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1.
andreas j. stylianides 《Educational Studies in Mathematics》2007,65(1):1-20
Despite increased appreciation of the role of proof in students’ mathematical experiences across all grades, little research has focused on the issue of understanding and characterizing the notion of proof at the elementary
school level. This paper takes a step toward addressing this limitation, by examining the characteristics of four major features
of any given argument – foundation, formulation, representation, and social dimension – so that the argument could count as
proof at the elementary school level. My examination is situated in an episode from a third-grade class, which presents a
student’s argument that could potentially count as proof. In order to examine the extent to which this argument could count
as proof (given its four major elements), I develop and use a theoretical framework that is comprised of two principles for
conceptualizing the notion of proof in school mathematics: (1) The intellectual-honesty principle, which states that the notion of proof in school mathematics should be conceptualized so that it is, at once, honest to mathematics
as a discipline and honoring of students as mathematical learners; and (2) The continuum principle, which states that there should be continuity in how the notion of proof is conceptualized in different grade levels so that
students’ experiences with proof in school have coherence. The two principles offer the basis for certain judgments about
whether the particular argument in the episode could count as proof. Also, they support more broadly ideas for a possible
conceptualization of the notion of proof in the elementary grades. 相似文献
2.
There is a documented need for more opportunities for teachers to learn about students’ mathematical reasoning. This article
reports on the experiences of a group of elementary and middle school mathematics teachers who participated as interns in
an after-school, classroom-based research project on the development of mathematical ideas involving middle-grade students
from an urban, low-income, minority community in the United States. For 1 year, the teachers observed the students working
on well-defined mathematical investigations that provided a context for the students’ formation of particular mathematical
ideas and different forms of reasoning in several mathematical content strands. The article describes insights into students’
mathematical reasoning that the teachers were able to gain from their observations of the students’ mathematical activity.
The purpose is to show that teachers’ observations of students’ mathematical activity in research sessions on students’ development
of mathematical ideas can provide opportunities for teachers to learn about students’ mathematical reasoning. 相似文献
3.
Matthew Inglis Juan Pablo Mejia-Ramos Adrian Simpson 《Educational Studies in Mathematics》2007,66(1):3-21
In recent years several mathematics education researchers have attempted to analyse students’ arguments using a restricted
form of Toulmin’s [The Uses of Argument, Cambridge University Press, UK, 1958] argumentation scheme. In this paper we report data from task-based interviews conducted with highly talented postgraduate
mathematics students, and argue that a superior categorisation of genuine mathematical argumentation is provided by the use
of Toulmin’s full scheme. In particular, we suggest that modal qualifiers play an important and previously unrecognised role
in mathematical argumentation, and that one of the goals of instruction should be to develop students’ abilities to appropriately
match up warrant-types with modal qualifiers. 相似文献
4.
Gabriel J. Stylianides 《International Journal of Science and Mathematics Education》2008,6(1):191-215
Despite widespread agreement that proof should be central to all students’ mathematical experiences, many students demonstrate poor ability with it. The curriculum
can play an important role in enhancing students’ proof capabilities: teachers’ decisions about what to implement in their
classrooms, and how to implement it, are mediated through the curriculum materials they use. Yet, little research has focused
on how proof is promoted in mathematics curriculum materials and, more specifically, on the guidance that curriculum materials
offer to teachers to enact the proof opportunities designed in the curriculum. This paper presents an analytic approach that
can be used in the examination of the guidance curriculum materials offer to teachers to implement in their classrooms the
proof opportunities designed in the curriculum. Also, it presents findings obtained from application of this approach to an
analysis of a popular US reform-based mathematics curriculum. Implications for curriculum design and research 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.
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
7.
Joshua Gisemba Bagaka��s 《International Journal of Science and Mathematics Education》2011,9(4):817-842
The study identified two dimensions of teacher self-efficacy and practices and five dimensions of students’ mathematics self-efficacy
and sought to determine the extent to which teacher characteristics and practices can enhance secondary school students’ self-efficacy.
Data were collected from 13,173 students in 193 teachers’ classrooms from 141 schools in the 10 districts of Lake Victoria
Region of Kenya. Two-level hierarchical linear model revealed that teachers’ frequent use of mathematics homework, their level
of interest and enjoyment of mathematics, as well as their ability and competence in teaching mathematics were found to play
a key role in promoting students’ mathematics self-efficacy. Teachers’ ability and competence in teaching were also found
to be effective in narrowing the gender gap in students’ self-confidence and competence in mathematics. The study recommends
that teacher training colleges emphasize such teacher practices and values in order to enhance students’ mathematics self-efficacy,
reduce their level of anxiety and fear of mathematics, and consequently, enhance their achievement in mathematics. Professional
development opportunities should also be made available to in-service teachers to continually update their knowledge and skills
and develop new strategies for teacher effectiveness. 相似文献
8.
Anita A. Wager 《Journal of Mathematics Teacher Education》2012,15(1):9-23
Connecting students’ cultural and community mathematical practices to school mathematics is a critical issue in mathematics
education. The goal of the study was to identify how teachers incorporate children’s cultural and out-of-school mathematics
in instruction. Four related practices were identified, and three drew on children’s cultural or out-of-school experiences:
(a) using these experiences as contexts for problems, (b) linking these experiences to school mathematics, and (c) identifying
embedded mathematical practices prominent in these experiences. A fourth category, teacher initiated situated settings, focused
on shared experiences using the classroom as a site of culture. Findings suggest that these practices represent varying levels
of complexity and that use of this framework might support teachers in better relating students’ cultural and out-of-school
experiences to mathematics. 相似文献
9.
Despina A. Stylianou 《Journal of Mathematics Teacher Education》2010,13(4):325-343
Current reform efforts call for an emphasis on the use of representation in the mathematics classroom across levels and topics.
The aim of the study was to examine teachers’ conceptions of representation as a process in doing mathematics, and their perspectives
on the role of representations in the teaching and learning of mathematics at the middle-school level. Interviews with middle
school mathematics teachers suggest that teachers use representations in varied ways in their own mathematical work and have
developed working definitions of the term primarily as a product in problem solving. However, teachers’ conception of representation
as a process and a mathematical practice appears to be less developed, and, as a result, representations may have a peripheral
role in their instruction as well. Further, the data suggested that representation is viewed as a topic of study rather than
as a general process, and as a goal for the learning of only a minority of the students—the high-performing ones. Implications
for mathematics teacher education, prospective and practicing, are discussed. 相似文献
10.
Colleen Vale Alasdair McAndrew Siva Krishnan 《Journal of Mathematics Teacher Education》2011,14(3):193-212
A professional learning program for teachers of junior secondary mathematics regarding the content and pedagogy of senior
secondary mathematics is the context for this study of teachers’ mathematical and pedagogical knowledge. The analysis of teachers’
reflections on their learning explored teachers’ understanding of mathematical connections and their appreciation of mathematical
structure. The findings indicate that a professional learning program about senior secondary mathematics can enable practicing
teachers to deepen and broaden their knowledge for teaching junior secondary mathematics and develop their practice to support
their students’ present and future learning of mathematics. Further research is needed about professional learning approaches
and tasks that may enable teachers to imbed and develop awareness of structure in their practice. 相似文献
11.
Joanna O. Masingila Samson M. Muthwii Patrick M. Kimani 《International Journal of Science and Mathematics Education》2011,9(1):89-108
This study examined standard 6 and 8 (Standards 6 and 8 are the sixth and eighth years, respectively, of primary level schooling
in Kenya.) students’ perceptions of how they use mathematics and science outside the classroom in an attempt to learn more
about students’ everyday mathematics and science practice. The knowledge of students’ everyday mathematics and science practice
may assist teachers in helping students be more powerful mathematically and scientifically both in doing mathematics and science
in school and out of school. Thirty-six students at an urban school and a rural school in Kenya were interviewed before and
after keeping a log for a week where they recorded their everyday mathematics and science usage. Through the interviews and
log sheets, we found that the mathematics that these students perceived they used outside the classroom could be classified
as 1 of the 6 activities that Bishop (Educ Stud Math 19:179–191, 1988) has called the 6 fundamental mathematical activities and was also connected to their perception of whether they learned
mathematics outside school. Five categories of students’ perceptions of their out-of-school science usage emerged from the
data, and we found that 4 of our codes coincided with 2 activities identified by Lederman & Lederman (Sci Child 43(2):53,
2005) as part of the nature of science and 2 of Bishop’s categories. We found that the science these students perceived that they
used was connected to their views of what science is. 相似文献
12.
Amy E. Ryken 《Journal of Mathematics Teacher Education》2009,12(5):347-364
This documentary account situates teacher educator, prospective teacher, and elementary students’ mathematical thinking in
relation to one another, demonstrating shared challenges to learning mathematics. It highlights an important mathematics reasoning
skill—creating and analyzing representations. The author examines responses of prospective teachers to a visual representation
task and, in turn, their examination of school children’s responses to mathematical tasks. The analysis revealed the initial
tendency of prospective teachers to create pictorial representations and highlights the importance of looking beyond the pictures
created to how prospective teachers use mathematical models. In addition, the challenges prospective teachers face in moving
beyond a ruled-based conception of mathematics and a right/wrong framework for assessing student work are documented. Findings
suggest that analyzing representations helps prospective teachers (and teacher educators) rethink their teaching practices
by engaging with a culture of teaching focused on reading for multiple meanings and posing questions about student thinking
and curriculum materials. 相似文献
13.
Albrecht Heeffer 《Science & Education》2011,20(9):863-880
Historical studies on the development of mathematical concepts will help mathematics teachers to relate their students’ difficulties
in understanding to conceptual problems in the history of mathematics. We argue that one popular tool for teaching about numbers,
the number line, may not be fit for early teaching of operations involving negative numbers. Our arguments are drawn from
the many discussions on negative numbers during the seventeenth and eighteenth centuries from philosophers and mathematicians
such as Arnauld, Leibniz, Wallis, Euler and d’Alembert. Not only does division by negative numbers pose problems for the number
line, but even the very idea of quantities smaller than nothing has been challenged. Drawing lessons from the history of mathematics,
we argue for the introduction of negative numbers in education within the context of symbolic operations. 相似文献
14.
The purpose of this article was to describe the task design and implementation of cultural artefacts in a mathematics lesson
based on the integration of modelling and conjecturing perspectives. The conceived process of integrating a soccer ball into
mathematics lessons via modelling- and conjecturing-based instruction was first detailed. Next, the paper analysed six students’
participation behaviours as they created mathematical problems, definitions, terms, representations and arguments during modelling
and conjecturing activities. Findings suggested students effectively engaged in the search for soccer ball models and solutions
to posed questions, especially the reason manufacturers prefer soccer balls constructed from regular pentagons and regular
hexagons to other types of regular polygons. 相似文献
15.
16.
The relations among students’ motivational beliefs, cognitive processes, and academic achievement were investigated. A 51-item
questionnaire together with a mathematics achievement test was administered to 459 fifth graders in Korean elementary school
mathematics classrooms. Results indicated that, in general, students’ cognitive processes related closely to competence beliefs,
task values, and achievement goals, and more importantly their success or failure in mathematics achievement was closely linked
to competence beliefs, performance-avoidance goals, and persistence strategies. Positive evidence of performance-approach
goals was observed in math learning relative to task goals. As expected, performance-avoidance goals turned out to be detrimental
to students’ math learning. These findings are generally congruent with the motivational theories and support the position
that students should be encouraged to adopt task goals and actively involve themselves in math class activities. However,
it also behooves us to recognize the potential benefits of performance-approach goals in different cultural contexts, such
as the Korean elementary school math classrooms. 相似文献
17.
Helen Forgasz 《Journal of Mathematics Teacher Education》2006,9(5):437-469
The findings presented in this article were derived from a 3- year study aimed at examining issues associated with the use of computers for secondary mathematics learning in Victorian (Australia) schools. Gender and other equity factors were of particular interest. In this article, the focus is on the participating mathematics teachers. Data on their perceived competence levels with technology, and their use of and beliefs about computers for their male and female students’ mathematics learning were gathered. A clear majority of teachers felt comfortable about, and did use, computers for teaching mathematics, and believed that computers helped students’ mathematical learning. Generally, the teachers considered boys to be more confident and capable than girls with computers. The results have implications for pre-service education programs and for the professional development of practicing secondary mathematics teachers.Various findings included in this article have been presented at a range of mathematics education conferences including: AAMT (2003), MAV (2003), and ICME 10 (2004). 相似文献
18.
Oliver F. Jenkins 《Journal of Mathematics Teacher Education》2010,13(2):141-154
A structured interview process is proffered as an effective means to advance prospective teachers’ understandings of students
as learners of mathematics, a key component of pedagogical content knowledge. The interview process is carried out in three
phases with the primary objective of developing listening skills for accessing students’ mathematical thinking. The interviews
adhere to clinical interview procedures for discovering cognitive activities and, accordingly, are initiated by presenting
an open-ended mathematics task. Three rounds of interviews were completed by undergraduates enrolled in a middle school mathematics
methods course. Anecdotal data generated by their interview reports suggest that the structured interview process engenders
an interpretive orientation to listening to students and furthers awareness of how students make sense of mathematics. Features
of the interview process that may limit its potential benefits are discussed; recommendations for further study are proposed. 相似文献
19.
20.
?se Hansson 《Educational Studies in Mathematics》2012,81(1):103-125
In the multilingual mathematics classroom, the assignment for teachers to scaffold students by means of instruction and guidance in order to facilitate language progress and learning for all is often emphasized. In Sweden, where mathematics education is characterized by a low level of teacher responsibility for students’ performance, this responsibility is in part passed on to students. However, research investigating the complexity of relations between mathematics teaching and learning in multilingual classrooms, as well as effect studies of mathematics teaching, often take the existence of teachers’ responsibility for offering specific content activities for granted. This study investigates the relations between different aspects of responsibility in mathematics teaching and students’ performance in the multilingual mathematics classroom. The relationship between different group compositions and how the responsibility is expressed is also investigated. Multilevel structural equation models using TIMSS 2003 data identified a substantial positive influence on mathematics achievement of teachers taking responsibility for students’ learning processes by organizing and offering a learning environment where the teacher actively and openly supports the students in their mathematics learning, and where the students also are active and learn mathematics themselves. A correlation was also revealed between group composition, in terms of students’ social and linguistic background, and how mathematics teaching was performed. This relationship indicates pedagogical segregation in Swedish mathematics education by teachers taking less responsibility for students’ learning processes in classes with a high proportion of students born abroad or a high proportion of students with low socio-economic status. 相似文献