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"Drijvers, Paul"
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The relation between graphing formulas by hand and students' symbol sense
Students in secondary school often struggle with symbol sense, that is, the general ability to deal with symbols and to recognize the structure of algebraic formulas. Fostering symbol sense is an educational challenge. In graphing formulas by hand, defined as graphing using recognition and reasoning without technology, many aspects of symbol sense come to play. In a previous study, we showed how graphing formulas by hand could be learned. The aim of the study we present here is to explore the relationship between students' graphing abilities and their symbol sense abilities while solving non-routine algebra tasks. A symbol sense test was administered to a group of 114 grade 12 students. The test consisted of eight graphing tasks and twelve non-routine algebra tasks, which could be solved by graphing and reasoning. Six students were asked to think aloud during the test The findings show a strong positive correlation between the scores on the graphing tasks and the scores on the algebra tasks and the symbol sense used while solving these tasks. The thinking-aloud protocols suggest that the students who scored high on the graphing tasks used similar aspects of symbol sense in both the graphing and algebra tasks, that is, using combinations of recognizing function families and key features, and qualitative reasoning. As an implication for teaching practice, learning to graph formulas by hand might be an approach to promote students' symbol sense.
Journal Article
The teacher and the tool: instrumental orchestrations in the technology-rich mathematics classroom
The availability of technology in the mathematics classroom challenges the way teachers orchestrate student learning. Using the theory of instrumental orchestration as the main interpretative framework, this study investigates which types of orchestrations teachers develop when using technology and to what extent these are related to teachers' views on mathematics education and the role of technology therein. Data consisted of videotapes of 38 lessons taught by three teachers, who also provided information on their views through questionnaires and interviews. Qualitative analysis of these data led to the identification of orchestration types and teacher profiles. The orchestration preferences of the three teachers proved to be related to their views. A detailed analysis of one exemplary episode suggests how other theoretical perspectives might complement the theory of instrumental orchestration.
Journal Article
Entering the Car Park – Primary Pupils’ Spontaneous Understanding of Programmed Control Systems in their Daily Life Environment
Programmed control systems are ubiquitous in the present-day world. In current educational practice, however, these systems are hardly being addressed, and little is known about children’s spontaneous understandings about such systems. Therefore, we explored pupils’ understandings prior to instruction in three concrete settings: a car park, an elevator, and an autonomous robot. We analysed written responses from 49 Grade 3 (aged 7 to 10) and Grade 6 pupils (aged 10 to 13) to assess their understandings from two perspectives: the user and the system programmer perspective. Results indicate that most pupils were capable describing programmed systems from a user perspective point of view but found it hard to describe the system programmer perspective. Substantial differences were found between the contexts. The car park context evoked richer descriptions for the user perspective and the system programmer perspective in comparison to the elevator and autonomous robot contexts.
Journal Article
Difficulties in initial algebra learning in Indonesia
Within mathematics curricula, algebra has been widely recognised as one of the most difficult topics, which leads to learning difficulties worldwide. In Indonesia, algebra performance is an important issue. In the Trends in International Mathematics and Science Study (TIMSS) 2007, Indonesian students' achievement in the algebra domain was significantly below the average student performance in other Southeast Asian countries such as Thailand, Malaysia, and Singapore. This fact gave rise to this study which aims to investigate Indonesian students' difficulties in algebra. In order to do so, a literature study was carried out on students' difficulties in initial algebra. Next, an individual written test on algebra tasks was administered, followed by interviews. A sample of 51 grade VII Indonesian students worked the written test, and 37 of them were interviewed afterwards. Data analysis revealed that mathematisation, i.e., the ability to translate back and forth between the world of the problem situation and the world of mathematics and to reorganise the mathematical system itself, constituted the most frequently observed difficulty in both the written test and the interview data. Other observed difficulties concerned understanding algebraic expressions, applying arithmetic operations in numerical and algebraic expressions, understanding the different meanings of the equal sign, and understanding variables. The consequences of these findings on both task design and further research in algebra education are discussed. [Author abstract]
Journal Article
Film as the Engine for Learning
Film has been used for education ever since educators recognized its powerful potential for learning. But its educational application has been criticized throughout the decades for underuse of the distinctive potential of film: to raise interest. To understand more fully film’s potential for learning, we propose a dynamic model of viewer interest and its underlying cognitive and emotional mechanisms (film’s interest raising mechanisms or FIRM model). In addition, we present an analysis method for assessing the interestingness of films in learning contexts. Our model marries interest theories from cognitive film theory and educational psychology and captures the dynamics of interestingness across a film as depending on a balance between challenge posed and coping potential provided.
Journal Article
The role of digital technologies in mathematics education: purposes and perspectives
Over the last decades, digital technologies (DTs) have become ubiquitous in mathematics education. Still, their integration into classroom teaching and learning varies enormously. In this narrative overview, we focus on the different purposes for which DTs are used in mathematics education in order to study how the effectiveness of DTs depends on researchers’ underlying goals and perspectives. We set up an experience- and literature-based framework including five different purposes. Applying this framework gave rise to the following results: (1) there is evidence for the benefit of using DTs for mathematics learning; (2) research on DTs leads to new theoretical developments and (3) to new design paradigms; (4) issues of equity with respect to access to and use of DTs are important but under-researched; and (5) DTs challenge curricula and teaching and assessment practices. While early research on the use of DTs focused on questions such as “does it work?” or “does it work better?”, the maturing of the field has shifted to more nuanced questions. As a future research agenda, we recommend further study of how the use of DTs in mathematics education impacts the time required for learning as well as the temporality of teaching and learning, how it changes the nature of doing mathematics and the relation to basic skills and higher-order skills in particular, how curricula, teaching practices, and assessment might change due to the availability of sophisticated mathematical tools, how DTs and other resources might be combined in teaching and learning, and how they may help to address equity issues in education. These questions will prompt the development of new theoretical constructs and approaches.
Journal Article
Student Difficulties in Mathematizing Word Problems in Algebra
To investigate student difficulties in solving word problems in algebra, we carried out a teaching experiment involving 51 Indonesian students (12/13 year-old) who used a digital mathematics environment. The findings were backed up by an interview study, in which eighteen students (13/14 year-old) were involved. The perspective of mathematization, i.e., the activity to transform a problem into a symbolic mathematical problem, and to reorganize the mathematical system, was used to identify student difficulties on the topic of linear equations in one variable. The results show that formulating a mathematical model--evidenced by errors in formulating equations, schemas or diagrams--is the main difficulty. This highlights the importance of mathematization as a crucial process in the learning and teaching of algebra.
Journal Article
Identifying students’ solution strategies in digital mathematics assessment using log data
Background
Students take many tests and exams during their school career, but they usually receive feedback about their test performance based only on an analysis of the item responses. With the increase in digital assessment, other data have become available for analysis as well, such as log data of student actions in online assessment environments. This paper explores how we can use log data to extend performance-related feedback with information related to the applied solution strategy.
Methods
First, we performed an exploratory model-based cluster analysis in order to identify the solution strategy of 802 students with a modal age of 14 in a pre-algebra item from the French national assessment
CEDRE
. Second, we related the students’ solution strategies to their mathematical ability based on the entire assessment.
Results
Five distinct groups of students with different in-assessment behavior were identified, of which one group had a significantly lower estimated mathematics ability than the other groups.
Conclusion
These findings can provide a basis for more in-depth feedback and further instruction on the level of an individual student and can inform teaching practices at the class level.
Journal Article
When bibliometrics met mathematics education research: the case of instrumental orchestration
Thanks to digital technology, methods for finding and analysing research literature have become dramatically more powerful over the last decades. Also, new bibliometric techniques have been developed and applied to the results of such literature search queries. The application of these bibliometric tools to mathematics education research, however, is rare. In this paper, we explore the value of these techniques for mathematics education research through triangulating bibliometrics and expert findings. To do so, we address the case of instrumental orchestration, and want to know how this notion developed over time and was used in research practices. The results show that bibliometric clustering techniques provided a sense-making sketch of the ‘landscape’ of instrumental orchestration research. Triangulating the bibliometric findings with expert interpretations seemed an appropriate method to set up compact ‘identity cards’. In the case of instrumental orchestration, we identified five main clusters in research literature, characterized by the following labels: Managing teaching complexity, Designing living resources, Teaching with technology, Adult learners, and Interacting with computers. The paper ends with some reflections on the potential of bibliometrics in our field and on future research on instrumental orchestration.
Journal Article