Action-Process-Object-Schema Research Articles

Acting on Totalities of infinite Processes: constructing facets of an Object conception

AbstractWe offer a new angle to explain the comprehension of mathematical infinity. We use Action—Process—Object—Schema (APOS) theory to explore the construction of different aspects of the cognitive Object related to this notion. We focus our attention on different ways of interacting with infinity, and how to act on infinite entities. Our aim is to shed light on the nature of mental constructions that take place when individuals deal with mathematical infinity. An infinite union of finite subsets of $${\mathbb{N}}$$ N provides the context for our theoretical analysis and the design of interviews conducted with graduate students and instructors. In this study we identify three types of Actions with different complexity levels. When these Actions are performed, they lead to the construction of different facets of the associated Object. We question the construction of an Object conception in infinity-related situations and offer an explanation that might change the way we think about the learning of certain mathematical concepts. We discuss the implications of our research for the advancement of APOS theory. We also offer pedagogical suggestions related to the comprehension of mathematical infinity.

Contributions to the characterization of the Schema using APOS theory: Graphing with derivative

This study contributes to Action, Process, Object, Schema (APOS) theory research by showing two approaches used by advanced mathematics students to construct relations between higher-order derivatives to solve complex problems. We show evidence of students’ ability to perform Actions on their graphing derivative Schema, that is, of its thematization. It also contributes to the literature on the learning of differential calculus by showing how advanced students use their knowledge to construct relations between concepts when facing complex situations. The work of three graduate students on transforming complex graphs and determining their properties and their relation to the domain structure is analyzed to determine their solution approaches. Their graphing derivative Schema is analyzed in depth in terms of the construction of relations among the Schema structures and assimilation and accommodation mechanisms involved in thematization in APOS theory. These findings are important in informing and developing didactic strategies to foster university students’ understanding of derivatives, which can smoothe the transition to the study of advanced mathematics courses.

Learning Eigenvalues and Eigenvectors with Online YouTube Resources: A Journey in the Embodied, Proceptual-symbolic, and Formal Worlds of Mathematics

Linear algebra is one of the service mathematics courses that students in many programs take as part of their undergraduate study. Previous studies indicate that linear algebra courses are often challenging for undergraduate students, leading some to seek help from other resources, including online resources such as YouTube. Through a multiple case study, I focused on identifying the opportunities that YouTube resources could offer students to learn about eigenvalues and eigenvectors, which are among the key concepts in introductory linear algebra. I utilized a combination of APOS (Action-Process-Object-Schema) and Tall’s three worlds of mathematics to analyze two highly viewed YouTube videos on this topic. The two analyzed files had different emphases (one more on the embodied world, and the other on the proceptual-symbolic world), catering to students with different levels of understanding of linear algebra. The findings suggest that, overall, these resources could provide students with opportunities to learn about eigen theory in the embodied and proceptual-symbolic worlds, with more focus on action. Lecturers and teacher assistants could benefit from familiarizing themselves with these available resources and considering the possible integration of these resources into their teaching and/or support they provide for students to meet their academic needs and preferences.

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.

Student-Teacher Level of Statistical Reasoning Ability Reviewed by Apos Theory (Action, Process, Object, Schema)

In learning statistics, there is the term statistical reasoning. Statistical reasoning arises because of the opinion that learning statistics using a traditional approach does not lead students to have statistical reasoning abilities or to think statistically. There are four statistical reasoning levels: idiosyncratic, transitional, quantitative, and analytical. Student-teachers must understand fundamental statistical concepts more deeply to develop statistical reasoning abilities. The mental construction of a statistical concept can be analyzed using APOS (Action, Process, Object, Schema) theory. Through the APOS theory, researchers can determine individual understanding of statistical concepts that cause individuals to reason or draw reasonable conclusions based on the statistical information obtained. This study aims to analyze the interrelationships and make conjectures regarding the level of statistical reasoning ability based on the APOS theory. The research subjects are. 19 informants. The research results obtained four conjectures, namely. Students at the idiosyncratic level are in the action stage; students at the transitional stage are in the action, process, or object stage; students at the quantitative stage are in the process or object stage; lastly, students at the analytical are in the schema stage.

Developing a Framework for Evaluating Student's Understanding at Figural Pattern Generalization

Figural patterns have a unique capacity to promote functional thinking. This study aimed to identify the mental constructs of 7th-grade students in Figural Pattern Generalization (FPG) by using the Action, Process, Object, Schema (APOS) theory in order to develop a framework for evaluating students' understanding of FPG. A sample of 220 students completed a test designed based on the APOS framework and 19 students participated in a semi-structured interview. Results showed that there are emergent and partial action levels before the action stage and pre-emergent, and partial process/object levels before the process/object stage.

Changes in students’ mental constructions of function transformations through the APOS framework

This study aims to examine the changes in students’ mental constructions of function transformation based on the designed activities, classroom discussions, exercises (ACE) teaching cycle in the light of the action, process, object, schema (APOS) theoretical framework. The study was conducted by seven volunteer students who were enrolled in mathematics teacher education program. The data consisted of video records of the designed teaching sessions, students’ worksheets and the researcher’s field notes. The results indicated that ACE teaching cycle was effective in developing students’ mental constructions of function transformation through sequential activities. Even though all the students initially did not reveal evidence for sufficient theoretical or practical knowledge on function transformation, which was evidence of their action level, through the process, it became evident that all the students provided evidence regarding at least their process level of function transformations. Some of the students also showed evidence related to their object levels of function transformations.

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.

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.

The spatial thinking process of the field-dependent students in reconstructing the geometrical concept

<p><span lang="EN-US">Reconstructing geometrical concepts requires a spatial thinking process, so the spatial thinking process will be correct and complete. The phenomena of cognitive style differences cause different perceptions and thinking activities to solve geometric problems. This qualitative-explorative research describes the spatial thinking process of students with field-dependent cognitive styles in reconstructing the concept of spatial geometry based on the theory of Action-Process-Object-Schema (APOS). The research subjects were 27 students and obtained five students with field-dependent cognitive styles. The researchers used a purposive sampling technique from the subjects with a certain consideration. The researchers selected a student that met the three elements of spatial thinking and the five indicators of spatial ability. This research collected the data with interviews, documentation, and group embedded figure test (GEFT). The analyzing techniques used data collection, data reduction, data presentation, and concluding. The spatial thinking process of the field-dependent students had a spatial category with three indications: i) Inaccuracy in the elements of representational thinking; ii) The inaccuracy of spatial perception indicators; and iii) Not using de-encapsulation mental mechanisms.</span></p>

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.

What’s new with APOS theory? A look into levels and Totality

This paper focusses on developments concerning transitional aspects of learning from the perspective of APOS (Action—Process—Object—Schema) theory. Recent investigations about levels between stages and Totality as a possible new structure are commented on, as well as offering related pedagogical suggestions and ideas for future research.

Student understanding of functions of two variables: A reproducibility study

In this Action-Process-Object-Schema (APOS) study, we aim to study the extent to which results from our previous research on student understanding of two-variable functions can be replicated in a different institutional context. We conducted the experience at a university in another country and with a different instructor than in previous studies. The experience consisted in comparing two sections of the same course; one taught through lectures and the other using activities designed with APOS theory and the ACE didactical strategy (Activities, Class discussion, Exercises). We show the results of a comparison of students’ performance in both groups and give evidence of the generalizability of our previously obtained research results and the possible replication of didactic aspects across institutions. We found that using the APOS theory didactical approach favors a deeper understanding of two-variable functions. We also found that factors other than the activity sets and teaching strategy affected the replication.

Critical thinking skills: Error identifications on students’ with APOS theory

<span>Identifying students' errors in solving cases of critical thinking skills from two variables of linear equation (TVLE). This was a qualitative study using a descriptive exploratory approach. The participants of the study were first-year students of mathematics education. The method used in this research is a test, interview, and triangulation. The findings showed that the students have low critical thinking skills; therefore, they could not complete the task correctly. Based on the Action-Process-Object-Schema (APOS) theory, students' mistakes in completing math problems consist of four elements, namely: i) The errors in interpreting; ii) The errors in understanding the concept; iii) The error in the procedures; and iv) The error in technical things. The student's response in this study was in the theoretical of APOS, so that they could not reach a correct schema. The study results are expected to be beneficial in developing the activities in teaching TVLE, so that the students will make less errors in completing critical thinking skills problems in mathematics. Therefore, further study in developing a teaching model for mathematics teaching to improve students' critical thinking skills is highly recommended.</span>

The Spatial Thinking Process of the Field-Independent Students based on Action-Process-Object-Schema Theory

<p style="text-align:justify">Spatial thinking has roles to facilitate learners to remember, understand, reason, and communicate objects and the connections among objects that are represented in space. This research aims to analyze the spatial thinking process of students in constructing new knowledge seen from the field-independent cognitive style learners based on Action-Process-Object-Schema (APOS) theory. APOS theory is used to explore spatial thinking processes which consist of mental structures of action, process, object, and schema. This research is qualitative research with an exploratory method. It provided the students' opportunity to solve problems alternately until the method found the most appropriate subjects for the research objectives. The subjects were 2 students of Mathematics Education in the fourth semester of Universitas Muria Kudus Indonesia. The data collection techniques were started by distributing the validated and reliable spatial thinking questions, the cognitive style question, and the interview. The applied data analysis consisted of data reduction, presentation, and conclusion. The findings showed (1) spatial thinking process of holistic-external representation typed learners were indicated by the representative thinking element, abstract-illustrative figure expression to communicate and complete the tasks correctly, (2) spatial thinking process of the holistic-internal representation typed learners were indicated by the representative means, having ideas, connecting with the previous knowledge in the forms of symbols and numbers, and finding the final results correctly although incomplete.</p>

Transitional points in constructing the preimage concept in linear algebra

From an APOS (Action–Process–Object–Schema) theory perspective, learning mathematics involves construction of knowledge through mental mechanisms, which evolves between different mental structures or stages. The focus of this study is to explore how transition occurs from an Action to a Process conception, in the context of a task related to the learning of the concept of linear transformation in general, and the notion of preimage in particular. A questionnaire was designed and applied to 31 students from three different higher education institutions in Mexico, who were enrolled in an introductory linear algebra course. For the first time, transitional points known as levels are explicitly identified in the aforementioned context and empirical evidence is presented. Some difficulties resulting from the intervention of constructions related to the function concept are also pointed out. Discussion about the characteristics of levels in the APOS framework provides a theoretical contribution to the field.

DEVELOPMENT OF MATHEMATICS TEACHINGS BASED ON APOS THEORY: CONSTRUCTION OF UNDERSTANDING THE CONCEPT OF STUDENT STRAIGHT LINE EQUATION

This research is an ADDIE-based research and development. Data on the development process were collected using tools validated by materiel experts, design experts, student response questionnaires and teacher responses. The results of the implementation of the use of textbooks are seen from the student activity sheets and student learning outcomes, in this study using participants from class VIII students of MTs YPI Kuala Enok totaling 28 students with a random sampling technique. The results of this development research are: (1) SMP / MTs mathematics textbooks based on the APOS theory on straight line equation material are declared suitable for use in learning activities based on the validation results of material experts and design experts. Based on the results of student and teacher responses to textbooks, they are ranked very well. (2) The effectiveness of using SMP/MTs mathematics textbooks based on the APOS (Action, Process, Object, Schema) theory of straight line equation material seen from the student activity sheet can be concluded that in the process of learning student activities. In addition, on the basis of the results of all students, the criteria for completeness were achieved, an average score of 85. Thus, from the results of this analysis, it can be concluded that textbooks based on the APOS theory are effective in constructing an understanding of the concept of straight line equations.