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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
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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
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