Action-Process-Object-Schema Research Articles

Cognitive trajectory of proof by contradiction for transition-to-proof students

History and research on proof by contradiction suggests proof by contradiction is difficult for students in a number of ways. Students’ comprehension of already-written proofs by contradiction is one such aspect that has received relatively little attention. Applying the cognitive lens of Action-Process-Object-Schema (APOS) Theory to proof by contradiction, we constructed and tested a cognitive model that describes how a student might construct the concept ‘proof by contradiction’ in an introduction to proof course. Data for this study was collected from students in a series of five teaching interventions focused on proof by contradiction. This paper will report on two participants as case studies to illustrate that our cognitive trajectory for proof by contradiction is a useful model for describing how students may come to understand the proof method.

Ways secondary mathematics teachers apply definitions in Taxicab geometry for a real-life situation: Midset

Definitions in mathematics are an integral part of understanding concepts and are often not used correctly by students in mathematical proofs and problem-solving situations. Research shows that by observing properties and making conjectures in non-Euclidean geometry, students can better develop their understanding of concepts in Euclidean geometry. For this study, Taxicab geometry (defined by Taxicab distance, or the L1 norm) was introduced to students enrolled in a College Geometry course at a university. Action-Process-Object-Schema (APOS) Theory was used as a guiding framework in the data analysis of responses from eleven secondary mathematics teachers to a real-life problem situated in Taxicab geometry. This report provides illustrations of the conceptual understanding of midset (known as a perpendicular bisector in Euclidean geometry) found among participants and suggestions for teaching material to help facilitate development of a deeper understanding of definitions in geometry.

On students’ understanding of implicit differentiation based on APOS theory

The Action-Process-Object-Schema (APOS) theory is applied to study student understanding of implicit differentiation in the context of functions of one variable. The APOS notions of Schema and schema development in terms of the intra-, inter-, and trans-triad are used to analyze semi-structured interviews with 25 students who had just finished taking a single-variable calculus course. Results suggest that the notions of chain rule and implicit function play a key role in the possibility of attaining implicit differentiation Schema coherence. For this, students need to construct at least a Process conception of implicit function and a chain rule Schema with coherence given by function composition. Students also need to construct relations between implicit function and each one of three components of the implicit differentiation Schema: explicit function, derivative, and differentiation rules. The study shows that students taking an introductory calculus course can be expected to have difficulty understanding the main ideas of implicit differentiation unless special activities are designed to help them make the necessary connections between components of the implicit differentiation Schema. The study suggests the need to further investigate the implementation of activities that foster the constructions proposed, in textbooks and classrooms.

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.

Complexities in university students’ understanding of parametric equations and curves

Despite the importance and many applications that parametric equations and curves have in mathematics and other sciences, there is a lack of research regarding this topic in mathematics education. The goal of this article is to explore university students’ understanding of parametric equations and curves based on two lenses: Action, Process, Object, Schema (APOS) and Onto-Semiotic Approach (OSA). Recent research showed that these two theories complement each other and the combination of them has good potential to analyse students’ performance of mathematical concepts. Three tasks of parametric equations and curves were proposed to characterize the schema development of twenty-seven university students in terms of triad (i.e. intra, inter and trans). Based on the results, most of the students participating in this study had difficulties regarding the practical tasks and developing the mental structures needed to solve parametric equation tasks, especially the mental structures that have to be made to draw parametric curves and identify their directions, and to compute the first and the second derivatives of parametric equations. Finally, we finish the article by making practical suggestions for instructors and researchers.

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.

An APOS Analysis of University Students’ Understanding of Derivatives: A Lesotho Case Study

The derivative is a central concept in calculus and has applications in many disciplines. This study explored students’ understanding of derivatives with a particular focus on the graphical (geometric) representation. The participants were four Mathematics Honours students from a university in Lesotho. Data were generated from the written responses to five tasks and interviews, with all collection methods included. The Action–Process–Object–Schema (APOS) theory was used as the framework of analysis. The findings of the study show that the mental constructions that students developed in learning the derivative concept were mainly actions. The coordinated process conception of the derivative as the limit was evident with respect to algebraic representation but not in the graphical representation. Only one student showed that they had the process conception of the secant lines tending to the tangent line. It was also found that the process of tending to the limit operator was interpreted as tending to and landing on rather than being in the neighbourhood of by some students. These findings suggest that students should be made aware of the interpretation of tending to as implying staying in the neighbourhood rather than landing on. Students should also be given activities based on derivatives that require translation from one mode of representation to another (algebraic to graphical in this case).

An APOS analysis of solving systems of equations using the inverse matrix method

The concept of determinant plays a central role in many linear algebra concepts and is also applied to other branches of mathematics and science. In this study, we focus on the application of determinant and inverse matrix concepts, in solving systems of equations by a group of 116 in-service mathematics teachers who were studying the topic at a Zimbabwean university as part of a first course in linear algebra. The purpose of the study was to explore the role of prerequisite concepts of determinant and matrix inverse in solving systems of equations using the inverse matrix method. The Action-Process-Object-Schema (APOS) theoretical framework was used to analyse the 116 participants’ written responses to a questionnaire together with interview responses of 13 of the participants. Across the group of participants, it was found that only a small number had constructed prerequisite concepts of determinant and that of matrix inverse as Objects. The difficulties they had with prerequisite concepts prevented them from working more efficiently with the higher level concept of solving systems of equations using the inverse matrix method. These results suggest that instructors should help students consolidate their understanding of prerequisite concepts so that they are better placed to work with applications of these concepts.

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.

Learning the concept of eigenvalues and eigenvectors: a comparative analysis of achieved concept construction in linear algebra using APOS theory among students from different educational backgrounds

International comparative studies of education have shown that the increase in mathematical attainment differs significantly among students in relation to the different school types in which their mathematical foundations were acquired. In tertiary education, the same differences are observed with respect to simple tasks that relate to the subject matter of secondary education. Therefore, we investigate the question of whether this observation holds equally for concept construction in the field of eigen theory, by applying Dubinsky’s action, process, object, schema (APOS) theory, and Tall’s worlds of mathematics. We focus on eigen theory because we regard it as particularly representative of linear algebra in terms of its expressibility in the embodied and symbolic worlds. In our empirical study, we investigated concept constructions as correlates of educational trajectories of 36 students who covered the entire course program of a first semester mathematics course for first-year engineering students. Although significant differences in procedural knowledge were observed, results indicate that the type of secondary school does not affect concept construction among students who complete the entire course program. In our final discussion, we present implications for teaching and learning of linear algebra in heterogeneous classes and the integration of deeper learning methods into the course design.

The development of a linear algebra schema: learning as result of the use of a cognitive theory and models

Learning is a complex phenomenon. Analyzing its development can provide knowledge about regularities and differences in students’ progress, which is needed to better understand learning. The aim of this study is to examine the development of students’ Linear Algebra Schema through an introductory one semester course. For that purpose, Action Process Object Schema (APOS) theory’s notions of Schema and Schema development were used to analyze students’ constructions at three different times throughout the course. Results show how students’ Schema are progressively constructed. They illustrate differences and commonalities in different students’ Schema development. Some concepts and factors were found to play an important role in promoting Schema development.

Mental constructions in linear algebra

The aim of this paper is on the one hand to discuss from an APOS (Action–Process–Object–Schema) theory perspective the mental constructions involved in the learning of linear algebra, through examples concerning the linear transformation concept and related notions. On the other hand, methodological issues related to the design of research instruments and implementation of didactic interviews are discussed, supported by empirical data. Detailed analysis of transcripts from an interview with a student focuses on strategies used in interviewing as well as the mental stages involved in the construction of some linear algebra concepts. Due to the strategies employed, during the interview it is possible to witness the transition between different conceptions. A discussion of the relationships and interactions between different mental structures and mechanisms that play a role in the development of knowledge is provided, including theoretical considerations on the matter. Recommendations about pedagogical strategies are included.

Pre-service mathematics teachers’ mental constructions when using Cramer’s rule

This study explored the mental constructions displayed by pre-service mathematics teachers (PMTs) when applying Cramer’s rule. The aim was to reveal the character of mental constructions made around the nature of the solution set of equations and the role of parameters in the solution of equations with parametric coefficients. These mental constructions occur within the Action-Process-Object-Schema (APOS) theory. In this study, data was generated from 31 first year pre-service teachers by means of an activity sheet and interviews. The interviews were used to clarify pre-service teachers’ responses to tasks from the activity sheet. The findings revealed that many PMTs displayed procedural understanding of Cramer’s rule. It was also found that they understood what the solution of system of linear equations means. This meant that they were operating at the action stage, in terms of APOS. Additionally, it revealed that the lack of construction of related schemata negatively impacted the PMTs’ attempt to construct the necessary mental constructions. Therefore, the researchers provided the genetic decomposition (GD) for the use of Cramer’s rule to assist teacher educators to analyse the mental constructions of students. This study explored a new application of APOS theory. Analysing the mental constructions of students by means of research is intended to assist in designing alternative teaching strategies. In this way, this study makes a significant contribution to the solution of system of equations pedagogy. Keywords: APOS; Cramer’s rule; parametric coefficients; pre-service mathematics teachers; system of linear equations

Analysing engineering students’ understanding of integration to propose a genetic decomposition

This paper reports on a study which explored engineering students’ understanding of the techniques of integration in calculus. There were 30 first year engineering students who participated in the project. The concepts were covered as part of a mathematics course at a university of technology in South Africa. Activity sheets, constructed with tasks based on integration were administered to the participants. Their written responses, which were used to identify the mental constructions of these concepts, were analysed using APOS (Action-Process-Object-Schema) theory and interviews were carried out to clarify the written responses. The discussions and written work indicated that students exhibit procedural tendencies in integration and that students could not define both definite and indefinite integrals. These findings raised some didactical implications for higher education and also provided applications of genetic decomposition design and modification.

Using APOS theory as a framework for considering slope understanding

In this paper a framework for slope is proposed using APOS (Action-Process-Object-Schema) Theory and conceptualizations of slope previously identified in research. The proposed APOS-slope framework allows for discussion of students’ cognitive development in relation to different conceptualizations of slope. As such, it may be adopted as a means to advance future research or as a way to plan instruction. In particular, the framework uses specific examples to consider interrelations between the ways of thinking about slope that have been reported to provide additional insight on how individuals understand this concept. The proposed framework contributes to the field by bringing together a number of past studies related to slope and providing a common ground under which these works might be interpreted.

An Exploratory Study on the Understanding of the Vector Subspace Concept

This exploratory study investigated 10 pre-service mathematics students’ understanding of vector subspaces at a university in Zimbabwe. The study involved an in-depth document analysis of the students’ written responses to five items based on the vector subspace concept. An APOS (Action–Process–Object–Schema) approach was used to identify the difficulties associated with developing the mental constructions of the vector subspace concept. The responses revealed that most of these pre-service teachers used reasoning associated with Action, or not even Action, conceptions of the prerequisite concepts of binary operations and sets, which hampered their engagement with the higher-level concepts of checking axioms. It is recommended that students should work with different sets and different types of binary operations, allowing them to routinise the Action of carrying out the binary operations on the sets first before moving to abstract applications. The data also revealed that students had difficulties in making links within the embodied, symbolic and formal worlds of thinking. It is suggested that instructors ensure that these links form an explicit focus of tasks.

Pre-service primary Mathematics teachers’ understanding of fractions: An action–process–object–schema perspective

Background: The concept of rational numbers is one of the learner’s first experiences with a Mathematics concept beyond the basic skills operations on whole numbers. The personal knowledge of fractions that teachers bring to the teaching context is important because teachers mediate the conceptions that their learners construct. Aim: This study was set up to apply Action–Process–Object–Schema theory to study primary teachers’ understanding of addition and subtraction of fractions. Setting: The participants of this study comprised 60 undergraduate full-time students, studying to become teachers. The participants were enrolled in a foundational course in Mathematics because they had not passed Mathematics at Grade 12 level. This course was intended to help deepen their understanding of basic numeracy, allowing them to continue with further courses if they wanted to specialise in teaching primary mathematics Methods: Data were collected using written responses of the pre-service students to two tasks that focused on operations with fractions. Ten students volunteered to be interviewed of which three are drawn upon in this article. Results: Many of the pre-service teachers coped well with addition and subtraction of common fractions with the same denominator. However, more than 52% struggled to carry out these operations on common fractions with different denominators, showing that their conceptions had not developed into object-level structures. Conclusion: It is evident that the incorrect procedures have become embedded in the students mental schema. It is crucial that programmes for upgrading pre-service teachers should include opportunities for teachers to interrogate their personal understandings of the basic mathematics concepts.

The Analysis of Pre-service Math Teachers’ Level of Understanding the Derivative Concept within the Context of APOS Theory

APOS (Action-Process-Object-Schema) learning theory is the result of the studies of a mathematics education research group named Research in Undergraduate Mathematics Education Community (RUMEC), whose aim is to examine students’ level of comprehending university mathematics subjects. This study aimed to investigate how secondary mathematics education pre-service teachers in a public university structured the geometric dimension of the concept of derivative in their minds in the context of the components of the APOS theory. As data collection tools, questions developed by Çekmez [1] based on the genetic decomposition of Asiala, Dubinsky, Cottrill and Schwingendorf [2] for the geometric dimension of derivatives were used. The study revealed that secondary mathematics education pre-service teachers did not have the mental structures related to this topic and could not learn it at the desired level.

Calculus students’ understanding of the vertex of the quadratic function in relation to the concept of derivative

ABSTRACTThe purpose of this study was to gain insight into 30, first year calculus students’ understanding of the relationship between the concept of vertex of a quadratic function and the concept of the derivative. APOS (action-process-object-schema) theory was applied as a guiding framework of analysis on student written work, think-aloud and follow up group interviews. Students’ personal meanings of the vertex, including misconceptions, were explored, along with students’ understanding to solve problems pertaining to the derivative of a quadratic function. Results give evidence of students’ weak schema of the vertex, lack of connection between different problem types and the importance of linguistics in relation to levels of APOS theory. A preliminary genetic decomposition was developed based on the results. Future research is suggested as a continuation to improve student understanding of the relationship between the vertex of quadratic functions and the derivative.

Student Understanding of the Relation between Tangent Plane and the Total Differential of two-Variable Functions

Action-Process-Object-Schema (APOS) theory and tools resulting from dialogue with the Anthropological Theory of the Didactic (ATD) were used to analyse data from semi-structured interviews and teaching materials to study students’ understanding of the relationship between tangent planes and the total differential. Results of the study show students’ difficulties relating these ideas and suggest a refinement of the initial genetic decomposition. They also underline aspects of the teaching materials that need to be considered to promote those constructions and development of a complete praxeology for the total differential. This study exemplifies how the dialogue between a cognitive theory and one that focuses on institutional aspects of mathematics education, can provide tools to deeply analyse the teaching and learning of a mathematical topic.