because it depends on the child's ability. Training for novelty in terms of problem formulation, novelty in one student may be a problem for another student. terms of solving strategies, or novelty of concepts This is also conveyed by Stanic and Kilpatrick who discussed in the problem [20]. Therefore reflective define a problem as a condition in which a person abstraction will be more likely to occur when students does a task that was not found in the previous time work on problem-solving problems than practice [20] [21]. This means, a task is a problem or does not questions. depend on the individual and time. So that a task is a Conjectures about the use of reflective problem for someone, but maybe not a problem for abstraction in problem-solving were hypothesized by someone else. Likewise, a task is a problem for researchers in Geneva in 1983 who suggested that someone at one time, if that person already knows students might use reflective abstraction in problem- how or the process of getting a solution to the problem. solving to explain the process of development [26]. In The characteristic that distinguishes between addition, Cohen also stated that reflective abstraction practice and problem is novelty, which has an impact occurs when a new problem is confronted [25]. The on the need for creativity to answer. Some of the discussion of reflective abstraction in problem- novelties that can arise in the problem are [20] [22]: solving is further found by turning to Cohen [25] and (1) novelty in problem formulation, so it requires Cifarelli [27]. careful interpretation In explaining the relationship between reflective (2) novelty in the type of strategy for finding abstraction and problem-solving, Cohen departed solutions to problems from the concept of equilibration, which is the means (3) novelty of the concept used by which reflective abstraction emerges. Through This novelty is in line with the opinion of NCTM equilibration, reflective abstraction is also a way of which states that problem-solving means involving forming something new, be it relationships, links, or correspondences. There are six stages to bring up this reflective abstraction, namely encoding, conflict or 40 Advances in Social Science, Education and Humanities Research, volume 574 contradiction, coordination, destructuring, and and Psychology. Springer Science + Business followed by two processes of reflective abstraction, Media, 1966. reflecting and reflection. [3] Dubinsky E. Constructive Aspects of Reflective Encoding is the process of identifying each Abstraction in Advanced Mathematics. In: Steffe element of the problem and taking attributes from LP (ed) Epistemological Foundation of long-term memory that are relevant to the solution. Mathematical Experience. New York: Springer, Individuals with good problem-solving abilities can 1991, pp. 160–202. be identified through attention at this stage. Encoding [4] Author A, Simon MA, Tzur R, et al. Explicating a Mechanism for Conceptual Learning: Elaborating in problem-solving is an individual attempt to the Construct of Reflective Abstraction. 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The difference with Cohen, if Cohen is more confidence in procedural and conceptual focused and detailed in describing the process of mathematics. Int J Math Educ Sci Technol 2011; reflective abstraction occurs when students solve 37–41. problems, Cifarelli looks more at the awareness and [18] Hiebert, Lavefre. Conceptual and Procedural anticipation made by students. Knowledge in Mathematics: An Introductory Analysis. In: Procedural and Conceptual REFERENCES Knowledge: The Case of Mathematics. 1986. [19] Department NYSE. New York State Learning [1] Piaget J. STUDIES IN REFLECTING Standard for Mathematics. ABSTRACTION. New York: Psychology Press, [20] Badger MS, Sangwin CJ, Hawkes TO, et al. 2001. Teaching Problem-solving in Undergraduate [2] Beth EW, Piaget J. Mathematical Epistemology Mathematics. Coventry: Coventry University, 2012. 41
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