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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 ...

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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
                    
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...Advances in social science education and humanities research volume proceedings of the th international conference on technology iset developing mathematical conceptual understanding through problem solving role abstraction reflective lulu choirun nisa st budi waluya universitas negeri semarang indonesia choirunnisa gmail com s b math unnes kartono scholastika mariani scmariani abstract is a mechanism that lower level to higher reflexion moves individuals from one rearranging structure this may be knowledge first study very builds novelty therefore important part how dominant process formed beth piaget explicitly states forms new or for development example dubinsky david tall advanced concepts because mitchelmore are some researchers who focus constructs processed concept formation also stated resulting mastery students played lot good mathematics product will able simon et al reason comprehend operate connect method supports animates large buildings idea play logic construction arnon ...

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