Genetic Decomposition Research Articles

University students’ mental construction when learning the Convergence of a Series concept

This article uses action-process-object-schema theory to study the mental constructions about the limit of series during a Calculus 1 course at a university. The researchers also used the theory’s teaching methodology to teach the topic. A plethora of research on the limit concept is available and suggests that the concept is on record as being difficult for students to learn and comprehend. The study proposed a genetic decomposition on how undergraduate students might demonstrate their mental constructions in learning the limit of a series. Students were taken through the activities-classroom discussions-exercises cycle. Thirty students participated in answering questions based on the convergence of a series. The students’ written responses together with the interviews were analysed and based on the findings a revision of the preliminary genetic decomposition was done. We found that there were students who did not display the predicted mental construction indicated by the preliminary decomposition in the application of the definition for the convergence of a series, but displayed the predicted mental construction for the application of the series convergence tests. We also found that certain schema were necessary for the achievement of a complete understanding of the convergence of a series concept. The mental constructions within the missing schema were included in the modified genetic decomposition. This empirically enriched model is now expected to inform pedagogy on the convergence of a series concept.

Construction of a genetic decomposition: theoretical analysis of the concept of the value theorem medium

In this study, from a cognitive perspective, we present the structures necessary for a student to construct the concept of the mean value theorem. The theoretical framework used for this research is based on the action-process-object-scheme theory. We propose, as a result, two genetic decompositions that describe two possible ways to construct the concept: a graphical path and an analytical path. Generic decompositions become a point of reference for the design of teaching strategies; in that sense, the results of this research are presented as a theoretical proposal for teaching and learning the model of the mean value theorem to undergraduate students of mathematics, physics, engineering or in whichever area this particular subject is studied, with the purpose of contributing to the strengthening of educational practices and student learning.

How are the process of abstraction of the division of fraction numbers by elementary school students?

The division of fractions is one of the difficult materials for elementary students. That is a concept that they must understand. Therefore, the abstraction process is needed to achieve it. The purpose of this study is to describe the abstraction process of dividing fractions by elementary school students. The subjects of this study were elementary school students in Bengkulu City, Indonesia. The subjects of this study were chosen 3 out of 85 elementary school students in Bengkulu City, Indonesia. There are two story questions submitted to research subjects about the distribution of mineral water after playing soccer. Subjects were interviewed in depth. That is a task-based interview. Data were analyzed through genetic decomposition. The result is students can be more meaningful in preparing propositions about fraction operations. Students can state that the division of fractions is a repetitive reduction. By providing contextual problems students can complete the process of abstraction, and can reach the concept of fraction division appropriately.

Students’ understanding of the concept of function and mapping

Function is a concept that is very much related to daily life. Likewise with the concept of mapping. The purpose of this study is to describe students’ understanding of the concept of function and mapping by students who are at a high level (ie extended-trans level). The student is the subject of this study. This research is exploring the ability of research subjects. The research subjects were taken from mathematics education students at Bengkulu University. We selected one subject of research, Mg. Student was interviewed in depth about the concept of function. Data were analyzed based on genetic decomposition. The results of this study are that students are able to present an overview of the concept of function with a broad extension level. The function from the set A to B is the set f whose members are sequential pairs of the two cross-sets of non-empty sets A and B such that (a, b) and (a, b ′) elements of f then b = b′. Also able to make a simplification in the form of mapping definition, which states that the domain of function f is the set A. The conclusion of this study is that high-level students have an in-depth understanding of the concept of function. He is able to compile extensive extensions. Also, make the exact simulations.

Multiplication and division of integers through cultural approaches of playing dakon

Playing dakon has become a culture in Bengkulu. Since I was in kindergarten, dakon has been a fun game. It has the character of sharing fruit in each dakon convex (or hole). It becomes the starting-point of learning multiplication and division. The purpose of this study was to determine students’ thought processes in understanding the principle of multiplication and the division of integers through the cultural approach to playing the dakon. To achieve the goal, achieved through qualitative-explorative research. The subjects of this study were elementary students in Bengkulu City who were selected based on assignments. It was selected from 30 students who were given assignments about multiplication and division of integers. Students are given instructions using worksheets. After students submit their assignments, they are analyzed to select 2 students who have the thought process in achieving the principle of division and multiplication appropriately. Subjects were interviewed in-depth by researchers. Data were analyzed based on the genetic decomposition of research subjects. The results of this study are the Subject-1 can use the dakon to show the multiplicative properties of positive integers. He demonstrated that 4x3 by inserting it into four small dakon curves and three dakon seeds each. She concluded that multiplication is a repeated sum. Subject-2 uses the dakon to find the concept of dividing two positive integers. Subject-2 shows that 12: 4 by taking dakon seeds from a large curvature containing twelve seeds. She concluded that the division is a repeat reduction. This research concludes that Subject-1 and Subject-2 can build an understanding of operations multiplication and division, including verbal communication, writing, and drawing based on real media from local culture, namely the dakon game.

The cognitive process of students in understanding the triangles in Geometry of Riemann through local content

Learning geometry is one that students fear. They have difficulty understanding the concepts of geometry. Students experience problems with Lobachevsky’s axiom alignment, as well as Riemann’s axiom alignment. Therefore, we need learning that makes it easy for students to imagine these axioms more concrete, although geometry cannot always be carried in a concrete world. The purpose of this study is to describe students’ cognitive processes in understanding the properties of triangles in Riemann Geometry through local content. This is a part of development research. We conducted a needs analysis of the ability of students to understand new concepts in geometry. The subject of this research was chosen from students of mathematics education at one of the tertiary institutions in Bengkulu City, Indonesia. Subjects were interviewed in-depth through task-based interviews. His job is to solve the problem of the properties of triangles in Riemann Geometry. Interview data were analyzed through genetic decomposition analysis. The results of this study were subjects able to analyze the interrelationship between concepts, axioms, and basic properties of geometry through orange peel media. He can attribute and organize. The attributes are obtained from three segments that meet at the ends. Subjects can organize the elements of the triangle so that the nature of a large number of angles in the triangle is more than 180. The conclusions of this study are the subjects can achieve the cognitive process of analysis, and can arrange the correct population.

Formulating a Modified Genetic Decomposition for the Concept of ‘Limit of a Sequence’

In this paper we focused on university mathematics students’ mental constructions on the concept limit of a sequence. The aim was to explore the nature of the mental constructions and to contribute to Action–Process–Object–Schema (APOS) theory in terms of instructional strategy for teaching limit of sequences. The belief that understanding students’ mental constructions in learning mathematics leads to improved instructional strategy guided this study. Maple software was applied in the learning of limit of sequences to foster students’ understanding of the topic. Thirty university students took part in this study. The data was collected through the test questions and interviews. The results revealed that in most cases, the preliminary genetic decomposition could be used to appropriately describe students’ constructions of their responses. Furthermore, many students operated at an action/process/object level with the exception of a few students who were not able to operate at these levels. It was also revealed that few students carried out procedures effectively without the conceptual understanding of the concept limit of a sequence. It is recommended that the preliminary genetic decomposition be extended to include aspects necessary for conceptual understanding of the limit of sequences in modified genetic decomposition, such as prerequisites, concepts which include rules of inequalities and properties of the modulus function. These concepts are required for the understanding of proving or disproving the given limits of a sequence.

New approaches for two-variable inequality graphs utilizing the Cartesian Connection and the APOS theory

This study utilizes the Cartesian Connection and the action, process, object, and schema (APOS) theory to investigate how preservice mathematics teachers constructed and justified graphs of two-variable inequalities. The analysis shows that most teachers were not equipped with the skills and knowledge that were prerequisites for action, process, and object conceptions of two-variable inequality graphs. Many teachers used informal rules, rather than mathematical ideas such as the Cartesian Connection, to construct and/or justify inequality graphs. They also perceived inequality graphs as shapes rather than as representations based on the meaning of inequalities. I present a genetic decomposition for inequality graphs, through the concepts of variable and of parameter, as well as tasks associated with the genetic decomposition. I call for more studies on inequalities, including those relating to the implementation of the suggested genetic decomposition and tasks.

Students’ understanding of Riemann sums for integrals of functions of two variables

In this paper we report on a study of how students understand some of the most fundamental ideas of Riemann integrals of functions of two variables. We apply Action-Process-Object-Schema (APOS) Theory to pose a preliminary genetic decomposition (GD), conjecturing mental constructions that students would need to relate Riemann sums to integrals of functions of two variables over rectangles. The genetic decomposition is informed by the researchers’ classroom experience, findings of a previous study that applied semiotic representation theory, and by a study on integrals of functions of one variable. We pay particular attention to the case of an integral of a continuous function over a rectangle and the simplest partition possible, that consisting only of the rectangle itself. We then explore students’ geometrical understanding of the relation between the single termfa,bΔxΔy, where a,b is a point on the rectangle, and the double integral over the rectangle. We tested the GD by performing student interviews with 10 students who had just finished taking a lecture-based multivariable calculus course. The findings underscore the importance of each of the mental constructions described in the genetic decomposition and suggests that students have difficulty in some mental constructions that may commonly be assumed to be obvious during instruction.

Implementation of APOS theory to encourage reflective abstraction on Riemann Sum

The purpose of this study is to explore the implementation of APOS (Action-Process-Object-Schema) in first-year students at higher education as a basis of instructional learning in Riemann Sum to emerge the reflective abstraction. This is qualitative research, and data were taken by tests, interviews, and observation. To construct the Riemann Sum understanding, students came through the Action by estimating the area under the curve with various shapes, looking for the area under the curve through several partitions, and make repetitions. The Process was built by interiorizing the previous action on n partitions. The Object was constructed through the encapsulation mechanism by determining the existence and value of the limit for the Riemann Sum. Schemes occurred through thematization by applying Riemann Sum to contextual situations involving functions of distance and time. The results showed that the success of APOS implementation was determined by genetic decomposition, assistance, reinforcement of preconditions and reinforcement of concepts by lecturers. Some effective scaffoldings to help the construction of reflective abstractions in Riemann Sum are worksheets and guided question from educator.

Analysis of difficulties for pre-service mathematics teacher in problem solving of division and divisibility based on theory of action, process, object, and schemes

Action, process, object, and schemes is an acronym of APOS. The difficulty in the perspective of APOS theory is the inability of students to access specific schemes in all situations, thereby inhibit the construction of other concept. Knowing student’s difficulties in using the concept of division and divisibility is necessary to design genetic decomposition of that concept. This article is a report from qualitative research that discusses conceptual issues about the idea of division and divisibility given by prospective mathematics teacher students who use the APOS theory. The data for the study were collected from an interview. The researcher took three interview subjects from 30 pre-service mathematics teachers who have attended and completed a test on the division and divisibility in an elementary number theory course. These participants were selected based on their performance in the test. However, the interview conducted showed that there are five difficulties involved and they include solving the divisibility problem on integers, algebraic manipulation and recognize structures in algebraic expressions that correspond to the division algorithm or the definition of divisibility., distinguishing between variables and parameters on division problem presented in algebraic expression, associating between division algorithm and definition, and application of divisibility properties and theorems.

The genetic decomposition of students about infinite series through the ethnomathematics of Bengkulu, Indonesia

The cognitive theory views individuals as active information processors, so that individuals were able to represent each information according to the level of knowledge they have. Student representation can be seen as its genetic decomposition. Ethnomathematics was a vehicle for unlimited series learning. The purpose of this study was to describe the genetical decomposition of students about infinite series through the Bengkulu ethnomathematics. This was the initial research from a series of development research. This stage we interviewed in depth 10 high school students in Bengkulu, Indonesia. The research instrument was the researchers themselves who were guided by interview guides about understanding concepts and the principle of infinite series. Interviews were conducted during and after Bengkulu’s ethnomathematics learning. Data were analyzed through fixed comparison techniques. The results of this study were found that students can coordinate two or more actions about convergence of sequences, but not for convergence of infinite series. Conversely, there were students who can describe a particular object about converging an infinite series but not coordinated with the processes that were built for converging sequences. Also, there were students who can coordinate the action-process related objects so that a schema of converging sequence was formed, but for infinite series only in the form of separate actions or processes. The conclusion of this study was the genetic decomposition of students in understanding the infinite series through ethnomathematics learning at the intermediate level (inter level). The student before ethnomathematics learning was at the lower level (intra and pre-intra levels).

Exploring the Conceptual Understanding of the Quadratic Function Concept in Teachers’ Colleges in Zimbabwe

This paper reports on an exploration of preservice teachers’ understanding of the quadratic function concept in Zimbabwe. These concepts were taught to preservice teachers studying for a diploma in Education wishing to specialize in the teaching of Ordinary level mathematics. Concerns about high Mathematics failure rate in Zimbabwean Secondary Schools have prompted this investigation into finding out if teachers’ understanding of quadratic function concept could be the cause. The study adopted the APOS (action-process-object-schema) to investigate their conceptual understanding of the concepts. Data were generated from students’ responses to a written task and follow up interviews were used to solicit information from the preservice teachers. A designed genetic decomposition for quadratic concepts was used as an analysis tool. The findings of the study also revealed majority of the pre-service teachers seemed to be operating at the action level of understanding, with very few teachers who have reached the object level. It was noted that the preliminary genetic decomposition failed to accommodate all students’ responses which lead to the development of a modified genetic decomposition.

What does ‘y is defined as an implicit function of x’ mean?: An application of APOS-ACE

Implicit functions is a basic and important topic in calculus and yet very little research has been done on students’ understanding of this notion. In a previous study we had documented students’ misconceptions and lack of understanding, and proposed a conjecture of mental constructions (called a genetic decomposition; GD) students may do in order to understand this topic. In the present study the proposed GD is tested with student interviews and refined. The study starts by using semi-structured interviews of fourteen students who had just finished a traditional one-variable calculus course, further exploring students’ understanding of implicit functions. Then using the GD, a cycle of Activities on computer, Classroom discussion, and Exercises (ACE cycle) was designed and implemented to help students do mental constructions found to be lacking. Finally, the fourteen students were again interviewed after they participated in the activities of the ACE cycle. The students’ answers in the second interview revealed that their understanding of implicit functions had improved. The results of this article are of practical importance as they underscore that activities based on a GD have the potential of improving student understanding of implicit functions. Further, the study also suggested refinements to the genetic decomposition for implicit functions.

Cognitive obstacles in interiorization of the Riemann’s Sum concept through APOS approach

The APOS approach in mathematics learning has been developed by Dubinsky and has been applied in many studies. Instructional to build the concept through the stages of Action - Process - Objects and Schema do not always run well as genetic decomposition that has been built. This study explores students’ cognitive difficulties in learning advanced mathematics on the Riemanns Sum, particularly in the process of interiorization Action into Processes. The subject of research is first year student of UIN Walisongo Semarang and the study was conducted in the first half of 2018. The data were taken using observation, tests and interviews. The results of the qualitative analysis show that students got the Action well but failed to interiorize it into Process due to the inability to connect two-dimensional knowledge with the given partition. It was showed when students could not find rectangular elements on a partition built from the area under the curve with a particular function. Unexpectedly, even though students understand the problem and the student could not use the given function y=f(x) to determine the partition height. Furthermore, students got an obstacle to imagine adding partitions when the number of partitions approach to infinity. The transition from school mathematics to advanced mathematics is perceived when students represent their knowledge partially, instead understanding the problem as a whole interrelated concept. This constraint also indicates the importance of mathematical connections and mastery of prerequisite materials to construct the further concepts.

Pre-Service Mathematics Teachers Divisibility Construction

Divisibility concept is fundamental to the development of another concept in elementary number theory. A study of the construction of the divisibility advanced mathematics teacher was done to provide empirical evidence of early genetic decomposition design. Through the question, we designed in the form of test sheets and interview, we explorer the construction of 3 students. The result of this qualitative study illustrated that construction of the divisibility concept on the problem solving of number theory represented in algebra.

The abstraction ability of students in understanding the concept of geometry

The purposes of this study are to determine the ability of learner’s abstraction in understanding concepts and principles of geometry. This study used a qualitative method. The research subjects were selected from undergraduate students of mathematics education. The selection technique is based on the ability to understand geometric concepts. We were collected data through task-based interviews. The analysis of data with the genetic decomposition techniques. Furthermore, the theoretical processes is carried out to determine the characteristics of each level of abstraction. The result was the six levels of students’ ability to understand geometrical concepts. The six levels were: Level 0 (concrete objects), Level 1 (model of semi-concrete), Level 2 (theoretical models), Level 3 (language in domain example), Level 4: (language of the math), and Level 5 (model of inference).

Pre-service mathematics teachers’ development process in using manipulatives in number operations

Background: Teaching using manipulatives is emphasised, especially in the early grades, to help learners conceptualise operations on whole numbers. Therefore, teachers’ competencies in using manipulatives is the key in helping learners master these basic operation skills.Aim: Drawing from the literature on using manipulatives to improve learners’ performance in mathematics, this study recounts foundation phase pre-service teachers’ conception of using manipulatives to enhance their competencies and reasoning skills to model the solution in number operations.Setting: Data presented here was collected from 31 participants. These pre-service teachers either passed mathematics or mathematical literacy with 40% at the grade 12 level.Methods: Data was collected from participants’ written work (e.g. classroom tasks, homework, tests and examinations) and during class discussions. Interviews were conducted with some students. We analysed their conception guided by the APOS theory, namely, Action-Process- Object-Schema.Results: We observed improvement in the conception of using manipulatives among pre-service teachers. In the first semester, most students display action conception of using manipulatives to either represent or model a solution. However, in the second semester, most students either display process or object conception as explained in the genetic decomposition. We attributed the improvement to change of instruction in the second semester as we taught in accordance with the APOS theory.Conclusion: It is evident that there are a number of contributing factors to pre-service teachers’ conception of mathematical concepts, and teacher educators need to pay particular attention to these to help pre-service teachers master the concepts they would teach at school.

The ability to Understanding of the Concept of Derivative Functions for Inter-Level Students During Ethnomathematics Learning

The mathematical abilities can be measured through students’ mental and physical activities. One of them is using genetic decomposition analysis of APOS theory. The purpose of this study was to describe the ability to understand the concept of derivative functions for interlevel students during ethnomathematics learning. The research is part of our development research.The subjects were high school students majoring in mathematics and science in Bengkulu. The research was carried out in a participatory manner in the regular learning process by applying the ethnomathematics approach. Subjects were interviewed based on the assignments given. Data were analyzed qualitatively by applying genetic decomposition. To obtain inter-level characteristics, the description of genetic decomposition continued with fixed-comparison methods. The result was able students to encapsulate the process of function properties or intervals in the domain so that the object is formed about the sketch of the function graph h.

THE TRANS LEVEL CHARACTERISTICS ABOUT INFINITE SERIES

Infinite series is one of the difficult calculus material for students. It needs to be analyzed the cognitive activities of students to make it easier for lecturers to arrange learning plans. The purpose of this study is to describe the characteristics of the trans level about infinite series. The subject of this research was a student of Mathematics Education, one of the universities in Bengkulu. A total of 5 people were selected from 29 students based on their cognitive abilities. Subjects were interviewed based on the assignment given. Data is analyzed through its genetic decomposition. The results of the study are that subjects can coordinate other objects and processes, so that the scheme is formed about the convergence of infinite series and sequences. Conclusion: the subject is able to do thematization so that forming a mature scheme is a trance level characteristic.