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Motivated by the observation that formal logic answers questions students have not yet asked, we conducted exploratory teaching experiments with undergraduate students intended to guide their reinvention of truth-functional definitions for basic logical connectives. We intend to reframe the relationship between reasoning and logic by showing how logic emerges within students’ mathematical activity. This activity entails reflecting on and systematizing their own language use across diverse semantic content. We present categories of students’ untrained strategies for assessing the truth-values for mathematical disjunctions. Students’ initial reasoning heavily reflected content-specific and pragmatic factors in ways inconsistent with the norms and conventions of mathematical logic. Despite this, all student groups reinvented the standard truth-functional definition for simple disjunctions. We demonstrate how this learning depended upon particular forms of reasoning about logic. We also contrast various strategies for assessing quantified disjunctions and their different affordances in students’ mathematical activity. 相似文献
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Eivind Kaspersen Birgit Pepin Svein Arne Sikko 《Educational Studies in Mathematics》2017,95(2):163-179
Studies on identity in general and mathematical identity in particular have gained much interest over the last decades. However, although measurements have been proven to be potent tools in many scientific fields, a lack of consensus on ontological, epistemological, and methodological issues has complicated measurements of mathematical identities. Specifically, most studies conceptualise mathematical identity as something multidimensional and situated, which obviously complicates measurement, since these aspects violate basic requirements of measurement. However, most concepts that are measured in scientific work are both multidimensional and situated, even in physics. In effect, these concepts are being conceptualised as sufficiently uni-dimensional and invariant for measures to be meaningful. We assert that if the same judgements were to be made regarding mathematical identity, that is, whether identity can be measured with one instrument alone, whether one needs multiple instruments, or whether measurement is meaningless, it would be necessary to know how much of the multidimensionality can be captured by one measure and how situated mathematical identity is. Accordingly, this paper proposes a theoretical perspective on mathematical identity that is consistent with basic requirements of measurement. Moreover, characteristics of students’ mathematical identities are presented and the problem of “situatedness” is discussed. 相似文献
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Research on identity has been a growing domain in the contexts of teacher education and mathematics education; however, identity work has been explored to a much lesser extent, with a future orientation overlooked. In addition, earlier studies have not provided sufficient knowledge on how different elementary teacher education programs might facilitate pre-service teachers’ identity work. In this study, we compare future-oriented mathematical identity work through a narrative framework considering six pre-service teachers undergoing two different teacher education programs. All pre-service teachers reported having had negative experiences with mathematics during their school years. Based on the results we conclude that despite the striking similarities in pre-service teachers’ mathematical backgrounds, the ways in which these cases are conducting their identity work differ substantially. It seems that the main reasons for these differences are different emphases and pedagogical practices in mathematics education courses. Additionally, we further elaborate on our earlier conceptualisation of identity work. 相似文献
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This project highlights preschool teachers’ views of toddlers’ learning in mathematics. The Swedish national curriculum covers even the youngest children who are 1–3?years old. Interesting questions are thus: what should mathematics be for this age group and how should preschool teachers work with maths to achieve the curriculum objectives? Data were collected through interviews with six preschool teachers working in four different preschools. The data show that the teachers emphasize the body as very important for the learning process, which means that for these children, it is not a matter of simply talking about mathematical concepts, but experiencing them bodily. The teachers also report that they now pay more attention than previously to what material the children use and how they interact with it. They are more aware of how they organize and offer the various materials in the preschool and how this influences the way children use them and, consequently, their learning processes. 相似文献
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Nesrin Cengiz Kate Kline Theresa J. Grant 《Journal of Mathematics Teacher Education》2011,14(5):355-374
Studies show that extending students’ mathematical thinking during whole-group discussions is a challenging undertaking. To
better understand what extending student thinking looks like and how teachers’ mathematical knowledge for teaching (MKT) supports
teachers in their efforts to extend student thinking, the teaching of six experienced elementary school teachers was explored.
During group discussions, all six teachers created opportunities for extending student thinking about important mathematical
ideas and solution methods. Findings on the nature of these episodes include identification of individual instructional actions
and the ways in which teachers’ MKT was connected to these actions. 相似文献
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In this paper, based on a project funded by the UK Economic and Social Research Council considering how people position themselves in relation to popular representations of mathematics and mathematicians, we explore constructions of mathematicians in popular culture and the ways learners make meanings from these. Drawing on an analysis of popular cultural texts, we argue that popular discourses overwhelmingly construct mathematicians as white, heterosexual, middle‐class men, yet also construct them as ‘other’ through systems of binary oppositions between those doing and those not doing mathematics. Turning to the analysis of a corpus of 27 focus groups with school and university students in England and Wales, we explore how such images are deployed by learners. We argue that while learners’ views of mathematicians parallel in key ways popular discourses, they are not passively absorbing these as they are simultaneously aware of the clichéd nature of popular cultural images. 相似文献
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Gabriel J. Stylianides Andreas J. Stylianides George N. Philippou 《Journal of Mathematics Teacher Education》2007,10(3):145-166
There is a growing effort to make proof central to all students’ mathematical experiences across all grades. Success in this goal depends highly on teachers’ knowledge
of proof, but limited research has examined this knowledge. This paper contributes to this domain of research by investigating
preservice elementary and secondary school mathematics teachers’ knowledge of proof by mathematical induction. This research
can inform the knowledge about preservice teachers that mathematics teacher educators need in order to effectively teach proof to preservice teachers. Our analysis is based
on written responses of 95 participants to specially developed tasks and on semi-structured interviews with 11 of them. The
findings show that preservice teachers from both groups have difficulties that center around: (1) the essence of the base
step of the induction method; (2) the meaning associated with the inductive step in proving the implication P(k) ⇒ P(k + 1) for an arbitrary k in the domain of discourse of P(n); and (3) the possibility of the truth set of a sentence in a statement proved by mathematical induction to include values
outside its domain of discourse. The difficulties about the base and inductive steps are more salient among preservice elementary
than secondary school teachers, but the difficulties about whether proofs by induction should be as encompassing as they could
be are equally important for both groups. Implications for mathematics teacher education and future research are discussed
in light of these findings.
相似文献
| George N. PhilippouEmail: |
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S. G. Dani 《Resonance》2012,17(9):824-846
Modern mathematics, and modern science in general, was embraced enthusiastically in the Indian subcontinent quite early, thanks to a large extent to the tradition of learning going back to the ancient times. By the early decades of the 20th century the Indian mathematical community had made important research contributions in diverse areas of mathematics, including number theory, real and complex analysis, diffrential geometry, diffrential equations, algebra, combinatorial mathematics and applied mathematics. Interaction with world leaders in the field, reforms in the educational system, establishment of societies for actively pursuing study and discussion of mathematics, publication of mathematical journals, were some of the major progressive steps taken. Apart from the indigenous leadership, participation of a few enlightened foreigners in the process also paved the way for the advancement of the subject in the country. We recount here the story of the early developments of the mathematical scene in India, and of the various players involved. 相似文献
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M. Lynn Breyfogle 《Teachers and Teaching》2013,19(2):151-167
A secondary school mathematics teacher and mathematics education researcher engaged in a collaborative professional development experience that focused the teacher’s reflection on videotapes of his instruction. The purpose was to document and explore the teacher’s experiences with respect to his reflections and classroom practice while trying to create inquiry‐based mathematical discourse. Data included videotaped classroom observations, audiotaped interviews, and audiotaped focused reflection sessions. Analysis of the data revealed that over the course of the semester‐long collaboration the teacher exhibited, at various times, four different reflective states (‘explain but not question’, ‘question but not explain’, ‘question and explore’, and ‘exploring’). The identification of these reflective states suggests a more complex relationship between reflection and changing teachers’ practice than previously thought. The results of this article suggest that the interaction of teachers’ reflective activities and reflective states contribute to various kinds of teacher change in the classroom. 相似文献
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Based on the notions of social and socio‐mathematical norms we investigate how these are established during the interactions of pre‐service teachers who solve mathematical problems. Norms identified in relevant studies are found in our case too; moreover, we have found norms related to particular aspects of the problems posed. Our results show that most of these norms, once established, enhance the problem‐solving process. However, exceptions do exist, but they have a local orientation and a relatively small influence.
En s'appuyant sur les concepts des normes sociales et ‘socio‐mathématiques’, nous avons étudié comment ces normes se sont établies au cours des interactions entre les enseignants et les étudiants en activité de résolution des problèmes mathématiques. Aux résultats de la recherche apparaissent d'une part les mêmes normes qui ont été déjà remarquées à d'autres recherches relatives et d'autre part des normes liées plus particulièrement aux problèmes posés. Les résultats de la recherche montrent que dans la majorité des cas les normes aident le processus de la resolution des problèmes. Il existe bien sûr des exceptions, mais elles ont une influence et une orientation locale.
Basierend auf den Begriffen der sozialen und sozio‐mathematischen Normen untersuchen wir, wie diese in die Interaktionen von angehenden Lehrern beim Lösen von mathematischen Problemen einfliessen. Normen, welche in relevanten Studien identifiziert werden, wurden in unserem Fall ebenfalls gefunden. Wir haben ausserdem Normen gefunden, welche sich auf bestimmte Aspekte der Fragestellungen beziehen. Unsere Resultate zeigen, dass die meisten Normen, sind sie einmal etabliert, die Problemlösungsprozess verbessern. Es bestehen zwar Ausnahmen, doch diese haben eine lokale Orientierung und einen relativ kleinen Einfluss.
Basados en las nociones de la norma social y sociomatemática, estamos investigando cómo se establecen dichas normas durante las interacciones de los profesores en pre‐servicio, que resuelven problemas matemáticos. Las normas identificadas en estudios relevantes también se encuentran en nuestro caso; de hecho, hemos hallado normas relacionadas con aspectos particulares de los problemas expuestos. Nuestros resultados demuestran que la mayor parte de estas normas, una vez establecidas, mejoran el proceso de solución de problemas. Sin embargo, existen también excepciones pero éstas tienen una orientación local y una relativamente menor influencia. 相似文献
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The relation between teacher-set performance goals for 361 individual students and these students’ mathematics achievement was investigated. High performance goals were found to strongly relate to student performance, with an effect size of d = 0.80. The performance goals were set by the teachers at the end of a step-by-step procedure, consisting of initial teacher expectations, the use of data, and team input. This procedure was expected to decrease negative expectancy bias. Higher teacher performance goals than teachers’ initial expectations, so-called positive changes, were positively associated with the performance of initially low achievers. Initially high achievers, for whom the teachers made a positive change, performed worse than comparable students for whom initial expectation and final goal were the same. 相似文献
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Trinidad García Celestino Rodríguez Paloma González-Castro Julio Antonio González-Pienda Mark Torrance 《Metacognition and Learning》2016,11(2):139-170
Calibration, or the correspondence between perceived performance and actual performance, is linked to students’ metacognitive and self-regulatory skills. Making students more aware of the quality of their performance is important in elementary school settings, and more so when math problems are involved. However, many students seem to be poorly calibrated, with a tendency towards over-confidence. The present study analyzes the relationship between post-performance calibration accuracy and the metacognitive process shown by 524 fifth- and sixth-grade students while solving two math problems. After calculating a calibration index and establishing the stability of students’ judgments and actual performance, differences in the metacognitive process exhibited by students with different calibration accuracy (Accurate vs. Inaccurate groups) were analyzed. The emergence of different calibration patterns and differences in the metacognitive process as a function of mathematics achievement and grade level were also examined. Results indicated that: (a) students in the overall sample were little calibrated and over-confident, showing high stability in their judgments and actual performance across problems; (b) inaccurate students reported using information representation sub-processes (drawing/summarizing) less frequently, but writing and reviewing (and also correcting mistakes) more frequently than their accurate peers; and (c) differences in calibration patterns and the metacognitive process were found when achievement level was considered, whereas grade level did not generate any important effect. These findings suggest the usefulness of process-based measures to examine the metacognitive processes involved in making post-performance judgments, considering achievement and its possible mediating role in this relationship. 相似文献
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Tonia J. Land Tonya G. Bartell Corey Drake Mary Q. Foote Amy Roth McDuffie Erin E. Turner 《课程研究杂志》2019,51(4):471-493
Elementary mathematics curriculum materials can serve as a lever for instructional change. In this paper, we promote a particular kind of instructional change: supporting teachers in learning to integrate children’s multiple mathematical knowledge bases (MMKB), including children’s mathematical thinking and children’s home and community-based mathematical funds of knowledge, in instruction. A powerful means of supporting pre-service teachers in integrating children’s MMKB in instruction may be to scaffold teachers’ noticing of potential spaces in elementary mathematics curriculum materials for connecting to children’s MMKB and then developing practices for leveraging these spaces during instruction. We focus on existing and potential spaces in written curriculum materials, or curriculum spaces, so as to better support teachers in enacting curriculum that opens spaces for connecting to children’s MMKB. 相似文献
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CAO Yan-long LIU Yu-sheng MAO Jian YANG Jiang-xim 《浙江大学学报(A卷英文版)》2006,7(11):1810-1818
INTRODUCTION Tolerance is imperative for seamless integration of computer-aided design (CAD) and computer aided manufacturing (CAM) (Bjorke, 1989; Wu and Yang, 1999) and influences greatly the quality, process planning, measurement, cost and assembly of the product. Many researches have been conducted on tolerance (Baer, 1979; Requicha, 1982; 1983; 1992; Clement, 1991). Most of these researches are focused on tolerance analysis and tolerance synthesis. In some commercial CAD/CAM sy… 相似文献
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In this article, we demonstrate how I poems, part of the Listening Guide method, can help researchers to better understand the multiple voices that might comprise a student’s mathematical identity. We provide an analysis of one student’s interview as an example of this method. We argue that the use of the Listening Guide method illuminates the complexity of students’ mathematical identities, especially their fluid, negotiated, and sometimes contested character. In the example shared here, we show how others’ discourses become part of a student’s developing mathematical identity, as well as highlight how the addition of drawings enhances the Listening Guide method. 相似文献