Task-based Interviews Research Articles

The abstraction ability of students in understanding the concept of geometry

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

Quantitative Reasoning Process in Mathematics Problem Solving: A Case on Covariation Problems Reviewed from APOS Theory

The focus of this study was to describe the students' quantitative reasoning process in solving covariate problems using APOS theory. This research was a case study conducted on 9th grade students of junior high schools in five different schools in Bima Regency, Indonesia. The data collection in this study was initially done by providing Covariation Problem Task (CPT) for students in a 60-minute task. Afterwards, it was followed up by holding task-based interviews with students who performed different processes. These interviews were intended to confirm the existing mental process. The data obtained were analyzed by combining retrospective and continuous analysis, namely by analyzing work results and student interviews simultaneously, then thinking structures were described based on APOS theory. The results of this study showed two different students' quantitative reasoning processes for covariation problems. First, the inductive quantitative reasoning process (IQRP) was conducted by linking the quantity involved in covariation problems and substituting it into the formula directly, without making any modeling. Second, the deductive quantitative reasoning process (DQRP) was conducted by connecting quantity through the process of mathematical modeling by producing formula equations. This study indicated that the students comprehended the situation caused by covariate quantity relationships and the process (reasoning) of how quantity values change together.

Fostering students geographic systems thinking by enriching causal diagrams with scale. Results of an intervention study

Several challenges faced by humanity, such as the world food problem, migration flows or climate change, are all taught about in geography courses worldwide. Behind these challenges are complex systems consisting of several interrelated causes and consequences. In order to better understand these highly complex geographical systems, a systems thinking approach is helpful. However, the spatial embeddedness of relations in a system is often not included in this approach. Although this spatial embeddedness is important to understand geographical systems, it also contributes to the complexity of systems. The aim of this study is to examine whether the enrichment of causal diagrams with spatial scales helps students to better understand these systems. A quasi experimental design with an intervention of four lessons and task-based think-aloud interviews is used. These results show that students who have used these enriched diagrams in class are more aware of the spatial embeddedness of relations and therefore achieve a higher level of geographical thinking. This study shows the potential of enriched causal diagrams to make them actively use this spatial component while processing information.

Calculus students’ quantitative reasoning in the context of solving related rates of change problems

ABSTRACTDespite the increasing amount of research on students’ quantitative reasoning at the secondary level, research on students’ quantitative reasoning at the undergraduate level is scarce. The present study used task-based interviews to examine 16 high-performing undergraduate calculus students’ quantitative reasoning in the context of solving three related rates of change problems-two geometric and one non-geometric. A qualitative analysis of verbal responses and work written by the students when solving the three problems revealed that 15 students created and used diagrams to support their quantitative reasoning in the geometric problems, and that the creation of these diagrams helped the students to solve the two problems. In addition, seven students exhibited lack of facility with the product or quotient rule of differentiation. We argue that the use of diagrams in calculus instruction at the undergraduate level could play an important role in supporting students’ quantitative reasoning in topics that require students to make sense of relationships between quantities such as related rates of change problems. Furthermore, we argue that helping students develop greater facility with rules of differentiation such as the product or quotient rule could increase students’ success in many calculus application problems, including related rates of change problems. Directions for future research are included.

Students’ Analytical Thinking Profile Based on Reflective Cognitive Style in Solving Mathematics Problem

The objective of this study is to describe students’ analytical thinking profile based on reflective cognitive style in solving a mathematics problem, especially in two-dimension materials. This research is a qualitative descriptive study that elaborates the characteristics of analytical thinking including; pre-analytical, analytical-partial, semi-analytical, and complete analytics. Data collection is employed through task-based interviews and data validation is done by time triangulation. The participants of this study are two students at the eleventh grade of State Senior High School (SMAN) 1 Kedungwaru, Tulungagung. They are selected based on the results in answering matching familiar figures test. The results reveal that the subjects have partial analytical thinking characteristics in solving the problem.

Cognitive processes of high intelligence student with autism spectrum disorder in understanding mathematical problems

The purpose of this study is to describe the cognition processes of an ASD student with a high intelligence quotient (IQ) in understanding mathematical problems. The processes of cognition being studied are the process of receiving, storing, recalling and processing information. The subject of this research is a high IQ autism spectrum disorder (ASD) student. Data was gathered from task-based interviews. The results of this research indicate that in the process of receiving information, the ASD characteristics of the subject were still very visible, such as reading with a flat expression, interspersed laughing and smiling to themselves, repeating certain words and muttering speaking an unclear sentence. In the process of storing and recalling information, the subject: (1) stored and recalled information in the problem in three visual representation forms: images, verbal language and arithmetic symbols, (2) stored and recalled important information, and (3) recalled information that has been stored in his long-term memory by linking it to the concepts in the problems. In processing information, the subject: (1) mentioned the information and the question in the problems, (3) understood the meaning of the symbol in the problems, (4) understood the adequacy of the information in the problems, and (5) was able to reduce the information in the problems.

Intuitive acceptance of proof by contradiction

The formal acceptance of a mathematical proof is based on its logical correctness but, from a cognitive point of view, this form of acceptance is not always naturally associated with the feeling that the proof has necessarily proved the statement. This is the case, in particular, for proof by contradiction in geometry, which can be linked to a loss of evidence in various ways, owing to its particular logical structure and to the difficulty in managing geometrical figures with contradictory properties. In this paper, we observe that students produce argumentation by starting with the assumption that the claim is false (indirect argumentation), and that they seem to accept this as more evident than the proofs by contradiction. On the basis of the notion of intuitive knowledge developed by Fischbein and through the analysis of task-based interviews, we investigate the intuitive acceptance of proof by contradiction and of indirect argumentation, underlining, in particular, that indirect argumentation can be produced as a compromise between a proof by contradiction and the need for a more evident argument.

Students’ Understanding of Institutional Practices: The Missing Dimension in Human Rights Education

This study used task-based group interviews with young adolescents in four countries to investigate their understanding of the causes of human rights violations, means for protecting human rights, and their own potential role in ensuring human rights. Although students recognized the role of personal and institutional factors in both violating and protecting human rights, their ideas for influencing human rights focused primarily on the personal contexts with which they were most familiar. Their understanding of political and economic mechanisms was much less elaborate. These findings suggest the need for curricula that equip students with the complex and specialized knowledge that would enable them to engage in a range of meaningful civic action, both in their lives now and as adults.

Children';s Geometric Understanding through Digital Activities: The Case of Basic Geometric Shapes

Early mathematics education bases a foundation of academic success in mathematics for higher grades. Studies show that introducing mathematical contents in preschool level is a strong predictor of success in mathematics for children during their progress in other school levels. Digital technologies can support children’s learning mathematical concepts by means of the exploration and the manipulation of concrete representations. Therefore, digital activities provide opportunities for children to engage with experimental mathematics. In this study, the effects of digital learning tools on learning about geometric shapes in early childhood education were investigated. Hence, this study aimed to investigate children progresses on digital learning activities in terms of recognition and discrimination of basic geometric shapes. Participants of the study were six children from a kindergarten in Kırşehir, Turkey. Six digital learning activities were engaged by children with tablets about four weeks in learning settings. Task-based interview sessions were handled in this study. Results of this study show that these series of activities helped children to achieve higher cognitive levels. They improved their understanding through digital activities.

Mathematics education for children under four years of age: a systematic review of the literature

ABSTRACT There is growing international research evidence about young children’s engagement with mathematics education. However, much of this evidence is drawn from children aged four years and upwards. This paper reports findings from a systematic review of peer-reviewed research concerning mathematics education for children aged under four. The majority of the studies took place within an early childhood education service such as a preschool or day care centre, and task-based interviews and standardised assessments were most commonly used. Most of the papers focused on either educators’ knowledge, attitudes and strategies, or children’s mathematical competencies. The literature presents compelling evidence that children do engage in mathematics education prior to four years of age, and that they possess many mathematical competencies. Findings suggest that educators play a critical role in shaping the mathematical learning opportunities available to children; however, there is some uncertainty among educators about how to support young children’s mathematics learning.

Making connections among representations of eigenvector: what sort of a beast is it?

Many studies provide insights into students’ conceptions of various linear algebra topics and difficulties they face with multiple modes of thinking needed for conceptualization. While it is important to understand students’ initial conceptions, students’ transfer of learning of these conceptions to subsequent courses can provide additional information to structure meaningful curricular materials. This study explores physics students’ transfer of learning of eigenvalues and eigenvectors from prerequisite experiences to quantum mechanics. Data analysis focused on three task-based interviews with undergraduate students, observations of physics courses, and students’ course artifacts. Existing studies on students’ conceptions of linear algebra topics indicate the necessity of developing flexible shifts between different modes of thinking in order to grasp linear algebra. This study’s participants, who had initial learning experiences of linear algebra, were also observed to struggle with such shifts prior to quantum courses. It seems that various contexts in quantum courses, and explicit instructional methods, provided opportunities for students to enhance this initial learning of eigenvalues and eigenvectors. In particular, the explicit reasoning of one of the quantum courses’ instructors concerning the choice of certain representations during problem solving in class, seemed to facilitate students’ construction of similarities, thus providing evidence for actor-oriented transfer. Results of this study align with goals for recently developed instructional materials and interventions that emphasize opportunities for students to inquire and connect multiple modes of thinking.

Area measurement: structuring with nonsquare units

ABSTRACTIn this article, we describe how students structured 2-dimensional space with square and nonsquare units. We employed a cross-sectional design, interviewing 5 students from each of 4 different grade groups: Grades 1, 3, 5, and 7 (ages 7, 9, 11, and 13) in structured, task-based interviews. Our findings about students’ ways of measuring area fit a hypothetical route of learning, validating 4 specific levels of thinking about area units across tasks, regardless of unit shape and dimension indicators. Implications for instruction and research are discussed.

Students’ Reasoning Processes While Constructing Causal Diagrams

The use of causal diagrams to externalize the mental representation of a problem is recognized to be an important step in solving complex problems. In geography education several global challenges taught about in class are highly complex due to the interconnectedness of many causes and consequences. A systems thinking approach might be helpful to better understand these global challenges. Former studies have shown the effectiveness of concept maps and causal diagrams to foster students’ systems thinking. However, it is not always obvious for students to construct proper causal diagrams. In order to optimize teaching strategies concerning these complex systems in geography education, this study analyzes students’ cognitive strategies while constructing a causal diagram. We used task-based think-aloud interviews to study their cognitive strategies. Four different cognitive strategies were observed. The different types of cognitive strategies all resulted in an acceptable constructed causal diagram by the students. The presented insights are explorative, but it reveals the thinking processes that are mostly tacit and therefore has the potential to contribute to better teaching strategies. After all, if we know what processes novices go through while carrying out a complex skill, which are often taken for granted by experts, in this case geography teachers, we can raise awareness among teachers to explicitly take those processes into account while designing lessons.

Microgenetic analysis of young children\u2019s shifts of attention in arithmetic tasks: underlying dynamics of change in phases of seemingly stable task performance

A key aim of mathematics teaching is for children to develop appropriate and efficient strategies for solving tasks. The analysis presented in this paper moves beyond the exploration of changes in the strategies that children employ to solve tasks and extends to observation and exploration of changes that occur when their overall solving approach remains seemingly stable. We present an analysis of data from two qualitative studies, each of which combined a microgenetic design with task-based interviews, to examine changes that occur in 5–6-year-old children’s verbal reports when solving an additive task, and in 9–10-year-old children’s verbal reports when solving fraction word problems. Children’s verbal reports were analysed through the lens of the theory of shifts of attention. We found that phases of stability are underlain by dynamic changes in how the same strategy is communicated and conceptualised over a number of sessions and these changes appear to be accounted for by changes related to shifts in the object and structure of children’s attention, i.e. what children attend to and how, when reporting on their solving approach. The paper extends the theory by revealing and studying microqualities that underlie different learners’ structure of attention during phases of stability in arithmetic tasks. The findings provide new, significant insights for understanding qualitative dynamics of change in learning. Sensitivity to differences and changes in learners’ shifts of attention is essential for teachers to make sense of what learners experience and identify opportunities conducive to further learning.

Cognitive Level Profile in Solving Mathematics Problem at Ten Grade of Senior High School Students with Low Ability

This research is motivated by the importance of problem solving skill for students in 21st century, whereas students' skill in solving mathematics problems is various in accordance their cognitive levels. Students’ cognitive levels in solving problems include; cognition; metacognition; and epistemic cognition. Cognitive level affects individual in understanding problem and deciding the right strategy to solve it. The purpose of this study is to describe cognitive levels of low-ability students. This study uses qualitative methods with task-based interviews. The material is three-variable linear equation system. The research subjects are two low-ability students at ten grade of State Senior High School in Pati Regency. The results show that the two subjects have weaknesses at each level, for example the two subjects are able to work on the given problems, but both subjects are unable to define equations, inequality, similarity, and dissimilarity correctly in the initial type of cognition level.

Profiles quantitative reasoning and students’ generalization ability on topic of direct proportion

This research was qualitative that aims to described the quantitative reasoning and students’ generalization ability in direct proportion. This research involved 35 grade VIII junior high schools; one of them with moderate math skills was chosen as the main subject. The data collected through the test, assignment, and task-based interviews. The results of this research showed that a generalization that was built by the subject begins with forming a relationship between two or more quantities by establishing ratio of the quantity which appears linking the two initial quantity, doing repetition to find the elements of common ground and further developed the pattern or relationship into a more general form of structure. The quantitative reasoning which belonged to the subject greatly affects the generalizations made. The results of this research are important as an input material for teachers, to develop a strong generalization in General, and then the student must construct the ratio of the quantity that appears with the linking between the quantities. In order to support the students’ ability to perform a generalizing correctly.

An onto-semiotic approach: Analyzing of field-independent and field dependent students’ understanding in solving statistical problems

This article describes to analyzing of field-independent and field dependent students’ understanding in solving statistical problems used to onto-semiotic approach. The subject matter of field dependent and field independent cognitive styles. Explorative descriptive research was undertaken involving 30 senior high school students. Data was obtained through task-based interviews. The result of this study indicates that there is a difference of students’ understanding between field dependent and field independent students’ in solving mathematical problems when using onto-semiotic approaches based on language, concepts and procedures, while based on computational aspect, proposition and argument. Field independent students in solving statistical problems tend to use mathematical terms and symbols consistently, more analytically and unaffected by the given problem, and using correct settlement measures. Field dependent students’ tend to use maturing terms and symbols of math, tend not to be able to explain the related elements of concepts used in solving statistical problems and tend to have some correct steps of resolution. Thus the onto-semiotic approach is one of the innovations in learning mathematics to overcome student difficulties in solving statistical problems.

Translation process of mathematics representation: From graphics to symbols and vice versa

The objective of this research is to examine the process of translation of mathematical representation from symbols to graphics and vice versa. Translation of representation refers to a basic skill to develop a concept, understanding, and to promote students’ mathematical thinking. This qualitative descriptive research employed task-assigning and task-based interview method. It is revealed that the students can perform every process of the translation of representation from symbol to graphic and vice versa. Translation among mathematical representations comprises four steps, i.e. unpacking the source, preliminary coordination, constructing the target, and determining equivalence. The process of translation of mathematical representations between graphics and symbols can be done without involving another form of representations. The students follow each process that is relatively similar, even though the construction of the translation output is somewhat different.

KEMAMPUAN MAHASISWA PGMI DALAM MEMECAHKAN MASALAH GEOMETRI DITINJAU DARI PERBEDAAN KEMAMPUAN MATEMATIKA

This research is an explorative study using qualitative approach that aims to describe the ability of students in solving geometry problems in terms of differences in mathematical skills. Students' problem-solving capabilities were reviewed based on Polya's troubleshooting techniques, namely: understanding the problem, devising a plan, carrying out the plan, and looking back to review the results obtained. The subjects of the study are three students from Department of Primary School Teacher Education (PGMI) of Faculty of Tarbiyah and Teacher Training, Universitas Islam Negeri Ar-Raniry Banda Aceh. The research process begins by giving a mathematics competency test to the student from a class unit, followed by selecting one student from each group, respectively from high math skills, medium math skills, and low math skills. These three selected subjects were assigned geometry-solving tasks and task-based interviews. The validity of data were checked using time triangulation. The results showed that the subject of high mathematical skill was able to use all 4 steps to solve the problem and provide correct answer. The subject of medium mathematical skill was capable of using the 4 steps of Polya's technique but also was less accurate in giving answers. Subjects of low math skill are able to use only 3 steps but are not correct in giving answers.

Investigating secondary students’ generalization, graphing, and construction of figural patterns for making sense of quadratic functions

An important aim in school mathematics is to help students experience algebra’s power to express generality. Researchers have been investigating figural growing pattern generalization as an early route to developing students’ understanding of functional relationships. More recently studies have been considering ‘open’ or ‘free’ tasks involving figural growing pattern construction. This study explored twelve Years 7–12 Australian students’ intuitions and connection of meanings for quadratic functions through growing pattern generalization, graphing, and construction activities during individual task-based interviews. The students who evidenced complementary correspondence and covariation views when generalizing appeared to handle pattern construction activities in productive ways, regardless of their prior formal study of quadratic functions. Areas of difficulty with graphing growing patterns and potential benefits of pattern construction tasks for exploring quadratic equations and for formative assessment are discussed.