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Showing papers in "Zdm in 2011"


Journal ArticleDOI
01 Feb 2011-Zdm
TL;DR: In this paper, the Learning to Learn from Mathematics Teaching (LTLMT) project is described, where video is used to develop pre-service teachers' orientations, knowledge and skills for analyzing and reflecting on mathematics teaching in ways that generate knowledge for improvement.
Abstract: Video is commonly used in teacher preparation programs. Teacher educators use video for various purposes. In this study, we describe the Learning to Learn from Mathematics Teaching project. In this project, video is used to develop pre-service teachers’ (PSTs) orientations, knowledge and skills for analyzing and reflecting on mathematics teaching in ways that generate knowledge for improvement. We discuss the ways we have used video in a course aimed at developing elementary PSTs’ abilities to learn from teaching. In addition, we report on a study that investigated PSTs’ changes in lesson analysis abilities as a result of participating in the course.

319 citations


Journal ArticleDOI
22 Feb 2011-Zdm
TL;DR: In this paper, the authors discuss the relationship between beliefs and emotions, investigating the interplay among the three dimensions in the proposed model of attitude, as emerging in the students' essays.
Abstract: Recent research in the field of affect has highlighted the need to theoretically clarify constructs such as beliefs, emotions and attitudes, and to better investigate the relationships among them. As regards the definition of attitude, in a previous study we proposed a characterization of attitude towards mathematics grounded in students’ experiences, investigating how students express their own relationship with mathematics. The data collected suggest a three-dimensional model of attitude towards mathematics that includes students’ emotional disposition, their vision of mathematics, and their perceived competence. In this paper, we discuss the relationship between beliefs and emotions, investigating the interplay among the three dimensions in the proposed model of attitude, as emerging in the students’ essays.

174 citations


Journal ArticleDOI
01 Feb 2011-Zdm
TL;DR: In this paper, the Problem-Solving Cycle (PSC) and Learning and Teaching Geometry (LTG) models are used for practice-based professional development (PD) to serve as a focal point for teachers to explore the central activities of teaching.
Abstract: This article explores how video can be used in practice-based professional development (PD) programs to serve as a focal point for teachers’ collaborative exploration of the central activities of teaching. We argue that by choosing video clips, posing substantive questions, and facilitating productive conversations, professional developers can guide teachers to examine central aspects of learning and instruction. We draw primarily from our experiences developing and studying two mathematics PD programs, the Problem-Solving Cycle (PSC) and Learning and Teaching Geometry (LTG). While both programs feature classroom video in a central role, they illustrate different approaches to practice-based PD. The PSC, an adaptive model of PD, provides a framework within which facilitators tailor activities to suit their local context. By contrast, LTG is a highly specified model of PD, which details in advance particular learning goals, design characteristics, and extensive support materials for facilitators. We propose a continuum of video use in PD from highly adaptive to highly specified and consider the affordances and constraints of different approaches exemplified by the PSC and LTG programs.

171 citations


Journal ArticleDOI
05 Jul 2011-Zdm
TL;DR: This paper explored a psychological concept they termed an engagement structure, with which beliefs are intertwined, describing complex affective and social interactions as students work on conceptually challenging mathematics, and suggested how beliefs are characteristically woven into their fabric and influence their activation.
Abstract: Beliefs influencing students’ mathematical learning and problem solving are structured and intertwined with larger affective and cognitive structures. This theoretical article explores a psychological concept we term an engagement structure, with which beliefs are intertwined. Engagement structures are idealized, hypothetical constructs, analogous in many ways to cognitive structures. They describe complex “in the moment” affective and social interactions as students work on conceptually challenging mathematics. We present engagement structures in a self-contained way, paying special attention to their theoretical justification and relation to other constructs. We suggest how beliefs are characteristically woven into their fabric and influence their activation. The research is based on continuing studies of middle school students in inner-city classrooms in the USA.

134 citations


Journal ArticleDOI
01 Feb 2011-Zdm
TL;DR: The authors argue that comics can be semiotic resources in learning to teach and suggest how information technologies can support experiences with comics in university mathematics methods courses that help learners see the mathematical work of teaching in lessons they observe, allow candidates to explore tactical decision-making in teaching, and support preservice teachers in rehearsing classroom interactions.
Abstract: This article situates comic-based representations of teaching in the long history of tensions between theory and practice in teacher education. The article argues that comics can be semiotic resources in learning to teach and suggests how information technologies can support experiences with comics in university mathematics methods courses that (a) help learners see the mathematical work of teaching in lessons they observe, (b) allow candidates to explore tactical decision-making in teaching, and (c) support preservice teachers in rehearsing classroom interactions.

114 citations


Journal ArticleDOI
04 Feb 2011-Zdm
TL;DR: The role of beliefs as they affect teachers' behavior can thus be described at a level of mechanism, but of necessity in interaction with resources and goals as mentioned in this paper, and they are interconnected, and why their evolution is necessarily slow.
Abstract: There is now robust evidence that teachers’ and others’ in-the-moment decision making can be modeled and explained as a function of the following: their knowledge and other intellectual, social, and material resources; their goals; and their orientations (their beliefs, values, and preferences). The role of beliefs as they affect teachers’ behavior can thus be described at a level of mechanism—but of necessity in interaction with resources and goals. This paper outlines and exemplifies how resources, goals, and orientations shape teachers’ behavior. It indicates how they are interconnected, and why their evolution is necessarily slow. It then suggests how these understandings can be used as a foundation for mathematics teachers’ professional development, and describes how they are being used to shape a course of participatory professional development for middle school mathematics teachers.

103 citations


Journal ArticleDOI
21 Jul 2011-Zdm
TL;DR: This article examined teachers' selection and implementation of cognitively challenging tasks at three points in time: before and after their participation in the professional development initiative and during a follow-up data collection 2 years later.
Abstract: In this article, we describe a task-centric approach to professional development for mathematics teachers in which teachers’ learning experiences are focused on the selection and implementation of cognitively challenging mathematical tasks. We examined teachers’ selection and implementation of cognitively challenging tasks at three points in time: before and after their participation in the professional development initiative and during a follow-up data collection 2 years later. Data included instructional tasks, samples of student work, and classroom observations, and were compared between the time points to identify changes in teachers’ task selection and implementation and to determine whether these changes were sustained over time. Results indicate that teachers increased and sustained their ability to select high-level instructional tasks and to maintain the level of cognitive demand during instruction. All teachers, however, did not exhibit this pattern. Portraits of teachers who continued to select and enact tasks at a high level are contrasted with those who did not, and factors are identified to account for teachers’ current practices.

64 citations


Journal ArticleDOI
11 Apr 2011-Zdm
TL;DR: In this article, the authors propose an epistemic model of techno-pedagogic mathematic task design which serves as a theoretical combined-lens to view mathematics knowledge acquisition, and conceptualize design of dynamic geometry tasks capitalizing on the unique drag-mode nature in DGE.
Abstract: Dynamic geometry environment (DGE) has been a catalytic agent driving a paradigm shift in the teaching and learning of school geometry in the past two decades. It opens up a pedagogical space for teachers and students to engage in mathematical explorations that niche across the experimental and the theoretical. In particular, the drag-mode in DGE has been a unique pedagogical tool that can facilitate and empower students to experiment with dynamic geometrical objects which can lead to generation of mathematical conjectures. Furthermore, the drag-mode seems to open up a new methodology and even a new discourse to acquire geometrical knowledge alternative to the traditional Euclidean deductive reasoning paradigm. This discussion paper proposes an epistemic model of techno-pedagogic mathematic task design which serves as a theoretical combined-lens to view mathematics knowledge acquisition. Three epistemic modes for techno-pedagogic mathematical task design are proposed. They are used to conceptualize design of dynamic geometry tasks capitalizing the unique drag-mode nature in DGE that opens up an explorative space for learners to acquire mathematical knowledge.

63 citations


Journal ArticleDOI
13 Aug 2011-Zdm
TL;DR: In this paper, a secondary mathematics teacher, taking part in a teacher professional development programme in 2002, was revisited in 2005 and 2010 to gather data regarding the sustainable impact of the programme.
Abstract: This paper deals with the sustainable effectiveness of professional development programmes. Based on a review of literature and research findings, the following questions are raised: What is regarded as an effective way of promoting mathematics teachers’ sustainable professional development? Which levels of impacts are aimed at? What are the factors promoting the effectiveness of professional development programmes? Regarding these questions, the article links theoretical considerations with research findings from a case study. A secondary mathematics teacher, taking part in a teacher professional development programme in 2002, was revisited in 2005 and 2010 to gather data regarding the sustainable impact of the programme. The case study’s results provide information about the teacher’s professional growth and lead to a discussion of implications for mathematics teachers’ professional development and teacher education in general.

53 citations


Journal ArticleDOI
01 Feb 2011-Zdm
TL;DR: In this article, the authors argue that the recent discussion about learning-as-participation is primarily led rather on an abstract level and therefore needs specific mathematics classroom research in order to substantiate how these general theoretical perspectives are represented in social practices at school.
Abstract: It will be argued that the recent discussion about “learning-as-participation” is primarily led rather on an abstract level and therefore needs specific mathematics classroom research in order to substantiate how these general theoretical perspectives are represented in social practices at school. Blumer’s differentiation between “sensitizing concepts” and “definite concepts” in theories of social science that had been adapted to everyday classroom situations of teaching and learning mathematics will be applied in order to develop an empirically definitive concept for the description of forms of participation that facilitate the learning of mathematics.

49 citations


Journal ArticleDOI
26 Jun 2011-Zdm
TL;DR: In this article, the authors investigated the different ways by which secondary school mathematics teachers view how advanced mathematics studies are relevant to expertise in classroom instruction, and found that teachers pointed out at least one specific feature that they viewed as relevant to their work: advanced mathematics courses (e.g., improving understanding about what mathematics is, and reminding teachers what learning mathematics feels like).
Abstract: This study investigates the different ways by which secondary school mathematics teachers view how advanced mathematics studies are relevant to expertise in classroom instruction. Data sources for this study included position papers and written notes from a group interview of 15 Israeli teachers who studied in a special master’s program, of which advanced mathematics courses comprise a sizeable share. Data analysis was iterative and comparative, aiming at identifying and characterizing teachers’ different perspectives. Overall, all participating teachers thought that the advanced mathematics studies in the program were relevant to their teaching of secondary school mathematics. Moreover, teachers specifically mentioned the importance of studying contemporary mathematics from research mathematicians. All teachers pointed out at least one specific feature that they viewed as relevant to their work: advanced mathematics courses (1) as a resource for teaching secondary school mathematics, (2) for improving understanding about what mathematics is, and (3) for reminding teachers what learning mathematics feels like.

Journal ArticleDOI
08 Mar 2011-Zdm
TL;DR: In this article, the structural properties of affect related to mathematics are captured by means of factor analysis, and seven dimensions are described by reliable scales, which allow outlining an average image of Finnish students' views of themselves as learners of mathematics.
Abstract: Students’ views of themselves as learners of mathematics are a decisive parameter for their engagement and success in school. We are interested in students’ experiences with mathematics encompassing cognitive, emotional and motivational aspects. In particular, we focus on capturing the structural properties of affect related to mathematics. Participants in our study were 1,436 randomized chosen students of secondary schools from overall Finland. In the Finnish upper secondary school, there are two different syllabi for mathematics: the general and the advanced one. Schools were invited to organize the survey by one of their year 2 general syllabus courses and one of their year 2 advanced syllabus courses in grade 11. By means of factor analysis, we obtained seven dimensions in which students’ hold beliefs and emotions about mathematics partly intertwined with their motivational orientations. These dimensions are described by reliable scales, which allow outlining an average image of Finnish students’ views of themselves as learners of mathematics. Moreover, we analyzed relations between the seven dimensions and what kind of structure they generate. Thereby, a core of three high correlating dimensions could be identified, yielding different accentuations with regard to course choice.

Journal ArticleDOI
01 Feb 2011-Zdm
TL;DR: In this article, the authors introduce and explore the work of making practice studyable by analyzing a case of practice-based professional development in which the professional development designers deliberately tried to mediate participants' learning in and from practice.
Abstract: A common way to situate professional learning in practice is to use representations of teaching, such as videos of classroom instruction or samples of student work. Using representations of teaching, however, does not automatically lead to teacher learning. Learning in and from practice also requires supports that make such practice studyable. The authors introduce and explore the work of “making practice studyable” by analyzing a case of practice-based professional development in which the professional development designers deliberately tried to mediate participants’ learning in and from practice. From this analysis, the authors identified five categories of work that can help make practice studyable: (1) engaging the content, (2) providing insight into student thinking, (3) orienting to the instructional context, (4) providing lenses for viewing, and (5) developing a disposition of inquiry. These categories are then applied to the use of a representation of mathematics teaching in a course for preservice elementary teachers.

Journal ArticleDOI
20 Sep 2011-Zdm
TL;DR: For example, this article evaluated and refined a previously developed learning trajectory in early length measurement, focusing on the developmental progressions that provide cognitive accounts of the development of children's strategic and conceptual knowledge of measure.
Abstract: Measurement is a critical component of mathematics education, but research on the learning and teaching of measurement is limited, especially compared to topics such as number and operations. To contribute to the establishment of a research base for instruction in measurement, we evaluated and refined a previously developed learning trajectory in early length measurement, focusing on the developmental progressions that provide cognitive accounts of the development of children’s strategic and conceptual knowledge of measure. Findings generally supported the developmental progression, in that children reliably moved through the levels of thinking in that progression. For example, they passed through a level in which they measured length by placing multiple units or attempting to iterate a unit, sometimes leaving gaps between units. However, findings also suggested several refinements to the developmental progression, including the nature and placement of indirect length comparison in the developmental progression and the role of vocabulary, which was an important facilitator of learning for some, but not all, children.

Journal ArticleDOI
26 Jul 2011-Zdm
TL;DR: In this article, the authors argue that dual design research is a fruitful way to promote and trace the development of a mathematics teacher's expertise and argue that this learning process could be attrib- uted to the characteristics of dual design, for instance the cyclic and interventionist character, the con- tinuous process of prediction and reflection that lies at its heart, and the process of co-designing complemented with stimulated recall interviews.
Abstract: In this paper, we argue that dual design research (DDR) is a fruitful way to promote and trace the development of a mathematics teacher's expertise. We address the question of how a teacher participating in dual design research can learn to scaffold students' development of the language required for mathematical learning in multilingual classrooms. Empirical data were collected from two teaching experiments (each with 8 lessons, and 21 and 22 students, aged 11-12 years), for which lesson series about line graphs were co-designed by the researchers and the teacher. The teacher's learning process was promoted (e.g. by conducting stimulated recall inter- views and providing feedback) and traced (e.g. by carrying out 5 pre- and post-interviews before and after the teaching experiments). An analytic framework for teachers' reported and derived learning outcomes was used to analyse pre- and post-interviews. The teacher's learning process was analysed in terms of changes in knowledge and beliefs, changes in practice and intentions for practice. Further analysis showed that this learning process could be attrib- uted to the characteristics of dual design research, for instance the cyclic and interventionist character, the con- tinuous process of prediction and reflection that lies at its heart, and the process of co-designing complemented with stimulated recall interviews.

Journal ArticleDOI
25 Jun 2011-Zdm
TL;DR: In this article, the authors present a re-framing of teacher development that derives from their convictions regarding the enactive approach to cognition and the biological basis of being, and exemplify these principles further through analysis of one group that met over 2-years as part of a research project focused on the work of Gattegno.
Abstract: In this article, we present a re-framing of teacher development that derives from our convictions regarding the enactive approach to cognition and the biological basis of being. We firstly set out our enactivist stance and then distinguish our approach to teacher development from others in the mathematics education literature. We show how a way of working that develops expertise runs through all mathematics education courses at the University of Bristol, and distil key principles for running collaborative groups of teachers. We exemplify these principles further through analysis of one group that met over 2 years as part of a research project focused on the work of Gattegno. We provide evidence for the effectiveness of the group in terms of teacher development. We conclude by arguing that the way of working in this group cannot be separated from the history of interaction of participants.

Journal ArticleDOI
27 Sep 2011-Zdm
TL;DR: This article examined ways of improving students' unit concepts across spatial measurement situations and reported data from their teaching experiment during a six-semester longitudinal study from grade 2 through grade 5.
Abstract: We examined ways of improving students’ unit concepts across spatial measurement situations. We report data from our teaching experiment during a six-semester longitudinal study from grade 2 through grade 5. Data include instructional task sequences designed to help children (a) integrate multiple representations of unit, (b) coordinate and group units into higher-order units, and (c) recognize the arbitrary nature of unit in comparison contexts and student’s responses to tasks. Our results suggest reflection on multiplicative relations among quantities prompted a more fully-developed unit concept. This research extends prior work addressing the growth of unit concepts in the contexts of length, area, and volume by demonstrating the viability of level-specific instructional actions as a means for promoting an informal theory of measurement.

Journal ArticleDOI
17 May 2011-Zdm
TL;DR: In this article, the authors investigated the effect of reading picture books to children on their general measurement performance and found a weak but significant effect for K1 children on the component of holistic visual recognition.
Abstract: This paper addresses: firstly, kindergartners' performance in length measurement, the components of their performance and its growth over time; secondly, the possibility to develop kindergartners' performance in length measurement by reading to them from picture books. To answer the research questions, an experiment with a pretest-posttest experimental control group design was carried out involving nine experimental classes and nine control classes. The children in the experimental group participated in a 3-month picture book program that, among other things, spotlighted the measurement of length situated in meaningful contexts. Before and after the intervention, the children's performance in length mea- surement was assessed in both groups. The responses of 308 kindergartners (4- to 6-year-olds) from two kinder- garten years (K1 and K2) were analyzed. Analysis of the pretest data showed that the measurement tasks included in the test were not easy to solve. However, the children belonging to K2 did better than the younger children belonging to K1. Within children's performance, three components could be identified: holistic visual recognition, ordering and unitizing. Finally, the effect of the interven- tion was investigated by comparing the performances of the experimental and control group in the pretest and the posttest. We found a weak but significant effect of reading picture books to children on their general measurement performance. However, this effect was only found for K1 children on the component of holistic visual recognition.

Journal ArticleDOI
28 Jun 2011-Zdm
TL;DR: The authors argue that the use of the interviews builds teacher expertise through enhancing teachers' knowledge of individual and group understanding of mathematics, and also providing an understanding of typical learning paths in various mathematical domains.
Abstract: In this paper, we outline the benefits to teachers’ expertise of the use of research-based, one-to-one assessment interviews in mathematics. Drawing upon our research and professional development work with teachers and students in primary and middle years in Australia and the research of others, we argue that the use of the interviews builds teacher expertise through enhancing teachers’ knowledge of individual and group understanding of mathematics, and also provides an understanding of typical learning paths in various mathematical domains. The use of such interviews also provides a model for teachers’ interactions and discussions with children, building both their pedagogical content knowledge and their subject matter knowledge.

Journal ArticleDOI
20 Apr 2011-Zdm
TL;DR: In this paper, a meta-emotion perspective is presented as an essential component of a conceptual framework on self-regulation that fully acknowledges the role of emotions against this background, a study is presented that attempts to contribute to the clarification of the relevance and functioning of students' metaemotional knowledge and emotional regulation skills in school-related mathematical activities.
Abstract: Over the past decade, the concept of self-regulated learning has broadened to include motivational, volitional, and emotional components next to (meta-)cognitive ones In this article, we present a meta-emotion perspective as an essential component of a conceptual framework on self-regulation that fully acknowledges the role of emotions Against this background, a study is presented that attempts to contribute to the clarification of the relevance and the functioning of students’ meta-emotional knowledge and emotional regulation skills in school-related mathematical activities It investigates the coping strategies that 393 students of the second (age 14) and fourth (age 16) year of secondary school report to use to regulate their emotions in three different mathematical school settings (ie, a mathematics test, a difficult mathematics homework, and a difficult mathematics lesson) More specifically, it aims (1) to document the nature and frequency of the reported coping strategies, and (2) to explore—for the three different mathematical school settings—relationships between these reported coping strategies and personal characteristics (ie, students’ familiarity with the particular school settings, their track in secondary education, their achievement level, their age, and gender) The results indicate that students report to know and to make use of several coping strategies in school-related mathematical activities, and reveal that the use of these strategies is related to specific person-related characteristics In conclusion, we elaborate on how schools and teachers can stimulate students to acquire appropriate strategies and skills to self-regulate their emotions

Journal ArticleDOI
17 Feb 2011-Zdm
TL;DR: In this paper, an innovative national teacher training program in France, Pairform@nce, designed to sustain ICT integration, is presented, whose objective is to foster the development of an inquiry-based approach in the teaching of mathematics, using investigative potentialities of dynamic geometry environments.
Abstract: We present a research work about an innovative national teacher training program in France: the Pairform@nce program, designed to sustain ICT integration. We study here training for secondary school teachers, whose objective is to foster the development of an inquiry-based approach in the teaching of mathematics, using investigative potentialities of dynamic geometry environments. We adopt the theoretical background of the documentational approach to didactics. We focus on the interactions between teachers and resources: teachers’ professional knowledge influences these interactions, which at the same time yield knowledge evolutions, a twofold process that we conceptualise as a documentational genesis. We followed in particular the work of a team of trainees; drawing on the data collected, we analyse their professional development, related with the training. We observe intertwined evolutions and stabilities, consistent with ongoing geneses.

Journal ArticleDOI
30 Mar 2011-Zdm
TL;DR: In this paper, the authors focus on special cases in which examples satisfy certain universal statements, either true or false in a special type of mathematical task, which they call "Is this a coincidence?" and examine how such tasks may trigger a need for proof.
Abstract: It is widely known that students often treat examples that satisfy a certain universal statement as sufficient for showing that the statement is true without recognizing the conventional need for a general proof. Our study focuses on special cases in which examples satisfy certain universal statements, either true or false in a special type of mathematical task, which we term “Is this a coincidence?”. In each task of this type, a geometrical example was chosen carefully in a way that appears to illustrate a more general and potentially surprising phenomenon, which can be seen as a conjecture. In this paper, we articulate some design principles underlying the choice of examples for this type of task, and examine how such tasks may trigger a need for proof. Our findings point to two different kinds of ways of dealing with the task. One is characterized by a doubtful disposition regarding the generality of the observed phenomenon. The other kind of response was overconfidence in the conjecture even when it was false. In both cases, a need for “proof” was evoked; however, this need did not necessarily lead to a valid proof. We used this type of task with two different groups: capable high school students and experienced secondary mathematics teachers. The findings were similar in both groups.

Journal ArticleDOI
22 Feb 2011-Zdm
TL;DR: In this paper, the authors report on a study aimed at describing the way secondary school teachers treat proof and at understanding which factors may influence such a treatment, i.e., beliefs that seem to drive the way each teacher treats proof.
Abstract: In this paper, we report on a study aimed at describing the way secondary school teachers treat proof and at understanding which factors may influence such a treatment. This study is part of a wider project on proof carried out for many years. In our theoretical framework, we combine references from research on proof with those from research on teachers in relation to their beliefs. The study was carried out through interviews with secondary school teachers aimed at learning how they describe their work with proof in the classroom, and to elicit beliefs and other factors that shape this work. Through the interviews we were able to detect reasons behind teachers’ choices in planning their work in the classroom. In the present paper, we concentrate on four cases that, among other factors, offer elements suitable to unravel the problem of inconsistencies using the construct of leading beliefs, i.e., beliefs (whose nature may vary from teacher to teacher) that seem to drive the way each teacher treats proof.

Journal ArticleDOI
02 Jul 2011-Zdm
TL;DR: In this paper, the authors support students in developing conceptions of statistics by positioning them to design measures of center and of variability for distributions that they had generated through repeated measure of a length.
Abstract: Students often learn procedures for measuring, but rarely do they grapple with the foundational conceptual problem of generating and validating coordination between a measure and the phenomenon being measured. Coordinating measures with phenomenon involves developing an appreciation of the objects and relations in each as well as establishing their mutual correspondence. We supported students’ developing conceptions of statistics by positioning them to design measures of center and of variability for distributions that they had generated through repeated measure of a length. After students invented and explored the viability of their measures individually, they participated in a public (whole-class conversation) forum featuring justification and reflection about the viability of their designed measures. We illustrate how individual invention enticed students to attend to, and to make explicit, characteristics of distribution not initially noticed or known only tacitly. Conceptions of statistics and of relevant characteristics of distribution were further expanded as students justified and argued about the utility and prospective generalization of particular inventions. Teachers supported student learning by highlighting prospective relations between characteristics of measures and characteristics of distribution as they emerged during the course of activity in each setting.

Journal ArticleDOI
18 Feb 2011-Zdm
TL;DR: In this paper, the role of examples in the proving process was analyzed and it was shown that examples are effective for the construction of a proof when they allow cognitive unity and structural continuity between argumentation and proof.
Abstract: In this paper, we analyze the role of examples in the proving process. The context chosen for this study was finding a general rule for triangular numbers. The aim of this paper is to show that examples are effective for the construction of a proof when they allow cognitive unity and structural continuity between argumentation and proof. Continuity in the structure is possible if the inductive argumentation is based on process pattern generalization (PPG), but this is not the case if a generalization is made on the results. Moreover, the PPG favors the development of generic examples that support cognitive unity and structural continuity between the argumentation and proof. The cognitive analysis presented in this paper is performed through Toulmin’s model.

Journal ArticleDOI
24 Sep 2011-Zdm
TL;DR: In this article, the authors present a collection of empirical research reports that have examined different aspects of the learning, teaching, and use of measurement in elementary and secondary mathematics, focusing on non-spatial quantities such as time, weight, and money.
Abstract: This issue presents a collection of empirical research reports that have examined different aspects of the learning, teaching, and use of measurement. The work reported addresses measurement as an important domain of school mathematics, including vocational education, and measurement in use in various occupations and workplaces. The collection is diverse in many ways, as characterized below. Though the focus of many articles is the measurement of space (length, area, or volume), attention is also given in some to non-spatial quantities such as time, weight, and money. The appearance of this issue in ZDM reflects the concern felt in many countries that measurement is an important elementary mathematical and scientific competence, but one that—as evidence considered below suggests—appears to be poorly learned. Weak learning of measurement—particularly of the conceptual principles that underlie measurement procedures—undermines students’ ability to learn and understand more advanced mathematical and scientific content and hence their access to important kinds of skilled work—both professional and not. The research reported in this issue will not solve that problem. Instead, the issue targets a more modest goal: That more researchers across the globe will reconsider the importance of measurement (in school and out), its place in elementary mathematics, and the need to pursue research that will produce partial answers the basic question, ‘‘why are we doing so poorly teaching and learning measurement?’’ We hope these partial answers, as they assemble, will help curriculum developers design more potent materials, teachers teach the measurement content more effectively, and assessment professionals develop more revealing assessments of learning. In this introduction, we seek to orient the reader to the collected articles in two ways. First, we briefly review some of the issues that make measurement ‘‘basic and fundamental’’ content in mathematics and science, in order to orient and frame the inquiries reported in the articles. We also identify some of the principal themes pursued and central results reported in the articles. While this overview is approximate, leaving out important messages particular to individual articles, it is offered to the reader as a partial ‘‘roadmap’’ to the issue—and as motivation to explore further.

Journal ArticleDOI
20 Mar 2011-Zdm
TL;DR: In this paper, two specially designed interactive activities for the teaching and learning of concepts of calculus based on dynamic geometry software are presented, which accentuate the aspect of change and the object aspect of functions using a double stage visualization.
Abstract: Calculus and functional thinking are closely related. Functional thinking includes thinking in variations and functional dependencies with a strong emphasis on the aspect of change. Calculus is a climax within school mathematics and the education to functional thinking can be seen as propaedeutics to it. Many authors describe that functions at school are mainly treated in a static way, by regarding them as pointwise relations. This often leads to the underrepresentation of the aspect of change at school. Moreover, calculus at school is mainly procedure-oriented and structural understanding is lacking. In this work, two specially designed interactive activities for the teaching and learning of concepts of calculus based on dynamic geometry software are presented. They accentuate the aspect of change and the object aspect of functions using a double stage visualization. Moreover, the activities allow the discovery and exploration of some concepts of calculus in a qualitative-structural way without knowing or using curve-sketching routines. The activities were used in a qualitative study with 10th grade students of age 15–16 in secondary schools in Berlin, Germany. Some pairs of students were videotaped while working with the activities. After transcribing, the interactions of the students were interpreted and analyzed focusing to the use of the computer. The results show how the students mobilize their knowledge about functions working on the given tasks, and using the activities to formulate important concepts of calculus in a qualitative way. Also, some important epistemological obstacles can be detected.

Journal ArticleDOI
05 Jan 2011-Zdm
TL;DR: In this article, the authors present a view of three central conceptual activities in the learning of mathematics: concept formation, conjecture formation and conjecture verification, which also take place in everyday thinking, in which the role of examples is crucial.
Abstract: The purpose of this paper is to present a view of three central conceptual activities in the learning of mathematics: concept formation, conjecture formation and conjecture verification. These activities also take place in everyday thinking, in which the role of examples is crucial. Contrary to mathematics, in everyday thinking examples are, very often, the only tool by which we can form concepts and conjectures, and verify them. Thus, relying on examples in these activities in everyday thought processes becomes immediate and natural. In mathematics, however, we form concepts by means of definitions and verify conjectures by mathematical proofs. Thus, mathematics imposes on students certain ways of thinking, which are counterintuitive and not spontaneous. In other words, mathematical thinking requires a kind of inhibition from the learners. The question is to what extent this goal can be achieved. It is quite clear that some people can achieve it. It is also quite clear that many people cannot achieve it. The crucial question is what percentage of the population is interested in achieving it or, moreover, what percentage of the population really cares about it.

Journal ArticleDOI
01 Feb 2011-Zdm
TL;DR: In this article, the authors focus on representations produced by prospective teachers when they were asked to generate a hypothetical classroom dialogue for the equality task: "What goes in the box: 8+4=[ ]+5?"
Abstract: Prospective teachers work with a variety of representations of mathematics teaching (i.e., narrative cases, transcripts, video clips) in teacher preparation courses. Generally, they are considered the audience, not producers, of those artifacts. In this article, however, we focus on representations produced by prospective teachers when they were asked to generate a hypothetical classroom dialogue for the equality task: “What goes in the box: 8+4=[ ]+5?” We discuss the nature and quality of the representations produced by four different cohorts of teacher preparation students—prior to admission, at the beginning, middle, and end of their program. Prospective teachers within and across all cohorts produced an unexpected diversity of representations of class discussions. Of special interest to us were their hybrid representations, those that combined multiple images of mathematics teaching practices. These representations not only provide a lens into prospective teachers’ development as mathematics teachers but could also become tools to support novices as they learn more complex forms of mathematics teaching.

Journal ArticleDOI
31 Jul 2011-Zdm
TL;DR: In this paper, the authors review the efforts of Singapore mathematics teacher educators in incorporating problem-solving competency in teacher education and PD programmes, and discuss conceptual and practical issues, actions taken and changes made in building teachers' capacity to enact a problem solving curriculum in a school-based design experiment project.
Abstract: Problem solving is at the heart of the Singapore Mathematics curriculum. However, it remains a challenge for teachers to realise this curricular goal in practice. Here, we review the efforts of Singapore mathematics teacher educators in incorporating problem-solving (teaching) competency in teacher education and PD programmes. We discuss conceptual and practical issues, actions taken and changes made in building teachers’ capacity to enact a problem-solving curriculum in a school-based design experiment project. In the project, teachers learnt problem solving, observed and then carried out lessons, using the “Mathematics Practical”—akin to the science practical—as key to instruction and assessment.