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Difficulties in solving context-based PISA mathematics tasks: An analysis of students’ errors
The intention of this study was to clarify students’ difficulties in solving context-based mathematics tasks as used in the Programme for International Student Assessment (PISA). The study was carried out with 362 Indonesian ninth- and tenth-grade students. In the study we used 34 released PISA mathematics tasks including three task types: reproduction, connection, and reflection. Students’ difficulties were identified by using Newman’s error categories, which were connected to the modeling process described by Blum and Leiss and to the PISA stages of mathematization, including (1) comprehending a task, (2) transforming the task into a mathematical problem, (3) processing mathematical procedures, and (4) interpreting or encoding the solution in terms of the real situation. Our data analysis revealed that students made most mistakes in the first two stages of the solution process. Out of the total amount of errors 38% of them has to do with understanding the meaning of the context-based tasks. These comprehension errors particularly include the selection of relevant information. In transforming a context-based task into a mathematical problem 42% of the errors were made. Less errors were made in mathematical processing and encoding the answers. These types of errors formed respectively 17% and 3% of the total amount of errors. Our study also revealed a significant relation between the error types and the task types. In reproduction tasks, mostly comprehension errors (37%) and transformation errors (34%) were made. Also in connection tasks students made mostly comprehension errors (41%) and transformation errors (43%). However, in reflection tasks mostly transformation errors (66%) were made. Furthermore, we also found a relation between error types and student performance levels. Low performing students made a higher number of comprehension and transformation errors than high performing students. This finding indicates that low performing students might already get stuck in the early stages of the modeling process and are unable to arrive in the stage of carrying out mathematical procedures when solving a context-based task.
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In this study, we investigated teachers’ teaching practices and their underlying beliefs regarding context-based tasks to find a possible explanation for students’ difficulties with these tasks. The research started by surveying 27 Junior High School teachers from seven schools in Indonesia through a written questionnaire. Then, to further examine teachers’ teaching practices related to context-based tasks, four teachers were observed and video recorded in two mathematics lessons in which they were asked to deal with context-based tasks. The questionnaire data revealed that the teachers had a tendency toward a view on teaching and learning mathematics which includes encouraging students to be actively involved in solving problems in various contexts. Although this finding suggests that the teachers may offer opportunities to learn context-based tasks to students, the questionnaire data also revealed that the teachers saw context-based tasks as plain word problems. Furthermore, the observations disclosed that their teaching was mainly teacher-centered and directive, which is not considered to be supportive for learning to solve context-based tasks. Combining the findings of this study with the results from our earlier study on Indonesian students’ errors when solving context-based tasks, we found a relationship between how Indonesian teachers teach context-based tasks and the errors Indonesian students make in solving these tasks. These findings support the conclusion that insufficient opportunity-to-learn to solve context-based tasks offered by teachers is a possible explanation for students’ difficulties in solving these tasks.
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Opportunity-to-learn context-based tasks provided by mathematics textbooks.
Based on the findings of an error analysis revealing that Indonesian ninth- and tenth-graders had difficulties in solving context-based tasks, we investigated the opportunity-to-learn offered by Indonesian textbooks for solving context-based mathematics tasks and the relation of this opportunity-to-learn to students’ difficulties in solving these tasks. An analysis framework was developed to investigate the characteristics of tasks in textbooks from four perspectives: the type of context used in tasks, the purpose of context-based tasks, the type of information provided in tasks, and the type of cognitive demands of tasks. With this framework, three Indonesian mathematics textbooks were analyzed. Our analysis showed that only about 10 % of the tasks in the textbooks are context-based tasks. Moreover, at least 85 % of these tasks provide exactly the information needed to solve them and do not leave room for students to select relevant information by themselves. Furthermore, of the context-based tasks, 45 % are reproduction tasks requiring performing routine mathematical procedures, 53 % are connection tasks requiring linking different mathematical curriculum strands, and only 2 % are reflection tasks, which are considered as tasks with the highest level of cognitive demand. A linkage between the findings of the error analysis and the textbook analysis suggests that the lacking opportunity-to-learn in Indonesian mathematics textbooks may cause Indonesian students’ difficulties in solving context-based tasks. Based on the results of this study, recommendations are given for improving the opportunities-to-learn to solve context-based tasks as well as for doing further research on this topic.
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Students’ information literacy: A perspective from mathematical literacy
Information literacy is mostly seen from the perspective of library science or information and communication technology. Taking another point of view, this study was aimed to explore students’ information literacy from the perspective of mathematical literacy. For this purpose, a test addressing Programme for International Student Assessment (PISA) mathematics tasks were administered to 381 eighth and ninth graders from nine junior high schools in the Province of Yogyakarta. PISA mathematics tasks which were used in this test had specific characteristics regarding information processing, i.e. containing superfluous information, having missing information, and requiring connection across information sources. An error analysis was performed to analyze students’ incorrect responses. The result of this study shows that students did not acquire three characteristics of information literacy; i.e. recognizing information needs, locating and evaluating the quality of information, and making effective and ethical use of information. This result indicates students’ low ability in information literacy.
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The present study investigates Indonesian fourth-graders low performance in dealing with fractions in TIMSS 2015. Furthermore, the present study also explores possible reasons for this low performance. The data for this study was drawn from TIMSS 2015 data which included test results and responses to Teacher Questionnaire. Descriptive statistics was used to analyze the data. Indonesian textbooks were also analyzed to portrait a broader scope of possible reasons for students’ low performance. The analysis of TIMSS test result reveals that Indonesian students, in comparison to students from other countries, had low understanding of the basic concepts of fractions. From the Teacher Questionnaire it was found that a possible reason for this low understanding was the Indonesian curriculum for third grade which gave low emphasis on the basic concepts of fractions and introduced operations of fractions rather early. Furthermore, the result of textbook analysis shows that Indonesian textbooks restricted only to one definition of fractions, i.e. fractions as parts of wholes. This finding might also explain Indonesian fourth graders’ low understanding of fractions.
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Exploring students’ modeling competences: A case of a GeoGebra-based modeling task
This study was aimed to explore undergraduate students’ modeling competences. A total of 36 undergraduate students were involved in the study. These students worked in groups to solve a modeling task which was presented in GeoGebra application. The students’ modeling competence was investigated before and after the use of metacognitive instruction. The study shows that the students could make a mathematical model or mathematical question of the modeling task. However, prior to the use of metacognitive prompt the students mainly focused on exploring the features of GeoGebra application rather than on identifying mathematical strategies. After the use of metacognitive prompt, the students shifted from feature-based strategies to mathematical strategies. The results of this study indicate that metacognitive prompt is helpful to direct students to think about mathematical concepts or strategies which are relevant to a modeling task.
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This paper reports an exploration into Indonesian fourth graders’ difficulties in fractions and their relation to the opportunity to learn fractions students got at schools. The concept of ‘opportunity to learn’ is often considered as a framework to investigate possible reasons for students’ difficulties. The data for this study was drawn from TIMSS 2015 that comprised test results and teachers’ responses to TIMSS Teacher Questionnaire. The test and questionnaire data were analyzed by using descriptive statistics. In addition to test and questionnaire, this study also included an analysis of Indonesian textbooks in order to get a broader scope of the opportunity to learn. Qualitative approach was used to analyse the extbooks. The analysis of the TIMSS results shows Indonesian students’ low conceptual understanding of fractions. Three possible reasons for students’ low conceptual understanding were revealed. First, the content of Indonesian curriculum that gave low emphasis on basic concepts of fractions and introduced operations of fractions too early. Second, the Indonesian mathematics textbooks only presented one definition of fractions, i.e. fractions as parts of wholes. Third, there is a limited use of models or representations of fractions in the classroom practices.
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Teachers’ perception or belief is seen as an important factor that influences teachers’ teaching practices and students’ learning. In this regard, this study aims to explore teachers’ perceptions on the use of Proofs without Words (PWWs) for visualizing mathematics concepts. This research involves mathematics teachers with a diverse teaching experience. This study employed a qualitative approach with interview as data collecting technique. A total of 12 teachers were interviewed in a semi structured way. The questions that asked to them during interview were focused on how their perception saw about PWWs. The result of the interviews revealed that the PWWs can be used as a source of problems for students for problem-solving activities. However, it also found that students’ mathematical knowledge and skills play an important role in process of interpreting the images.
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This study investigated whether providing opportunity-to-learn can improve Indonesian students’ performance in solving context-based mathematics tasks. On the basis of an inventory of Indonesian students’ difficulties with these tasks and an analysis of textbooks and classroom practices, an intervention program for mathematics teachers was developed. This program contained tasks with relevant and essential contexts with missing or superfluous information, but without explicitly given mathematical procedures. The program also comprised guidelines for a consultative teaching approach with metacognitive prompts and questions for discussion to promote reflection in class. A field experiment with a pretest-posttest control-group design was carried out in six junior high schools in Indonesia involving 299 eightgraders. Students in the experimental group made significantly more progress on solving context-based mathematics tasks than students in the control group. Furthermore, an analysis of students’ errors revealed that experimental students made significantly fewer task comprehension errors than control students. These results show that providing opportunity-to-learn, that is offering context-based tasks to students, which require mathematical modeling, and having teachers knowing the characteristics of such tasks and using a consultative teaching approach, can improve students’ ability in solving context-based tasks.
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This study aims to explore junior high school students’ adaptive reasoning and the Van Hiele level of geometric thinking. The present study was a quasi-experiment with the non-equivalent control group design. The participants of the study were 34 seventh graders and 35 eighth graders in the experiment classes and 34 seventh graders and 34 eighth graders in the control classes. The students in the experiment classes learned geometry under the circumstances of a Knisley mathematical learning. The data were analyzed quantitatively by using inferential statistics. The results of data analysis show an improvement of adaptive reasoning skills both in the grade seven and grade eight. An improvement was also found for the Van Hiele level of geometric thinking. These results indicate the positive impact of Knisley learning model on students’ adaptive reasoning skills and Van Hiele level of geometric thinking.