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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
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Juan Pablo Mejia-Ramos Evan Fuller Keith Weber Kathryn Rhoads Aron Samkoff 《Educational Studies in Mathematics》2012,79(1):3-18
Although proof comprehension is fundamental in advanced undergraduate mathematics courses, there has been limited research
on what it means to understand a mathematical proof at this level and how such understanding can be assessed. In this paper,
we address these issues by presenting a multidimensional model for assessing proof comprehension in undergraduate mathematics.
Building on Yang and Lin’s (Educational Studies in Mathematics 67:59–76, 2008) model of reading comprehension of proofs in high school geometry, we contend that in undergraduate mathematics a proof is
not only understood in terms of the meaning, logical status, and logical chaining of its statements but also in terms of the
proof’s high-level ideas, its main components or modules, the methods it employs, and how it relates to specific examples.
We illustrate how each of these types of understanding can be assessed in the context of a proof in number theory. 相似文献
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In this paper, we examine the support given for the ‘theory of formal discipline’ by Inglis and Simpson (Educational Studies
Mathematics 67:187–204, 2008). This theory, which is widely accepted by mathematicians and curriculum bodies, suggests that the study of advanced mathematics
develops general thinking skills and, in particular, conditional reasoning skills. We further examine the idea that the differences
between the conditional reasoning behaviour of mathematics and arts undergraduates reported by Inglis and Simpson may be put
down to different levels of general intelligence in the two groups. The studies reported in this paper call into question
this suggestion, but they also cast doubt on a straightforward version of the theory of formal discipline itself (at least
with respect to university study). The paper concludes by suggesting that either a pre-university formal discipline effect
or a filtering effect on ‘thinking dispositions’ may give a better account for the findings. 相似文献
6.
The paper addresses the apparent lack of impact of ‘history in mathematics education’ in mathematics education research in
general, and proposes new avenues for research. We identify two general scenarios of integrating history in mathematics education
that each gives rise to different problems. The first scenario occurs when history is used as a ‘tool’ for the learning and
teaching of mathematics, the second when history of mathematics as a ‘goal’ is pursued as an integral part of mathematics
education. We introduce a multiple-perspective approach to history, and suggest that research on history in mathematics education
follows one of two different avenues in dealing with these scenarios. The first is to focus on students’ development of mathematical
competencies when history is used a tool for the learning of curriculum-dictated mathematical in-issues. A framework for this
is described. Secondly, when using history as a goal it is argued that an anchoring of the meta-issues in the related in-issues
is essential, and a framework for this is given. Both frameworks are illustrated through empirical examples. 相似文献
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Current reform-driven mathematics documents stress the need for teachers to provide learning environments in which students
will be challenged to engage with mathematics concepts and extend their understandings in meaningful ways (e.g., National
Council of Teachers of Mathematics, 2000, Curriculum and evaluation standards for school mathematics. Reston, VA: The Council). The type of rich learning contexts that are envisaged by such reforms are predicated on a number
of factors, not the least of which is the quality of teachers’ experience and knowledge in the domain of mathematics. Although
the study of teacher knowledge has received considerable attention, there is less information about the teachers’ content
knowledge that impacts on classroom practice. Ball (2000, Journal of Teacher Education, 51(3), 241–247) suggested that teachers’ need to ‘deconstruct’ their content knowledge into more visible forms that would
help children make connections with their previous understandings and experiences. The documenting of teachers’ content knowledge
for teaching has received little attention in debates about teacher knowledge. In particular, there is limited information
about how we might go about systematically characterising the key dimensions of quality of teachers’ mathematics knowledge
for teaching and connections among these dimensions. In this paper we describe a framework for describing and analysing the
quality of teachers’ content knowledge for teaching in one area within the domain of geometry. An example of use of this framework
is then developed for the case of two teachers’ knowledge of the concept ‘square’. 相似文献
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Debra Panizzon John Pegg 《International Journal of Science and Mathematics Education》2008,6(2):417-436
Aligned with recent changes to syllabuses in Australia is an assessment regime requiring teachers to identify what their students
‘know’ and ‘can do’ in terms of the quality of understanding demonstrated. This paper describes the experiences of 25 secondary
science and mathematics teachers in rural schools in New South Wales as they explore the changing nature of assessment and
its implications on their classroom practice. To help reconceptualise these changes, teachers were introduced to a cognitive
structural model as a theoretical framework. Throughout the 2-year study, teachers attended a series of professional development
sessions and received ongoing consultative support. Each session was taped and transcribed while interviews were conducted
with each teacher at the end of both years. Analysis of these data using a grounded theory approach identified seven major
components of teacher practice impacted by the study. The core component was questioning while the six contributing components
were teachers’ pedagogical practices, attention to cognition, teaching strategies, assessment linked to pedagogy, classroom
advantages for students, and classroom advantages for teachers. These findings represent a major shift in teachers’ perceptions
of assessment from a focus on the accumulation of students’ marks to one of diagnosis as a means of directing teaching to
enhance students’ scientific and mathematical understandings. 相似文献
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Barbara Jaworski 《Journal of Mathematics Teacher Education》2007,9(2):187-211
In this paper I address the challenge of developing theory in relation to the practices of mathematics teaching and its development.
I do this by exploring a notion of ‘teaching as learning in practice’ through overt use of ‘inquiry’ in mathematics learning,
mathematics teaching and the development of practices of teaching in communities involving teachers and educators. The roles
and goals of mathematics teachers and educators in such communities are both distinct and deeply intertwined. I see an aim
of inquiry in teaching to be the ‘critical alignment’ (Wenger, 1998) of teaching within the communities in which teaching
takes place. Inquiry ‘as a tool’ and inquiry ‘as a way of being’ are important concepts in reflexive developmental processes
in which inquiry practice leads to better understandings and development of theory. 相似文献
11.
Luciana Bazzini 《Educational Studies in Mathematics》2001,47(3):259-271
The mutual relationship between real objects and mathematical constructions is at the very base of studies concerned with
making sense in mathematics. In this wider perspective recent research studies have been concerned with the cognitive roots
of mathematical concepts. Human perception and movement and, more generally, interaction with space and time are recognized
as being of crucial importance for knowledge construction. A new approach to the cognitive science of mathematics, based on
the notion of ‘embodied cognition’ assumes that mathematics cannot be considered as mind free. Accordingly, mathematical concepts
derive from the cognitive activities of subjects and are highly influenced by the body structure. This article reports some
examples of teaching experiments based on body-related metaphors. Some of them are carried out by means of technological devices.
A call for legitimacy in school mathematics is made, both for an embodied cognition perspective and for a related use of technology.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
12.
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
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Building on the papers in this special issue as well as on our own experience and research, we try to shed light on the construct
of example spaces and on how it can inform research and practice in the teaching and learning of mathematical concepts. Consistent with our
way of working, we delay definition until after appropriate reader experience has been brought to the surface and several
‘examples’ have been discussed. Of special interest is the notion of accessibility of examples: an individual’s access to example spaces depends on conditions and is a valuable window on a deep, personal,
situated structure. Through the notions of dimensions of possible variation and range of permissible change, we consider ways in which examples exemplify and how attention needs to be directed so as to emphasise examplehood (generality)
rather than particularity of mathematical objects. The paper ends with some remarks about example spaces in mathematics education
itself. 相似文献
14.
What makes a problem mathematically interesting? Inviting prospective teachers to pose better problems 总被引：1，自引：0，他引：1
School students of all ages, including those who subsequently become teachers, have limited experience posing their own mathematical
problems. Yet problem posing, both as an act of mathematical inquiry and of mathematics teaching, is part of the mathematics
education reform vision that seeks to promote mathematics as an worthy intellectual activity. In this study, the authors explored
the problem-posing behavior of elementary prospective teachers, which entailed analyzing the kinds of problems they posed
as a result of two interventions. The interventions were designed to probe the effects of (a) exploration of a mathematical
situation as a precursor to mathematical problem posing, and (b) development of aesthetic criteria to judge the mathematical
quality of the problems posed. Results show that both interventions led to improved problem posing and mathematically richer
understandings of what makes a problem ‘good.’ 相似文献
15.
Mathematics and Virtual Culture: an Evolutionary Perspective on Technology and Mathematics Education
This paper suggests that from a cognitive-evolutionary perspective, computational media are qualitatively different from many
of the technologies that have promised educational change in the past and failed to deliver. Recent theories of human cognitive
evolution suggest that human cognition has evolved through four distinct stages: episodic, mimetic, mythic, and theoretical.
This progression was driven by three cognitive advances: the ability to ‘represent’ events, the development of symbolic reference,
and the creation of external symbolic representations. In this paper, we suggest that we are developing a new cognitive culture:
a ‘virtual’ culture dependent on the externalization of symbolic processing. We suggest here that the ability to externalize
the manipulation of formal systems changes the very nature of cognitive activity. These changes will have important consequences
for mathematics education in coming decades. In particular, we argue that mathematics education in a virtual culture should
strive to give students generative fluency to learn varieties of representational systems, provide opportunities to create
and modify representational forms, develop skill in making and exploring virtual environments, and emphasize mathematics as
a fundamental way of making sense of the world, reserving most exact computation and formal proof for those who will need
those specialized skills.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
16.
Kim Beswick 《International Journal of Science and Mathematics Education》2011,9(2):367-390
The professional literature in mathematics education is replete with calls to use tasks that are ‘authentic’, ‘relevant’ and
related to ‘real life’ and the ‘real world’. Such activities are frequently advocated for their potential to motivate and
engage students, but evidence of their ability to do so is rarely presented. This paper examines evidence in relation to the
effectiveness of context problems in achieving their intended purposes and thereby contributing to enhanced student participation,
engagement and achievement in mathematics education. It is argued that context problems are not a panacea and that categorising
problems as contextualised or de-contextualised is less helpful than the consideration of more salient aspects of tasks that
impact on their effectiveness. Such aspects also relate to the purposes for and affordances and limitations of particular
tasks in relation to the purposes they are intended to serve, along with attention to the contexts in which students learn
mathematics. Examples of theoretical and empirical programs built on these considerations are reviewed in terms of their potential
to enhance participation, engagement and achievement in school mathematics. 相似文献
17.
The attitude construct is widely used by teachers and researchers in mathematics education. Often, however, teachers’ diagnosis
of ‘negative attitude’ is a causal attribution of students’ failure, perceived as global and uncontrollable, rather than an
accurate interpretation of students’ behaviour, capable of steering future action. In order to make this diagnosis useful
for dealing with students’ difficulties in mathematics, it is necessary to clarify the construct attitude from a theoretical viewpoint, while keeping in touch with the practice that motivates its use. With this aim, we investigated
how students tell their own relationship with mathematics, proposing the essay “Me and maths” to more than 1,600 students
(1st to 13th grade). A multidimensional characterisation of a student’s attitude towards mathematics emerges from this study.
This characterisation and the study of the evolution of attitude have many important consequences for teachers’ practice and
education. For example, the study shows how the relationship with mathematics is rarely told as stable, even by older students:
this result suggests that it is never too late to change students’ attitude towards mathematics. 相似文献
18.
The Practices of Mathematicians: What do They Tell us About Coming to Know Mathematics? 总被引：1，自引：0，他引：1
Leone Burton 《Educational Studies in Mathematics》1998,37(2):121-143
In 1997, an interview-based study of 70 research mathematicians was undertaken with a focus on how they ‘come to know’ mathematics,
i.e. their epistemologies. In this paper, I discuss how these mathematicians understand their practices, locating them in
the communities of which they claim membership, identifying the style which dominates their organisation of research and looking
at their lived contradictions. I examine how they talk about ‘knowing’ mathematics, the metaphors on which they draw, the
empiricist connections central to the work of the applied mathematicians and statisticians, and the importance of connectivities
to the construction of their mathematical Big Picture. I compare the stories of these research mathematicians with practices
in mathematics classrooms and conclude with an appeal for teachers to pay attention to the practices of research mathematicians
and their implications for coming to know mathematics.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
19.
Mathematics teaching in Burkina Faso is faced with major challenges (high illiteracy rates, students’ difficulties, and high
failure rates in mathematics, which is a central topic in the curriculum). As evidenced in many of these studies, mathematics
is reputed to be tough, inaccessible, and far from what students live daily. Students here look as though they are living
in two seemingly distant worlds, school and everyday life. In order to better understand these difficulties and to contribute
in the long run to a more adapted teaching of mathematics, we tried to document and elicit the “mathematical resources” mobilized
in various daily life social practices. In this paper, we focus on one of them, the counting and selling of mangoes by unschooled
peasants. An ethnographic approach draws on the observation of the situated activity of counting and selling mangoes (during
harvesting) and on “eliciting interviews” of the involved actors. The analysis of results highlights a richness of structuring
resources mobilized and distributed through this practice, related to what Lave (1988) call “the experienced lived-in-world” and “constitutive order.” The mathematical resources take the form of “knowledge in
action” and “theorems in action” (Vergnaud, Rech Didact Math 10(23):133–170, 1990), embedded in the social, economic, and even cultural structures of actors. 相似文献
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Lene Møller Madsen Carl Winsløw 《International Journal of Science and Mathematics Education》2009,7(4):741-763
We examine the relationship between research and teaching practices as they are enacted by university professors in a research-intensive
university. First we propose a theoretical model for the study of this relationship based on Chevallard’s anthropological
theory. This model is used to design and analyze an interview study with physical geographers and mathematicians at the University
of Copenhagen. We found significant differences in how the respondents from the two disciplines assessed the relationship
between research and teaching. Above all, while geography research practices are often and smoothly integrated into geography
teaching even at the undergraduate level, teaching in mathematics may at best be ‘similar’ to mathematical research practice,
at least at the undergraduate level. Finally, we discuss the educational implications of these findings. 相似文献