- Tel Aviv, Tel Aviv, Israel
Michal Tabach
Tel Aviv University, School of Education, Faculty Member
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Patterning activities, specifically those related to repeating patterns, may encourage young children’s appreciation for underlying structures. This paper investigates preschool teachers’ knowledge and self-efficacy for defining, drawing,... more
Patterning activities, specifically those related to repeating patterns, may encourage young children’s appreciation for underlying structures. This paper investigates preschool teachers’ knowledge and self-efficacy for defining, drawing, and continuing repeating patterns. Results indicated that teachers were able to draw and continue various repeating patterns but had difficulties defining repeating patterns. In general, teachers had a high self-efficacy for all tasks. However, teachers’ had a significantly lower self-efficacy for defining repeating patterns than for drawing and continuing repeating patterns.
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Research Interests: Teacher professional development, Digital Resources, Documentational geneses, Operational invariants, Curriculum Materials, and 11 moreTeachers collective work, Resource systems, חומרים קוריקולריים, משאבים דיגיטאליים, מסע יצירת התיעוד, אינווריאנטים אופראציונאליים, מערכות משאבים, משאבים להוראה, עבודה שיתופית של מורים, התפתחות מקצועית של מורים, and Resources in/for teaching
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The aim of this study is to investigate the geometrical knowledge as well as the geometrical self-efficacy of kindergarten children, including abused and neglected kindergarten children. Individual interviews were conducted with 141... more
The aim of this study is to investigate the geometrical knowledge as well as the geometrical self-efficacy of kindergarten children, including abused and neglected kindergarten children. Individual interviews were conducted with 141 kindergarten children, ages 5–6 years old, of which 69 children were labeled as abused and neglected by the social welfare department of their municipality. Results indicated that both groups of kindergarten children had high self-efficacy beliefs related to identifying geometrical figures which were not significantly related to knowledge. In addition, significant differences in knowledge were found between the two groups.
Patterning activities, specifically those related to repeating patterns, may encourage young children’s appreciation for underlying structures. This paper investigates preschool teachers’ knowledge and self-efficacy for defining, drawing,... more
Patterning activities, specifically those related to repeating patterns, may encourage young children’s appreciation for underlying structures. This paper investigates preschool teachers’ knowledge and self-efficacy for defining, drawing, and continuing repeating patterns. Results indicated that teachers were able to draw and continue various repeating patterns but had difficulties defining repeating patterns. In general, teachers had a high self-efficacy for all tasks. However, teachers’ had a significantly lower self-efficacy for defining repeating patterns than for drawing and continuing repeating patterns.
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Qualitative research on classroom-based mathematics learning in inquiry-oriented classrooms is scarce. This paper presents a methodology aimed at developing a rich understanding of the interplay of mathematical progress in the different... more
Qualitative research on classroom-based mathematics learning in inquiry-oriented classrooms is scarce. This paper presents a methodology aimed at developing a rich understanding of the interplay of mathematical progress in the different settings in which learning in such classrooms occurs - individuals, small groups, and the whole class. For this purpose, we enhance a theoretical-methodological approach of coordinating Documenting Collective Activity and the RBC-model of Abstraction in Context that has been developed in earlier studies. We do this using an intact lesson on the area and perimeter of the Sierpiński triangle in a mathematics education master’s level course on Chaos and Fractals. The enhancement of the methodology allowed integrating the Collective and Individual Mathematical Progress (CIMP) by Layering the Explanations (LE) provided by the two approaches and thus exhibiting the complexity of learning processes in inquiry-oriented classrooms.
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This paper describes kindergarten children’s engagement with two patterning activities. The first activity includes two tasks in which children are asked to choose possible ways for extending two different repeating patterns and the... more
This paper describes kindergarten children’s engagement with two patterning activities. The first activity includes two tasks in which children are asked to choose possible ways for extending two different repeating patterns and the second activity calls for comparing different pairs of repeating patterns. Children’s recognition of the unit of repeat and their recognition of the structure of the repeating patterns are investigated. Findings suggest differences between children’s responses to patterns that end with a complete unit of repeat and those that end with a partial unit. In addition, the issue of presenting repeating patterns using different media is discussed.
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We focus on teachers’ ways of leading whole class discussions (WCDs) in mathematics, with the goal of uncovering their traces (if any) in their students’ responses (a) while participating in the WCDs and (b) in the written responses in a... more
We focus on teachers’ ways of leading whole class discussions (WCDs) in mathematics, with the goal of uncovering their traces (if any) in their students’ responses (a) while participating in the WCDs and (b) in the written responses in a final test. For this purpose, two 8th-grade probability classes learning a 10-lesson unit with different teachers were observed. Our data sources include (1) video-recordings of the WCDs and (2) the responses of students to final test items. We analyzed the teachers’ talk-moves, students’ accountable participation, and students’ reasoning in the final test items. Interweaving the findings from all analyses we found differences between the classes in students’ ways of participation in WCDs and in their corresponding final test responses. The teachers’ ways of leading the WCDs contribute to the explanation of these differences.
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Research Interests: Computer Science, Philosophy, Humanities, Art, Mathematics Education, and 15 moreDigital Resources, Documentational geneses, Operational invariants, Encyclopedia, Curriculum Materials, Dijital Kaynaklar, Öğretmenlerin Mesleki Gelişimi, Resource systems, Springer Ebooks, Müfredat materyalleri, Dokümantal oluşumlar, işlevsel sabitler, kaynak sistemleri, öğretim için kaynaklar, and öğretmenlerin kolektif çalışmaları
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we report on two consecutive randomized controlled studies that tested the implementation of a state-of-the-art neural network-based algorithm for personalizing the sequencing of content to learners based on predictive subjective... more
we report on two consecutive randomized controlled studies that tested the implementation of a state-of-the-art neural network-based algorithm for personalizing the sequencing of content to learners based on predictive subjective difficulty level. Performance of the students who followed the algorithm recommendations were first compared to those of students who followed an expert teacher-based recommendations (study 1); then, based on the findings, we compared the impact of the algorithm recommendations to that of a baseline (non-personalized) sequence set-up by human experts (study 2). In the second study, the algorithm was successful in preparing the students to the post-test tasks equally well as the human experts were, however without knowing what these tasks were. We highlight the advantages and the limitations of the expert teacher, as well as the algorithm's ability to do no worse than the human experts.
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Many studies have examined the teaching of mathematics in technological environments that are accessible both to the teacher and to the students. Nevertheless, some classrooms are equipped with only one computer and a data projector. This... more
Many studies have examined the teaching of mathematics in technological environments that are accessible both to the teacher and to the students. Nevertheless, some classrooms are equipped with only one computer and a data projector. This study examined case studies of four different teachers who had previously worked in the high-tech industry and then became high school mathematics teachers that used technology in the classroom. Two technological environments were examined: (1) an environment in which teachers used a computer and a projector and (2) an environment that also included an interactive whiteboard (IWB). The study aimed at characterizing teaching practices and teacher knowledge in these two environments. An innovative framework was developed, based on three lenses: (1) the teachers’ goals; (2) the technological resources used; and (3) the way these resources were used. Findings indicate that teachers used a whole-class lecture style of teaching, mostly for explaining con...
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In this report we further develop the notion of knowledge agent and analyse knowledge agency in an 8 grade mathematics classroom learning probability. By knowledge agency we mean the many ways and variations in which knowledge agents act.... more
In this report we further develop the notion of knowledge agent and analyse knowledge agency in an 8 grade mathematics classroom learning probability. By knowledge agency we mean the many ways and variations in which knowledge agents act. We also observe the teacher as an orchestrator of the learning process who as such invests efforts to create a learning environment that enables students to be active and become knowledge agents. In our previous work we have identified mainly a single student who acted as knowledge agent. Here we show how four students acted as a group of knowledge agents and that knowledge agency may appear in different forms: as one student and his followers, as two students, and as group of students.
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The research goal was to test the suitability of the character of the graphic design on the student's achievements while engaging with mathematical applets for elementary grades. This quantitative research compares two pairs of... more
The research goal was to test the suitability of the character of the graphic design on the student's achievements while engaging with mathematical applets for elementary grades. This quantitative research compares two pairs of mathematical applets, which differ only by the type of graphic design; in the extent of detail and the amount of distraction, while the mathematical problems are identical. The first applet has animated graphics based on designs by Matific, a repository of mathematical applets. The second applet was designed specifically for this research in a schematic and visually simple version of the same activities. Students in the schematic graphics group made fewer mistakes and needed less time to complete the activity than students in the animated graphics group. In addition, students with lower mathematic ability succeeded better in the schematic group. No differences between the two groups were observed regarding the level of students’ enjoyment.
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The aim of this closing chapter is to reflect on the content of this book and on its overall focus on the development of mathematical proficiencies through the design and use of digital technology and of teaching and learning with and... more
The aim of this closing chapter is to reflect on the content of this book and on its overall focus on the development of mathematical proficiencies through the design and use of digital technology and of teaching and learning with and through these tools. As such, rather than making an attempt to provide an overview of the field as a whole, or trying to define overarching theoretical approaches, we chose to follow a bottom-up approach in which the chapters in this monograph form the point of departure. To do so, we reflect on the book’s content from four different perspectives. First, we describe a taxonomy of the use of digital tools in mathematics education, and set up an inventory of the different book chapters in terms of these types of educational use. Second, we address the learning of mathematics with and through technology. Third, the way in which the assessment of mathematics with and through digital technology is present in this monograph is reflected upon. Fourth, the topic of teachers teaching with technology is briefly addressed. We conclude with some final reflections, including suggestions for a future research agenda.
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High schools commonly use a differential approach to teach minima and maxima geometric problems. Although calculus serves as a systematic and powerful technique, this rigorous instrument might hinder students’ ability to understand the... more
High schools commonly use a differential approach to teach minima and maxima geometric problems. Although calculus serves as a systematic and powerful technique, this rigorous instrument might hinder students’ ability to understand the behavior and constraints of the objective function. The proliferation of digital environments allowed us to adopt a different approach involving geometry analysis combined with the use of the inequality of arithmetic and geometric means. The advantages of this approach are enhanced when it is integrated with dynamic e-resources tailored by the instructor. The current study adopts the abstraction in context framework to trace students’ knowledge construction processes while solving extremum problems in an e-resource GeoGebra-based environment using a non-differential approach. We closely monitored the learning of 5 pairs of high-track yet low achieving 17-year-old students for several lessons. We further assessed the students’ understanding at the end of the learning unit based on their explanations of extrema problems. Our findings allowed us to pinpoint the contributions (and pitfalls) of the e-resources for student learning at the micro level. In addition, the students demonstrated the ability to solve extrema problems and were able to explain their reasoning in ways that reflect the e-resources with which they worked.
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In this chapter, we link between modelling activity and affect through the concept of “identifying” or identity construction, as conceptualized within the communicational framework (Heyd-Metzuyanim and Sfard 2012; Sfard 2008). Our aim is... more
In this chapter, we link between modelling activity and affect through the concept of “identifying” or identity construction, as conceptualized within the communicational framework (Heyd-Metzuyanim and Sfard 2012; Sfard 2008). Our aim is to trace the development of modelling abilities through following the development of routines and the changes in identifying talk that co-occur along this development. For this aim, we follow a group of five prospective teachers as they worked on two model-eliciting tasks. Their working process was video recorded and transcribed. The participants’ discourse was analyzed to identify changes in routines while working on the two modelling tasks along with changes in their subjectifying talk (communication about themselves and others). We were able to trace changes in both these measures. Regarding the mathematical talk, we identify a change from a nonsystematic choosing-routine to systematic-choosing-routines and from routines that focus on choosing specific cases to routines that focus on eliciting criterions for choosing. Regarding their identifying activity, we show how participants initially build on their everyday roles in real life (such as mother, citizen and student), to justify their claims in the modelling activity. Later, when routines become more systematic and established, there is much less identifying talk, and claims are justified based on mathematical narratives. We link these findings to previous findings regarding the interaction of mathematizing and identifying activities in mathematical learning.
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The slow uptake of technology by mathematics teachers is in contrast with the rapid growth in the availability of different digital resources specifically designed to help teaching and learning mathematics. We refer to platforms that were... more
The slow uptake of technology by mathematics teachers is in contrast with the rapid growth in the availability of different digital resources specifically designed to help teaching and learning mathematics. We refer to platforms that were designed to permit for mathematical communication between multiple users. We seek to explore the affordances of such digital platforms to support mathematics teachers who wish to integrate technology as part of their practice, when planning and enacting technology-based mathematical activity. Specifically, we ask: What are the affordances and constraints of the platforms that may support instrumentation and instrumentalization processes leading to the development of teacher’s didactic instrument for planning and enacting a mathematical activity in a digital environment? The four platforms we chose for analysis are STEP, DESMOS, WIMS and Labomep. Our analysis shows on the one hand that the platforms afford support to the teacher while enacting techn...
Recent official education-related policy documents at both the international and the national levels acknowledge the need for integrating information and communication technology (ICT) into education, and especially into mathematics... more
Recent official education-related policy documents at both the international and the national levels acknowledge the need for integrating information and communication technology (ICT) into education, and especially into mathematics education. These documents also concur that the teacher is central to such integration. Yet research on teacher education points to dissatisfaction with the results of most pre- and in-service teacher education initiatives to encourage ICT use. The main reason cited for this dissatisfaction is the disparity between teachers’ expectations and the contents of these initiatives. This disparity calls attention to the issue of ICT standards for teachers. The aim of this chapter is twofold: (1) to expand the investigation of existing ICT standards for (mathematics) teachers by examining national policies and institutional frameworks in several OECD countries and in Australia; and (2) to define a conceptual framework to capture various dimensions of teachers’ professional knowledge and skills oriented toward the use of digital technology. Our analysis highlights the need to consolidate the terminology used and to develop standards and competency frameworks geared to specific subjects (in our case, mathematics) and age levels.
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Collaborative work in small groups is often a suitable context for yielding substantial individual learning outcomes. Indeed, small-group collaboration has recently become an educational goal rather than a means. Yet, this goal is... more
Collaborative work in small groups is often a suitable context for yielding substantial individual learning outcomes. Indeed, small-group collaboration has recently become an educational goal rather than a means. Yet, this goal is difficult to attain, and students must be taught how to learn together. In this paper, we focus on how to prepare teachers to become facilitators of small-group collaboration. The current case study monitors a group of six prospective teachers and their instructor during a one-semester course. The instructor was a skilled mathematics teacher with strong beliefs about what is entailed in establishing a mini-culture of learning to learn together and about how to facilitate student group work in problem-solving situations. We describe the learning path followed by the instructor, including the digital environment. The findings show that by the end of the course, the students became more competent facilitators of learning to learn together.
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Personalizing the use of educational mathematics applets to fit learners’ characteristics poses a great challenge. The present study adopted a unique approach by comparing personalization processes implemented by a machine to those... more
Personalizing the use of educational mathematics applets to fit learners’ characteristics poses a great challenge. The present study adopted a unique approach by comparing personalization processes implemented by a machine to those implemented by a human teacher. Given the different affordances—the machine’s access to historical log file data, computation and automatization, and the teacher’s mathematical knowledge, pedagogical approach and personal acquaintance—the study hypothesized that different considerations would lead to different personalization and learning outcomes. Mathematical applets were assigned to 77 students in the 4th and 5th grades either by an expert teacher or by an algorithm. The assignment took place in a controlled setting in which the teacher was unaware which students were eventually assigned according to her recommendations. The teacher and the machine each recommended an ordered sequence of ten applets per student. The findings suggest that the teacher-assigned group outperformed the machine-assigned group among 5th graders when the applets were sequenced in increasing order of difficulty. In the 4th grade, only the machine recommended a sequence of increasing difficulty and both groups achieved equal performance. The study concludes that in the case of data-driven personalization processes, machines and teachers should learn from each other’s affordances and considerations.
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We study content recommendation in an online learning environment for mathematics (N=77, 4-5 grade student). We compare an expert teacher’s recommendation to that of a neural network algorithm, implementing collaborative filtering... more
We study content recommendation in an online learning environment for mathematics (N=77, 4-5 grade student). We compare an expert teacher’s recommendation to that of a neural network algorithm, implementing collaborative filtering ranking. We do so using a double-blind randomized controlled experiment. We find that when the difficulty of the teacher's sequence of recommendation was overall increasing, the teacher was superior to the algorithm regarding students’ performance. Taken together, our findings indicate on how the algorithm and the expert teacher can benefit from each other.
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In this report, we highlight the epistemic actions and concomitant discursive shifts of four students as they reinvent the fundamental idea and technique in Euler's method. We use this case to further the theoretical and... more
In this report, we highlight the epistemic actions and concomitant discursive shifts of four students as they reinvent the fundamental idea and technique in Euler's method. We use this case to further the theoretical and methodological coordination of the Abstraction in Context (AiC) approach, with its associated model commonly used for the analysis of processes of constructing knowledge by individuals, and small groups and the Documenting Collective Activity (DCA) approach, with its methodology commonly used for identifying norma-tive ways of reasoning with groups of students. In this report, we show students' first steps towards re-inventing Euler's method and explicate the theoretical and meth-odological commonalities of AiC and DCA.