Exploring Pre-service Teachers' Mental Constructions When Solving Problems Involving Infinite Sets
08 Jun 2015-International Journal of Embedded Systems (Kamla Raj Enterprises)-Vol. 09, Iss: 03, pp 273-281
TL;DR: In this article, the authors explore the meaning of the, is equal to symbol by exposing pre-service teachers (n = 38) to two tasks: baseequivalence of two countable infinite sets and comparison of the two numbers 0,9 and 1.
Abstract: KEYWORDS APOS. Functions. Itemized Genetic Decomposition ABSTRACT In this study, the researchers explore the meaning of the, is equal to symbol by exposing pre-service teachers (n = 38) to two tasks. The first task was baseequivalence of two countable infinite sets. The second task focused on the comparison of the two numbers 0,9 and 1.The study was carried out at a South African university, where activity sheets comprising the two tasks were completed by pre-service teachers specializing to become high school mathematics teachers. Their written responses were analyzed against the initial itemized genetic decompositions formulated in accordance with APOS (Action-Process-Object-Schema) Theory. The findings of the exploration led to modified itemized genetic decompositions. These modified genetic decompositions have pedagogical implications. They also contribute to the APOS theory.
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TL;DR: In this article, the authors describe the types of mental structure and mechanism of mathematics students upon the truth-seeking process in solving mathematical problems, and find that all mental structures have been constructed by all research subjects and two types of cognitive mechanisms are evident among the subjects, including the process of interiorization coupled with coordination and another process encompassing interiorization, coordination, and reversal.
Abstract: The Mathematics students who perform truth-seeking process upon solving mathematical problems were unique. Therefore, the study deems it necessary to know students’ mental structure and mechanism so that they can make the right decision by performing truth-seeking. However, no research has delved into the mental structures and mechanisms of Mathematics students, who tend to grapple with truth-seeking processes extensively. This study was explorative qualitative because the aims to describe the types of mental structure and mechanism of Mathematics students upon the truth-seeking process in solving mathematical problems. The research subjects are four Mathematics students at the University of Jember who perform truth-seeking and can communicate fluently when performing think-aloud. Their responses in the answer sheets drove the determination of research subjects' tendency in truth-seeking. Afterward, the results of think-aloud and task-based interview were put under analysis, so as to determine the types of mental structure and mechanism. The research findings have indicated that (1) all mental structures have been constructed by all research subjects and (2) two types of mental mechanism are evident among the subjects, including the process of interiorization coupled with coordination and another process encompassing interiorization, coordination, and reversal.
12 citations
TL;DR: In this article, the authors reveal the nature of mental constructions made by pre-service teachers when learning the determinant concepts and contribute to APOS theory (Action, Process, Object and Schema).
Abstract: This study aims to reveal the nature of mental constructions made by pre-service teachers whenlearning the determinant concepts and to contribute to APOS theory (Action, Process, Object and Schema)...
11 citations
TL;DR: In this article, the authors investigated the processes involved in problem-solving in a mathematics class and found that learners working in groups demonstrated most of the stages of the Polya linear problem solving model.
Abstract: This paper reports on a small-scale action research, which investigated the processes involved in problem-solving in a mathematics class. Grade ten learners (n = 47) at a South African middle school were involved in the study. The participants from two classes attempted the solution of tasks involving the fraction concept. In one class, the learners worked in groups and in the other class, the learners worked individually. A qualitative method was adopted for data capture and analysis. Social constructivism was adopted as a theoretical framework and the stages advocated by Polya were interrogated when analyzing the learner responses on their problem solutions. The results revealed that those learners working in groups demonstrated most of the stages of the Polya linear problem-solving model. The findings helped identify which stages of the model promote effective problemsolving and some recommendations are made for classroom practitioners engaging their learners in problemsolving.
7 citations
TL;DR: In this paper, the impact of e-learning support materials have on a group of engineering students at a South African University of Technology (SAT) has been explored, where the authors designed a basic mathematics pre-test to identify areas of weaknesses and strengths.
Abstract: A great emphasis is placed on improving students' mathematics performance levels in the South African higher education institutions. In the main study, the researchers will develop and explore t he impact e- learning support materials have on a group of engineering students at a South African University of Technology. The researchers designed a basic mathematics pre-test to identify areas of weaknesses and strengths. In this paper (which is part of a broader study), the researchers identify those areas of concern and provide a rationale for e- learning as a sustainable communication tool. The researchers adopted the domains of sustainable learning as a conceptual framework. It was found that the previous study findings, the availability of e-learning resources and the analysis of data all satisfy the demands of the three domains of sustainable learning. With these findings, the researchers deduced that e-learning can be adopted as a sustainable tool to communicate effective ma thematics learning.
3 citations
01 Jan 2020
TL;DR: In this paper, the authors summarize the many research studies in Southern Africa which address the teaching and learning of Mathematics in Higher Education and confine the studies to those which adopt APOS (Actions-Process-Object-Schema) as a theoretical analysis.
Abstract: Presently, a major desire of many African nations is to be technologically developed. In Southern Africa there tends to be an acute shortage of skilled manpower in the field of science, engineering, and technology. The interrelationship between Mathematics and development of humans to advance the cause of humans is a fundamental importance of Mathematics to humans. In this chapter, the author draws on the many empirical studies which address quality in teaching and learning. The focus of the chapter is to summarize the many research studies in Southern Africa which address the teaching and learning of Mathematics in Higher Education. The author shall confine the studies to those which adopt APOS (Actions-Process-Object-Schema) as a theoretical analysis. So far no research seems to have collated such African studies in a cohesive manner and this chapter intends to do that.
3 citations
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Book•
01 Jan 1980TL;DR: In this article, the context of educational research, planning educational research and the styles of education research are discussed, along with strategies and instruments for data collection and research for data analysis.
Abstract: Part One: The Context Of Educational Research Part Two: Planning Educational Research Part Three: Styles Of Educational Research Part Four: Strategies And Instruments For Data Collection And Researching Part Five: Data Analysis
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01 Jan 2015
1,015 citations
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TL;DR: The purpose in this chapter is to propose that the concept of reflective abstraction can be a powerful tool in the study of advanced mathematical thinking and can provide a theoretical basis that supports and contributes to the authors' understanding of what this thinking is and how it can help students develop the ability to engage in it.
Abstract: Our purpose in this chapter is to propose that the concept of reflective abstraction can be a powerful tool in the study of advanced mathematical thinking, that it can provide a theoretical basis that supports and contributes to our understanding of what this thinking is and how we can help students develop the ability to engage in it. To make such a case completely, it would be necessary to do at least several things:
751 citations
Journal Article•
TL;DR: This article found that the majority of students thought that 0.999 was less than one, and that the answers represented their own rationalisations made in an attempt to resolve conflicts inherent in the students' previous experience of limiting processes.
Abstract: The majority of students thought that 0.999 . . . was less than one. It may be that a few students had been taught using infinitesimal concepts, or that the phrase “just less than one” had connotations for the students different from those intended by the questioner; but it seems more likely that the answers represent the students’ own rationalisations made in an attempt to resolve conflicts inherent in the students’ previous experience of limiting processes.
150 citations
TL;DR: In this paper, the authors used the action-process-object-schema (APOS) theoretical framework to investigate university students' understanding of derivatives and their applica-tions and found that those students had difficulty in applying the rules for derivatives and this was possibly the result of many students not having appropriate mental structures at the process, object and schema lev-els.
Abstract: This article reports on a study which used the APOS (action-process-object-schema) theoretical framework to investigate university students’ understanding of derivatives and their applica-tions. Research was done at the Westville Campus of the University of KwaZulu-Natal in South Africa. The relevant rules for finding derivatives and their applications were taught to under-graduate science students. This paper reports on the analysis of students’ responses to six types of questions on derivatives and their applications. The findings of this study suggest that those students had difficulty in applying the rules for derivatives and this was possibly the result of many students not having appropriate mental structures at the process, object and schema lev-els. Keywords : action, APOS, chain rule, derivatives, process, object, schema, teaching
63 citations