Genetic Decomposition Research Articles

Students' conception of partial derivatives of a function of several variables

The study aimed to explore and assess students' conceptual understanding of partial derivatives in functions of several variables. A case study research design was employed, with four classes randomly selected for participation. Qualitative data were collected through interviews and open-ended questions developed by the researchers. The analysis was conducted using APOS theory, based on the proposed genetic decomposition. The findings reveal that the majority of students’ understanding falls under action conception, interpreting partial derivatives similarly to single-variable derivatives. Additionally, many students struggled to extend their knowledge from single-variable derivatives to partial derivatives and faced difficulties in converting between algebraic/symbolic and graphical representations. Based on these results, the researchers recommend that educators adopt instructional strategies that foster deeper conceptual understanding, rather than relying on traditional teaching methods. Keywords: APOS Theory; conceptual understanding; function of several variables; partial derivatives

Construction and mental mechanisms for learning about the exponential function in a school setting

[Objective] The objective of this study was to examine, in the light of the APOS theory (action, process, object, and schema), the results produced by secondary school students when addressing 11 items related to the exponential function. [Methodology] The approach was qualitative, using content analysis. The sample consisted of 15 students between 15 and 18 years old, chosen in a non-probabilistic way to obtain the greatest amount of information. Based upon the theory, an instrument was designed which consisted of a hypothetical genetic decomposition of the concept of exponential function as a cognitive object, according to structures (action, process, object and schema) and mental mechanisms (interiorization, coordination, encapsulation, decapsulation, and investment), to interpret the mental construct that students create about the function being studied, based on its historical epistemological development, its presentation in school textbooks and the experience of the researcher. [Results] At the level of results, it was found that 13 students showed an action conception of the function being studied and 2 students built the concept at the process level. [Conclusion] Students most often built the concept of an exponential function at the action level, that is, everything related to repetitive and mechanical processes with powers, image calculation, graphic representation, solution of exponential equations; some students even responded to the items without explicitly performing all the steps required for their resolution. These results demonstrated the absence of a mental process related to the exponential function and its consequent encapsulation into an exponential function object.

INVESTIGATING PRE-SERVICE MATHEMATICS TEACHERS' CONCEPTIONS OF THE PROPERTY OF COMPLETENESS OF REAL NUMBERS

In recent years, the emphasis on concept teaching and conceptual understanding has played a central role in mathematics education curricula. The conceptualization of number systems is important to understand all areas of mathematics. Considering the difficulties experienced in the conceptualization of real numbers, it is seen that students have difficulty in understanding the relationship between real numbers and other number systems. This study aims to investigate pre-service mathematics teachers' conceptions of the property of completeness, which is the most basic element that distinguishes a real number set from other number systems. To this end, the APOS theory, which reveals the mental structures and mechanisms of the individual regarding a concept, was used. Data were collected via semi-structured interviews with three pre-service teachers. The data were analyzed using the descriptive analysis method, and the findings were presented under two themes, which are the epistemology of the concept of the property of completeness of real numbers and the mental structures in the schemas related to the property of completeness of real numbers. The findings revealed that the pre-service teachers' conceptions of the property of completeness of real numbers was mostly at the action level. The study found some mental structures such as the representation of rational numbers on a line, which should be included in the genetic decomposition to be created in the context of the APOS theory regarding the completeness of real numbers.

Subject competency and teacher knowledge: An exploration of second-year pre-service mathematics teachers’ difficulties in solving logarithmic problems using basic rules for logarithm

Pre-service mathematics teachers’ (PMTs) subject competency continues to engage scholars and researchers. Understanding level of knowledge of concepts that PMTs bring to their learning in university is crucial to developing their teacher knowledge. This article examines genetic decomposition of schemas PMTs in one university in South Africa build (to know about rules) for solving logarithmic equations. A mixed methods approach, and the action-process-object-schema (APOS) theory were employed to examine mental construction the 19 purposively selected PMTs that responded to a 90-minute simple logarithm research task (LRT) made while solving problems. Analysis of task scripts using percentage score forms the basis of the qualitative phase of the research. Individual interview was useful to elicit PMTs’ views and perceptions of their encountered difficulties in solving LRT problems. One common difficulty was proving the logarithmic equation. This highlights gaps in PMTs’ prior knowledge of logarithmic concepts and basic rules. Implications of the findings for PMT subject competency were discussed.

The role of topology in the construction of students’ optimization schema for two-variable functions

This study uses Action-Process-Object-Schema theory (APOS) to examine students’ understanding of two-variable function optimization. A genetic decomposition (GD) based on the notion of Schema is proposed. This is a conjecture of mental structures and relations that students may construct to understand the optimization of these functions. The GD was tested with semi-structured interviews with eleven students who had just finished an introductory multivariable calculus course. Results show that giving explicit attention during instruction to the topological structure of the domain of the function to be optimized and the use of GD-based activities was effective in promoting students’ understanding of two-variable function optimization. On the theoretical side, the study contributes to a better understanding of the APOS notions of Schema, Schema-triad, and types of relations between Schema components that have not been used extensively in the literature and that proved to be a powerful tool to model students’ learning.

Advanced-Level Students’ Understanding of Exponential Equations:

This paper reports on a study which explored high school students’ conceptual understanding of the techniques of exponential functions. Thirty-one advanced-level students participated in the study. The study used APOS (Action-Process-Object-Schema) theory, a constructivist theory framework, to investigate participants’ conceptual understanding of exponential functions. Activity sheets constructed with tasks based on exponential equations were administered to the participants. The written responses were used to identify participants’ mental constructions of these concepts. Furthermore, interviews were carried out to clarify participants’ written responses. The written responses and interview discussions pointed out that participants exhibited procedural tendencies in exponential functions. Most of the participants could not solve exponential equations, especially the radioactive‑decay functions. In addition, many participants did not have appropriate mental constructions at the process, object and schema levels, since most of them could not coordinate processes and encapsulate them into an object. This paper raises some important implications for mathematics education and further provides applications of genetic decomposition design and modification.

Students’ geometric understanding of partial derivatives and the locally linear approach

We use Action-Process-Object-Schema (APOS) theory to study students’ geometric understanding of partial derivatives of functions of two variables. This study contributes to research on the teaching and learning of differential multivariable calculus and its didactics. This is an important area due to its multiple applications in science, mathematics, engineering, and technology (STEM). The study tests a previously proposed model of mental constructions students may use to understand partial derivatives through a set of activities designed to help students make the conjectured constructions. The model is based on the local linearity of differentiable two-variable functions, and the model-based activities explore the relationship between partial derivatives and tangent plane in different representations. We used semi-structured interviews with eleven students whose teacher used the three-part cycle—Activities designed with the genetic decomposition; collaborative work in small groups and Class discussion; and Exercises for home (ACE)—as pedagogical strategy. The model-based activity set based on local linearity and the ACE strategy helped students construct a geometric understanding of partial derivatives. Results led to reconsider and further refine the model. This study also resulted in improving activity sets and obtaining information on students’ construction of second-order and mixed partial derivatives.

Students' Spatial Ability During Geometry Learning through a Realistic Approach In Terms of Their Genetic Decomposition

Geometry is a part of mathematics that students must learn. Spatial ability as a basic ability that students must have to learn more geometry. The realistic mathematical approach provides horizontal processing towards formal understanding. The purpose of this study is to interpret the characteristics of students' spatial abilities during learninggeometry through a realistic mathematical approach in terms of APOS Theory. This research uses quantitative qualitative and descriptive methods. The focus of this research is to explore students' spatial abilities after learning geometry through a realistic approach in understanding geometry concepts reviewed from APOS (Action-Process-Object-Schema) Theory. To comply with the COVID-19 health protocol, the implementation of learning was carried out online with as many as 30 students in the Mathematics Education Program in Rejang Lebong Regency, Bengkulu Province, Indonesia. This study also has exploratory characteristics. The researcher acts as a teacher as well as the main instrument in this study. Researchers use spatial ability test instruments to explore qualitative data. The test result data is used as the basis for determining the research subjects to be interviewed in depth to explore students' spatial abilities during geometry learning through a realistic mathematical approach. The subjects of this study were three students selected from three categories of test results, namely the high, medium and low groups of one person each. Interviews of subjects were conducted using whatsapp media via video call. These interviews are recorded to obtain complete and accurate data. Research data are analyzed through genetic decomposition analysis (description of their actions, objects, processes, schemes, and relationships that individuals have for concepts and principles in spatial geometry). The result of this study was that it was found that students had a mature scheme about the properties of building space. He was able to perform the tematization of objects about the diagonal of the plane and the diagonal of space can be represented in good mental activity, he was a student of the trance level. Inter students are able to carry out action-process-object-and schematic activities, but the embedding is incomplete. Intra students are able to do perceptions of space, but are unable to do interiorization and encapsulation so they fail to solve the given problem. The conclusion of this study is that there is an increase in the spatial understanding of students who are taught through a realistic mathematical approach. There are three levels of students' spatial ability after learning geometry through a realistic mathematical approach. Those are the trance level, the inter level and the lowest intra level.

Genetic Decomposition of the Heritable Component of Reported Childhood Maltreatment

BackgroundDecades of research have shown that environmental exposures, including self-reports of trauma, are partly heritable. Heritable characteristics may influence exposure to and interpretations of environmental factors. Identifying heritable factors associated with self-reported trauma could improve our understanding of vulnerability to exposure and the interpretation of life events. MethodsWe used genome-wide association study summary statistics of childhood maltreatment, defined as reporting of abuse (emotional, sexual, and physical) and neglect (emotional and physical) (N = 185,414 participants). We calculated genetic correlations (rg) between reported childhood maltreatment and 576 traits to identify phenotypes that might explain the heritability of reported childhood maltreatment, retaining those with |rg| > 0.25. We specified multiple regression models using genomic structural equation modeling to detect residual genetic variance in childhood maltreatment after accounting for genetically correlated traits. ResultsIn 2 separate models, the shared genetic component of 12 health and behavioral traits and 7 psychiatric disorders accounted for 59% and 56% of heritability due to common genetic variants (single nucleotide polymorphism–based heritability [h2SNP]) of childhood maltreatment, respectively. Genetic influences on h2SNP of childhood maltreatment were generally accounted for by a shared genetic component across traits. The exceptions to this were general risk tolerance, subjective well-being, posttraumatic stress disorder, and autism spectrum disorder, identified as independent contributors to h2SNP of childhood maltreatment. These 4 traits alone were sufficient to explain 58% of h2SNP of childhood maltreatment. ConclusionsWe identified putative traits that reflect h2SNP of childhood maltreatment. Elucidating the mechanisms underlying these associations may improve trauma prevention and posttraumatic intervention strategies.

The process of achieving the principles of the triangle area of middle school students through YouTube assisted learning during a pandemic

The purpose of this study was to describe the achievement of the triangular principle of junior high school students through YouTube-assisted learning during the COVID-19 pandemic. It is exploratory research. The subjects of this study were middle school students in Bengkulu City, Indonesia. We conducted task-based interviews with twenty-one people. The data from the completion of the task was analyzed early to determine their level of thinking. It was using its genetic decomposition. We classified them into five levels of schema development. We selected students who were at the trance level for further in-depth interviews. We use audio-visual recorders to get complete and accurate data. Data were analyzed by applying qualitative data analysis techniques. The results of this study were 19% of the research subjects were at the intra level, 33% were at the semi-inter level, 33% were at the inter level, and 10% of the subjects were at the inter level, and only one person who was at the trance level (= 5%). The conclusion of this study is that students who are at the highest level are able to build relationships between actions, processes, objects, and previous schemas so that a mature schema is formed about the area of the triangle.

IN SERVICE TEACHERS MENTAL CONSTRUCTION OF LINEAR ALGEBRA CONCEPTS. AN UNDERGRADUATE CASE STUDY

This study investigated how undergraduate in-service mathematics teachers understand linear algebra concepts. The significance of the study is to reveal the difficulties and type of mental constructions they make when they are solving linear independence concepts by using inspection. The action, process, object, schema (APOS) theory was used to analyse the data collected. A descriptive qualitative case study design was used. The participants were 73 in-service teachers studying a Bachelor of Science Education Honours Degree in mathematics. Data was collected through a structured activity sheet and semi-structured interviews. The results showed that the teachers had an inadequate understanding of linear independence concepts. Many of the students developed the action conceptions of linear algebra as they were engrossed in step-by-step procedures trying to show that given vectors are linearly independent. The findings indicated that the schemata for determinants and the various theorems for determining linear independence promote learning. The modified genetic decomposition proposed has implications for teaching the concept and students should be given different kinds of sets to manipulate so that they can develop their mental constructions at the process level.

Cognitive obstacles in the learning of complex number concepts: A case study of in-service undergraduate physics student-teachers in Zimbabwe

Physics and mathematics are interrelated as part of the science, technology, engineering, and mathematics (STEM) disciplines. The learning of science is supported by mathematical skills and knowledge. The aim of this paper is to determine the cognitive obstacles of in-service undergraduate physics student-teachers’ understanding of the concept of complex numbers which is part of linear algebra. A case study is presented involving 10 undergraduate student-teachers at a university in Zimbabwe studying for a Bachelor of Science Education Honours Degree in physics. Data were generated from the 10 participants’ answers to structured activity sheets and interviews. Action, process, object, schema (APOS) theory was used to explore the possible ways that students may follow to understand the concepts of complex numbers and how they concur with the preliminary genetic decomposition. It was observed that most of the participants were operating at the action level, with a few operating at the process and object levels of understanding. Recommendations are made in this study that instructors should pay more attention to the prerequisite concepts and the “met afters” so that students can encapsulate processes into object understanding of division of complex numbers and polar form.

Students' cognitive processes in understanding the application of derivatives

Derivatives are one of the objects of calculus that is difficult for students to learn. The purpose of this study was to describe the cognitive processes of students in understanding the application of derivatives. This is a qualitative research involving one subject with the initials L. The main instrument of this research is the researcher who is guided by an assignment sheet and an interview guide. Data collection was carried out through task-based interviews. To get complete and accurate data recorded through audiovisual. Data were analyzed descriptively through genetic decomposition based on APOS (action-process-object-schema) theory. The results of this study are that L can coordinate the process-object of all the properties of a given function, with adjacent or overlapping intervals in all h domains so that a mature scheme of function graph sketches is formed. So that an accurate function graph sketch is obtained. The conclusion is that L's cognitive process of applying derivatives is at a high level. It's a trance level.

High school student understanding of exponential and logarithmic functions

We use Action-Process-Object-Schema theory (APOS) to study high school student understanding of exponentiation and their construction of exponential and logarithmic functions. We extend didactic materials similar to those of Ferrari-Escolá et al. (2016) to include exponentials on the rational numbers and to help students construct logarithms as numbers. Qualitative data from the problem-solving activities of two groups of eight students each during a series of teaching episodes suggests that some students can use these materials successfully. The data analysis enabled us to give specific suggestions on how to help other students do some of the constructions needed to understand these functions. Research shows these constructions are difficult for students. The findings of our study led to contributing a new and detailed genetic decomposition that can be tested and improved in future research cycles.

Content Analysis of the APOS Theory Studies on Mathematics Education Conducted in Turkey and Internationally: A Meta-Synthesis Study

Studies on the APOS Theory, which was developed within the context of conceptual understanding as one of the main aims of mathematics education, have been increasing in recent years. The content analysis of these studies can be useful both for the development of the APOS Theory and for identifying the needs for future research. This study aimed to review the APOS Theory studies in the field of mathematics education. For this purpose, the studies with national and international samples related to the APOS Theory were subjected to descriptive content analysis, and a systematic summary was presented. As a result of the analysis, three themes, namely the purpose of using the theory, the use of genetic decomposition, and the aim of the study were identified. It was determined that genetic decomposition, which is the main component of the APOS Theory, was not used in 13 percent of the studies reviewed. Although the theory was mainly used for data analysis, it can be said that not using genetic decomposition in data analysis is not compatible with the purpose of the theory. On the other hand, there are descriptive studies (39%) in the literature that determine the mental structures and mechanisms predicted by the APOS Theory rather than the structure of mathematical concepts. It can be said that such studies are weak in terms of the theory’s aim of reinforcing conceptual understanding. As a result, it has been revealed that, the APOS Theory has an important place in concept teaching.

Hybrid Temperature Compensation Model of MEMS Gyroscope Based on Genetic Particle Swarm Optimization Variational Modal Decomposition and Improved Backpropagation

Hybrid Temperature Compensation Model of MEMS Gyroscope Based on Genetic Particle Swarm Optimization Variational Modal Decomposition and Improved Backpropagation

Un enfoque cognitivo en la comprensión de la comparación entre los cardinales del conjunto de los números naturales y el conjunto de los números enteros en estudiantes de secundaria

This research studies the mental structures and mechanisms of 164 Chilean students of third and fourth half (17 and 18 years old), when comparing the cardinal of the set of natural numbers (ℕ) and the cardinal of integers (ℤ). It is based on a provisional genetic decomposition of infinity within the framework of APOS theory, and shows that students establish only an injective function from ℕ to ℤ to coordinate the infinite iterative processes and, furthermore, they do not manage to present the structure of a single infinite iterative process that would allow them to conclude that they have the same cardinal number.

A preliminary genetic decomposition for conceptual understanding of the indefinite integral

This paper reports on the use of APOS theory to investigate conceptual understanding of the indefinite integral amongst undergraduate students at the University of KwaZulu-Natal, South Africa. We present a Preliminary Genetic Decomposition (PGD) for the indefinite integral, which predicts the mental constructions and mechanisms that may facilitate conceptual understanding. In this pilot study, the analysis of students’ written responses to the research instrument suggested that more than half of the participants lacked the prerequisite knowledge of the concepts of function, derivative of functions and the chain rule. The findings confirmed that students experience greater difficulty dealing with transcendental functions than with algebraic functions. The analysis indicated that the cognitive mechanism of reversal, to recognise the inverse nature of differentiation and integration, was inconsistent or absent in many students. Interview data from a subset of participants was employed for triangulation with the document analysis. The empirical data suggested refinement of the PGD and modification of the research instrument, and further fine-grained interviews with study participants to investigate their conceptual understanding more deeply, and for the purposes of data triangulation.

Implementing GeoGebra integrated with multi-teaching approaches guided by the APOS theory to enhance students’ conceptual understanding of limit in Ethiopian Universities

The notion of limit is one of the fundamental concepts which underpins advanced calculus of one or more variables in the field of analysis. However, understanding the concept of limit has been an impenetrable problem for many students in Ethiopian Universities. Only very few literatures were documented focusing on overcoming the difficulty of learning the concept of limit. For this reason, the overarching aim of the present study is to enhance students' conceptual understanding of limit by empowering their visualization skills using GeoGebra integrated with multi-teaching approaches. The study employed mixed methods experimental (intervention) design within an APOS paradigm. Both qualitative and quantitative data were collected. Qualitative data was collected using students' reflections and interviews, whereas quantitative data was collected through pretest and posttest using diagnostic tests. The results of the qualitative data analysis revealed that the learning milieu created a positive impact on students' understanding of the concept of limit. Additionally, students provided coherent and viable reasons while making mental constructions and their coordination in the learning process based on the genetic decomposition grounded in APOS theory. Furthermore, the results of the quantitative (posttest) data analysis proved that students' mean scores on conceptual understanding of limit in the experimental group was significantly better than those in the control group. Thus, it could be possible to conclude that students’ conceptual understanding of limit is improved using GeoGebra integrated with multi-teaching approaches within an APOS paradigm. The findings open a great opportunity to suggest technology integrated mathematics curriculums for the teaching and learning of mathematics.

The cognitive process of students understanding quadratic equations

Algebra is mandatory learning in junior high school. That is a continuation of arithmetic with the bridge being variable. Students often have difficulty in understanding it, especially quadratic equations. The purpose of this study is how students’ cognitive processes in understanding the principles of quadratic equations. It was an exploratory study conducted on research subjects. The subjects were junior high school students selected based on their ability to understand concepts and principles about quadratic equations. That’s done through tests about understanding quadratic equations. There were 16 selected research subjects. They were interviewed in-depth by researchers using interview guides. Interview data were analyzed qualitatively based on genetic decomposition. The results of this study are that students can use square, rectangular and unit square cards to determine the principle of factorization of quadratic equations. Students can generalize until completing a perfect quadratic equation. The conclusion is that the cognitive process of students who learn to use learning media based on square and rectangular able to reach a high level (trans-level).