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1.
Cecile Ouvrier-Buffet 《Educational Studies in Mathematics》2006,63(3):259-282
The definition of ‘definition’ cannot be taken for granted. The problem has been treated from various angles in different
journals. Among other questions raised on the subject we find: the notions of concept definition and concept image, conceptions of mathematical definitions, redefinitions, and from a more axiomatic point of view, how to construct definitions.
This paper will deal with ‘definition construction processes’ and aims more specifically at proposing a new approach to the
study of the formation of mathematical concepts. I shall demonstrate that the study of the defining and concept formation
processes demands the setting up of a general theoretical framework. I shall propose such a tool characterizing classical
points of view of mathematical definitions as well as analyzing the dialectic involving definition construction and concept
formation. In that perspective, a didactical exemplification will also be presented. 相似文献
2.
One could focus on many different aspects of improving the quality of mathematics teaching. For a better understanding of
children’s mathematical learning processes or teaching and learning in general, reflection on and analysis of concrete classroom
situations are of major importance. On the basis of experiences gained in a collaborative research project with elementary
school teachers, several ideas about a professional reflection on one’s own instruction activities are explained. The paper
focuses on joint reflection between teachers and researchers on the participating teacher’s own classroom interaction by means
of concrete examples. It becomes clear that changes of one’s own interaction behavior will take place only in the long-term.
Nevertheless such a joint professional reflection should be an essential component of teachers’ professional knowledge in
a natural way.
An erratum to this article can be found at 相似文献
3.
Claire Margolinas Lalina Coulange Annie Bessot 《Educational Studies in Mathematics》2005,59(1-3):205-234
Our research is concerned with teacher’s knowledge, and especially with teacher’s processes of learning, in the classroom,
from observing and interacting with students’ work. In the first part of the paper, we outline the theoretical framework of
our study and distinguish it from some other perspectives. We argue for the importance of distinguishing a kind of teacher’s
knowledge, which we call didactic knowledge. In this paper, we concentrate on a subcategory of this knowledge, namely observational didactic knowledge, which grows from teacher’s observation and reflection upon students’ mathematical activity in the classroom. In modeling
the processes of evolution of this particular knowledge in teachers, we are inspired, among others, by some general aspects
of the theory of didactic situations. In the second part of the paper, the model is applied in two case studies of teachers
conducting ordinary lessons. In conclusion, we will discuss what seems to be taken into account by teachers as they observe
students’ activity, and how in-service teacher training can play a role in modifying their knowledge about students’ ways
of dealing with mathematical problems. 相似文献
4.
This paper investigates the role of tools in the formation of mathematical practices and the construction of mathematical
meanings in the setting of a telecommunication organization through the actions undertaken by a group of technicians in their
working activity. The theoretical and analytical framework is guided by the first-generation activity theory model and Leont’ev’s
work on the three-tiered explanation of activity. Having conducted a 1-year ethnographic research study, we identified, classified,
and correlated the tools that mediated the technicians’ activity, and we studied the mathematical meanings that emerged. A
systemic network was generated, presenting the categories of tools such as mathematical (communicative, processes, and concepts)
and non-mathematical (physical and written texts). This classification was grounded on data from three central actions of
the technicians’ activity, while the constant interrelation and association of these tools during the working process addressed
the mathematical practices and supported the construction of mathematical meanings that this group developed from the researchers’
perspective. Technicians’ emerging mathematical meanings referred to place value, spatial, and algebraic relations and were
expressed through personal algorithms and metaphorical and metonymic reasoning. Finally, the educational implications of the
findings are discussed. 相似文献
5.
“Accepting Emotional Complexity”: A Socio-Constructivist Perspective on the Role of Emotions in the Mathematics Classroom 总被引：1，自引：0，他引：1
Peter Op ’t Eynde Erik De Corte Lieven Verschaffel 《Educational Studies in Mathematics》2006,63(2):193-207
A socio-constructivist account of learning and emotions stresses the situatedness of every learning activity and points to the close interactions between cognitive, conative and affective factors in students’ learning and problem solving. Emotions are perceived as being constituted by the dynamic interplay of cognitive, physiological, and motivational processes in a specific context. Understanding the role of emotions in the mathematics classroom then implies understanding the nature of these situated processes and the way they relate to students’ problem-solving behaviour. We will present data from a multiple-case study of 16 students out of 4 different junior high classes that aimed to investigate students’ emotional processes when solving a mathematical problem in their classrooms. After identifying the different emotions and analyzing their relations to motivational and cognitive processes, the relation with students’ mathematics-related beliefs will be examined. We will specifically use Frank’s case to illustrate how the use of a thoughtful combination of a variety of different research instruments enabled us to gather insightful data on the role of emotions in mathematical problem solving. 相似文献
6.
Salvador Llinares Ana Isabel Roig 《International Journal of Science and Mathematics Education》2008,6(3):505-532
This study focussed on how secondary school students construct and use mathematical models as conceptual tools when solving
word problems. The participants were 511 secondary-school students who were in the final year of compulsory education (15–16
years old). Four levels of the development of constructing and using mathematical models were identified using a constant-comparative
methodology to analyse the student’s problem-solving processes. Identifying the general in the particular and using the particular
to endow the general with meaning were the key elements employed by students in the processes of construction and use of models
in the different situations. In addition, attention was paid to the difficulties that students had in using their mathematical
knowledge to solve these situations. Finally, implications are provided for drawing upon student’s use of mathematical models
as conceptual tools to support the development of mathematical competence from socio-cultural perspectives of learning. 相似文献
7.
Erin E. Turner Corey Drake Amy Roth McDuffie Julia Aguirre Tonya Gau Bartell Mary Q. Foote 《Journal of Mathematics Teacher Education》2012,15(1):67-82
Research repeatedly documents that teachers are underprepared to teach mathematics effectively in diverse classrooms. A critical
aspect of learning to be an effective mathematics teacher for diverse learners is developing knowledge, dispositions, and
practices that support building on children’s mathematical thinking, as well as their cultural, linguistic, and community-based
knowledge. This article presents a conjectured learning trajectory for prospective teachers’ (PSTs’) development related to
integrating children’s multiple mathematical knowledge bases (i.e., the understandings and experiences that have the potential to shape and support children’s mathematics learning—including
children’s mathematical thinking, and children’s cultural, home, and community-based knowledge), in mathematics instruction.
Data were collected from 200 PSTs enrolled in mathematics methods courses at six United States universities. Data sources
included beginning and end-of-semester surveys, interviews, and PSTs’ written work. Our conjectured learning trajectory can
serve as a tool for mathematics teacher educators and researchers as they focus on PSTs’ development of equitable mathematics
instruction. 相似文献
8.
Pnina S. Klein Esther Adi-Japha Simcha Hakak-Benizri 《Educational Studies in Mathematics》2010,73(3):233-246
The objective of this study was to examine gender differences in the relations between verbal, spatial, mathematics, and teacher–child
mathematics interaction variables. Kindergarten children (N = 80) were videotaped playing games that require mathematical reasoning in the presence of their teachers. The children’s
mathematics, spatial, and verbal skills and the teachers’ mathematical communication were assessed. No gender differences
were found between the mathematical achievements of the boys and girls, or between their verbal and spatial skills. However,
mathematics performance was related to boys’ spatial reasoning and to girls’ verbal skills, suggesting that they use different
processes for solving mathematical problems. Furthermore, the boys’ levels of spatial and verbal skills were not found to
be related, whereas they were significantly related for girls. The mathematical communication level provided in teacher–child
interactions was found to be related to girls’ but not to boys’ mathematics performance, suggesting that boys may need other
forms of mathematics communication and teaching. 相似文献
9.
Cristina Frade Oto Borges 《International Journal of Science and Mathematics Education》2006,4(2):293-317
This paper reports on study that investigated the tacit-explicit dimension of the learning of mathematics. The study was carried out in a secondary school and consisted of an episode analysis related to a class discussion about the difference between plane figures and spatial figures. The data analysis was based on integration between some aspects of Polanyi’s theory on tacit knowledge and Ernest’s model of mathematical knowledge, with reference to its mainly explicit and mainly tacit components. This integration has involved not only the types of knowledge – mainly explicit or mainly tacit – the students used in a psychological way to perform a mathematical task involving conversation, but also and particularly how much the projection of those types of knowledge on the task were manifest tacitly or formalized by the students. Among the results of the research, a strong finding was that the lack of correspondence between the students’ utterances and their original understandings is directly related to the manner in which the tacit co-operates with the explicit in the process of articulation.This paper is a development of three short papers presented at ICME-10, Copenhagen, Denmark, 4–11 July 2004, and PME-28, Bergen, Norway, 14–18 July 2004. 相似文献
10.
The purpose of this paper is to contribute to the debate about how to tackle the issue of ‘the teacher in the teaching/learning
process’, and to propose a methodology for analysing the teacher’s activity in the classroom, based on concepts used in the
fields of the didactics of mathematics as well as in cognitive ergonomics. This methodology studies the mathematical activity
the teacher organises for students during classroom sessions and the way he manages1 the relationship between students and mathematical tasks in two approaches: a didactical one [Robert, A., Recherches en Didactique
des Mathématiques 21(1/2), 2001, 7–56] and a psychological one [Rogalski, J., Recherches en Didactique des Mathématiques 23(3),
2003, 343–388]. Articulating the two perspectives permits a twofold analysis of the classroom session dynamics: the “cognitive
route” students are engaged in—through teacher’s decisions—and the mediation of the teacher for controlling students’ involvement
in the process of acquiring the mathematical concepts being taught. The authors present an example of this cross-analysis
of mathematics teachers’ activity, based on the observation of a lesson composed of exercises given to 10th grade students
in a French ‘ordinary’ classroom. Each author made an analysis from her viewpoint, the results are confronted and two types
of inferences are made: one on potential students’ learning and another on the freedom of action the teacher may have to modify
his activity. The paper also places this study in the context of previous contributions made by others in the same field. 相似文献
11.
This case study deals with a solitary learner’s process of mathematical justification during her investigation of bifurcation
points in dynamic systems. Her motivation to justify the bifurcation points drove the learning process. Methodologically,
our analysis used the nested epistemic actions model for abstraction in context. In previous work, we have shown that the
learner’s attempts at justification gave rise to several processes of knowledge construction, which develop in parallel and
interact. In this paper, we analyze the interaction pattern of combining constructions and show that combining constructions
indicate an enlightenment of the learner. This adds an analytic dimension to the nested epistemic actions model of abstraction
in context. 相似文献
12.
Rivka Feldhay 《Science & Education》2006,15(2-4):151-172
This paper attempts to draw attention to non-conventional but popular modes of transmitting scientific knowledge in Jesuit
institutions in the 17th century. The particular case study focuses on a fictive dialogue between Galileo, Mersenne and Paulus
Guldin on the power needed for moving the huge globe of the earth by mechanical means. The dialogue was written by a Jesuit
mathematician named Paolo Casati (1617–1707) and published in 1655. Apparently, Casati offers his readers an idealized representation
of a real event that took place at the Collegio Romano, where explanation of mathematical problems in a kind of public ritual
used to take place once or twice a month in presence of philosophers, theologians, visitors and students. My analysis of some
parts of Casati’s Terra Machinis Mota exemplifies the Jesuits’ success to accommodate the project of Renaissance practical mathematicians – the fusion of the pseudo-Aristotelian
interest in machines with the mathematical approach of Archimedes – to the framework of the traditional mixed mathematical
science that legitimized it and spread it among wide audiences. Casati’s text demonstrates how at least some Jesuit mathematicians
were ready to adopt Galileo’s early mechanical project. However, moving from an analysis of the contents of mechanical knowledge
popularized in this text to its analysis on the rhetorical level reveals the unbearable tensions by which Jesuit scientific
culture was actually torn. The rhetorical choice to construct a representation of a seemingly friendly dialogue between the
quasi-heretic Galileo, the Minim friar Mersenne and the suspected character of the Jesuit Guldin reveals the strategies by
which Galileo’s heretic image was tamed in order to fit the Jesuits’ needs to construct themselves an enlightened public image.
*Many thanks to Dr. Ido Yavetz, Professor Sabetai Unguru and Mr. Daniel Spitzer, who have read earlier versions of my paper
and helped me clarify both the text I was trying to interpret and my own thoughts about it. 相似文献
13.
This case study of a preservice secondary mathematics teacher focuses on the teacher’s beliefs about his role as mathematics
teacher. Data were collected over the final 5 months of the teacher’s university teacher education program through interviews,
written course assignments, and observations of student-teaching. Findings indicate that the preservice teacher valued classroom
roles in which students, rather than the teacher, explained traditional mathematics content. As his student-teaching internship
progressed, the teacher began to develop new roles and engaged students in additional mathematical processes. These results
emphasize the need for preservice teachers to recognize how teacher and student roles impact interrelationships between understanding
and mathematical activity, and illustrate the nature of teacher learning that can occur during an internship. 相似文献
14.
M.C. O'Connor 《Educational Studies in Mathematics》2001,46(1-3):143-185
This case study examines two days of teacher-led large group discussion in a fifth grade about a mathematical question intended
to support student exploration of relationships among fraction and decimal representations and rational numbers. The purpose
of the analysis is to illuminate the teacher’s work in supporting student thinking through the use of a mathematical question
embedded in a position-driven discussion. The focus is an examination of the ways that the emergence of mathematical ideas
is partially shaped by complex interactions among the mathematical contents of the question, the inherent properties of the
discourse format and participant structure, and the available computational methods. The teacher’s work is conceptualized
in terms of actions and practices that coordinate these diverse tools, in constant response to students’ concurrent use of
them.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
15.
This paper offers a typology of forms and uses of abduction that can be exploited to better analyze abduction in proving processes.
Based on the work of Peirce and Eco, we describe different kinds of abductions that occur in students’ mathematical activity
and extend Toulmin’s model of an argument as a methodological tool to describe students’ reasoning and to classify the different
kinds of abduction. We then use this tool to analyze case studies of students’ abductions and to identify cognitive difficulties
students encounter. We conclude that some types of abduction may present obstacles, both in the argumentation when the abduction
occurs and later when the proof is constructed. 相似文献
16.
Eugenia Vomvoridi-Ivanovi? 《Journal of Mathematics Teacher Education》2012,15(1):53-66
This paper explores Mexican–American prospective teachers’ use of culture—defined as social practices and shared experiences—as
an instructional resource in mathematics. The setting is an after-school mathematics program for the children of Mexican heritage.
Qualitative analysis of the prospective teachers’ and children’s interactions reveals that the nature of the mathematical
activities affected how culture was used. When working on the “binder activities,” prospective teachers used culture only
in non-mathematical contexts. When working on the “recipes project,” however, culture was used as a resource in mathematical
contexts. Implications for the mathematics teacher preparation of Latinas/os are discussed. 相似文献
17.
This study investigates the mechanisms of scaffolding in a synchronous network-based environment – the ‘collaborative virtual
workplace’. A theoretical ‘multi-actor’ scaffolding model was formulated. The study itself focused on the role and inter-relations
of verbal scaffolding by tutor and peers during a collaborative process of making decisions about environmental issues. The analysis drew on data from the decision-making
discussions of 31 groups – material that was saved automatically by the learning environment software. The age of the 62 students
ranged from 14 to 17. Discourse act categories were devised to describe the tutor’s and the students’ task-related, supportive
and social communicative acts. The scaffolding situation was characterized through a causal discourse act interaction approach.
Tutor and students appeared to be elaborating and replacing each other’s process scaffolding acts in the collaborative decision-making
situation. The influence of certain tutor’s and students’ inter-related scaffolding patterns on students’ decision-making
provided empirical support for the ‘multi-actor’ scaffolding model.
in final form: 12 May 2005 相似文献
18.
Marilena Pantziara Athanasios Gagatsis Iliada Elia 《Educational Studies in Mathematics》2009,72(1):39-60
The Mathematics education community has long recognized the importance of diagrams in the solution of mathematical problems.
Particularly, it is stated that diagrams facilitate the solution of mathematical problems because they represent problems’
structure and information (Novick & Hurley, 2001; Diezmann, 2005). Novick and Hurley were the first to introduce three well-defined types of diagrams, that is, network, hierarchy, and matrix,
which represent different problematic situations. In the present study, we investigated the effects of these types of diagrams
in non-routine mathematical problem solving by contrasting students’ abilities to solve problems with and without the presence
of diagrams. Structural equation modeling affirmed the existence of two first-order factors indicating the differential effects
of the problems’ representation, i.e., text with diagrams and without diagrams, and a second-order factor representing general
non-routine problem solving ability in mathematics. Implicative analysis showed the influence of the presence of diagrams
in the problems’ hierarchical ordering. Furthermore, results provided support for other studies (e.g. Diezman & English, 2001) which documented some students’ difficulties to use diagrams efficiently for the solution of problems. We discuss the findings
and provide suggestions for the efficient use of diagrams in the problem solving situation. 相似文献
19.
We ground Cultural-Historical Activity Theory (CHAT) in studies of workplace practices from a mathematical point of view.
We draw on multiple case study visits by college students and teacher-researchers to workplaces. By asking questions that
‘open boxes’, we ‘outsiders and boundary-crossers’ sought to expose contradictions between College and work, induce breakdowns
and identify salient mathematics. Typically, we find that mathematical processes have been historically crystallised in ‘black
boxes’ shaped by workplace cultures: its instruments, rules and divisions of labour tending to disguise or hide mathematics.
These black boxes are of two kinds, signalling two key processes by which mathematics is put to work. The first involves automation,
when the work of mathematics is crystallised in instruments, tools and routines: this process tends to distribute and hide
mathematical work, but also evolves a distinct workplace ‘genre’ of mathematical practice. The second process involves sub-units
of the community being protected from mathematics by a division of labour supported by communal rules, norms and expectations.
These are often regulated by boundary objects that are the object of activity on one side of the boundary but serve as instruments
of activity on the other side. We explain contradictions between workplace and College practices in analyses of the contrasting
functions of the activity systems that structure them and that consequently provide for different genres and distributions
of mathematics, and finally draw inferences for better alignment of College programmes with the needs of students. 相似文献