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数学证明在数学教学中占据着重要的作用．在数学教学中,数学证明要教些什么呢？本文从让学生顺利地从实用性证明过渡到理性证明,要淡化形式、注重实质,要善于揭示过程、培养推理能力以及把握好合情推理和演绎推理的关系四个方面进行了阐述． 相似文献
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一个不等式的推广 总被引：2，自引：0，他引：2
从许多相关杂志上都能见到如下不等式 :若ｘ、ｙ∈Ｒ+,则 (ｘ2 +ｙ2 ) 12 >(ｘ3+ｙ3) 13. ( 1 )下面笔者给出式 ( 1 )的两个推广 :推广 1 :若ｘ、ｙ∈Ｒ+,ｍ、ｎ∈Ｎ且ｎ >ｍ ,则 (ｘｍ+ｙｍ) 1ｍ >(ｘｎ+ｙｎ) 1ｎ . ( 2 )推广 2 :若ａ1,ａ2 ,… ,ａｎ∈Ｒ+,且ｓ>ｔ>0 ,则事实上 ,式 ( 3 )又是式 ( 2 )的推广 ,因此我们只证明式 ( 3 ) .证明 :所证不等式等价于下列不等式∑ｎｉ=1ａｔｉ1ｔ∑ｎｉ=1ａｓｉ1ｓ>1 ,即 ａｓ1∑ａｓｉｔｓ +… +ａｓｎ∑ａｓｉｔｓ1ｔ >1 .( 4)令 ａｓ1∑ａｓｉ1ｓ =ｂ1,… ,ａｓｎ∑ａｓｉ1ｓ =ｂｎ,则ｂｉ… 相似文献
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一个有趣不等式的推广 总被引：1，自引：0，他引：1
对于任意实数a,b都有 ((a+b)/(2))((a2+b2)/(2))((a3+b3)/(2))≤(a6+b6)/(2),(1)当且仅当a=b时取等号. 相似文献
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不等式的证明是高中数学的一个重点,也是一个难点.不等式的证明方法灵活多样,其中比较法、综合法、分析法是证明不等式最基本的方法.高考中不等式的证明经常出现在与其它知识如函数、数列、解析几何的综合题中,许多考生显得极不适应,觉得尤为困难.本文将通过具体的实例与读者一起探讨不等式的证明中经常用到的若干技巧: 相似文献
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一个三角形不等式的巧证 总被引：1，自引：0，他引：1
《数学教学》2 0 0 1年第 2期问题 532是 :在△ ABC中 ,∠ A,∠ B,∠ C的对边 BC=a,CA= b,AB=c,试证明 :2 bcos C2 +2 ccos B2 >a+b+c. (1 )这是一个形式优美的不等式 ,第 3期给出化边为角的常规的证明方法 ,下面我们给出另一种简便证法 .分析 观察不等式 (1 ) ,我们设想 ,如果能够构造出以 2 bcos C2 ,2 ccos B2 ,a+b+c为边长的三角形 ,则 (1 )式成立就不言而喻了 ,于是我们自然得到如下证法 .图 1证明 过 A点作直线 l∥ BC,BB′平分∠ABC,CC′平分∠ ACB,且 BB′∩ l=B′,CC′∩ l=C′.再过点 B作 BD∥ CC′,BD∩ l=D… 相似文献
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Although studies on students’ difficulties in producing mathematical proofs have been carried out in different countries,
few research workers have focussed their attention on the identification of mathematical proof schemes in university students.
This information is potentially useful for secondary school teachers and university lecturers. In this article, we study mathematical
proof schemes of students starting their studies at the University of Córdoba (Spain) and we relate these schemes to the meanings
of mathematical proof in different institutional contexts: daily life, experimental sciences, professional mathematics, logic
and foundations of mathematics. The main conclusion of our research is the difficulty of the deductive mathematical proof
for these students. Moreover, we suggest that the different institutional meanings of proof might help to explain this difficulty.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
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Michal Tabach Ruthi Barkai Pessia Tsamir Dina Tirosh Tommy Dreyfus Esther Levenson 《International Journal of Science and Mathematics Education》2010,8(6):1071-1090
According to reform documents, teachers are expected to teach proofs and proving in school mathematics. Research results indicate
that high school students prefer verbal proofs to other formats. We found it interesting and important to examine the position
of secondary school teachers with regard to verbal proofs. Fifty high school teachers were asked to prove various elementary
number theory statements, to write correct and incorrect proofs that students may use, and to evaluate given justifications
to statements from elementary number theory. While all the participants provided correct proofs to the statements, our findings
indicate that teachers are not aware of students’ preference for verbal justifications. Also, about half of the teachers rejected
correct verbal justifications. They claimed that these justifications lacked generality and are mere examples. 相似文献
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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
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Pessia Tsamir 《Educational Studies in Mathematics》2007,65(3):255-279
This paper indicates that prospective teachers’ familiarity with theoretical models of students’ ways of thinking may contribute
to their mathematical subject matter knowledge. This study introduces the intuitive rules theory to address the intuitive,
same sides-same angles solutions that prospective teachers of secondary school mathematics come up with, and the proficiency they acquired during
the course “Psychological aspects of mathematics education”. The paper illustrates how drawing participants’ attention to
their own erroneous applications of same sides-same angles ideas to hexagons, challenged and developed their mathematical knowledge. 相似文献
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This paper addresses the accumulating knowledge of prospective teachers of secondary school mathematics and their acquired
proficiency during the course “Psychological aspects of mathematics education,” in which we discussed theoretical models including
the intuitive rules theory. Participants’ performances are examined by means of an extensive report of two episodes, one during
the course and one afterwards. These episodes marked different stages in the prospective teachers’ analysis of their own and
of students’ solutions, which led me to conclude that exposing prospective teachers to the intuitive rules theory is important,
since their familiarity with the theory provided them with a tool to reflect on their own mathematical solutions (subject
matter knowledge; SMK), on others’ solutions, and on the tasks (pedagogical content knowledge; PCK). 相似文献
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Konstantinos A. Sdrolias Triandafillos A. Triandafillidis 《Educational Studies in Mathematics》2008,67(2):159-169
Students mainly perceive the transition to secondary school as an opportunity to start their school career anew. Reality often
proves them wrong, especially in the case of mathematics. In our paper, we briefly discuss children’s transition to secondary
school as both an opportunity and a problem, with reference to the Greek context. In discussions about the transition to secondary
school and its effect on the teaching and learning of mathematics, primary and secondary school teachers in Greece often depict
school mathematics as a “chain” of concepts and procedures. With this metaphor as our reference point, we discuss how ideologies
about the teaching of mathematics are enacted in both school levels. We will base our discussion on the analysis of extracts
taken from dialogues in primary and secondary school mathematics classrooms in Greece. In our analysis, we employ a Peircean
view of semiotics in an attempt to conceptualize students’ rushed introduction to rigor in justifying mathematical statements
in secondary school. These extracts are part of a longitudinal study that aimed, on the one hand, to pinpoint discontinuities
and continuities in the teaching of geometry between primary and secondary schools and, on the other, to investigate whether
a set of curriculum material that we designed could serve as a link in the teaching of geometry between the two school levels. 相似文献
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This study is grounded in the theoretical position that solving problems in different ways creates mathematical connections
when learning and teaching mathematics. It acknowledges the central role teachers play in providing students with learning
opportunities, and it is based on the empirical finding that mathematics teachers are reluctant to solve problems in different
ways in the classroom. In this paper we address the contradiction between theory-based recommendations and school mathematics
practice. Based on analysis of individual interviews and two group meetings with 12 Israeli secondary school mathematics teachers,
we demonstrate that in the context of multiple-solution connecting tasks this discrepancy is caused by the situated nature
of the teachers’ knowledge. We also reveal the complex relationship between different types of teacher knowledge and argue
the significance of developing a common language between members of the mathematics education community, including teacher
educators and researchers.
The names of the teachers have been changed to protect their privacy. 相似文献
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Teachers' beliefs about school mathematics and mathematicians' mathematics and their relationship to practice 总被引：1，自引：0，他引：1
Kim Beswick 《Educational Studies in Mathematics》2012,79(1):127-147
There is broad acceptance that mathematics teachers’ beliefs about the nature of mathematics influence the ways in which they
teach the subject. It is also recognised that mathematics as practised in typical school classrooms is different from the
mathematical activity of mathematicians. This paper presents case studies of two secondary mathematics teachers, one experienced
and the other relatively new to teaching, and considers their beliefs about the nature of mathematics, as a discipline and
as a school subject. Possible origins and future developments of the structures of their belief systems are discussed along
with implications of such structures for their practice. It is suggested that beliefs about mathematics can usefully be considered
in terms of a matrix that accommodates the possibility of differing views of school mathematics and the discipline. 相似文献
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In this study we investigate a strategy for engaging high school mathematics teachers in an initial examination of their teaching in a way that is non-threatening and at the same time effectively supports the development
of teachers’ pedagogical content knowledge [Shulman (1986). Educational Researcher, 15(2), 4–14]. Based on the work undertaken by the QUASAR project with middle school mathematics teachers, we engaged a group
of seven high school mathematics teachers in learning about the Levels of Cognitive Demand, a set of criteria that can be
used to examine mathematical tasks critically. Using qualitative methods of data collection and analysis, we sought to understand
how focusing the teachers on critically examining mathematical tasks influenced their thinking about the nature of mathematical
tasks as well as their choice of tasks to use in their classrooms. Our research indicates that the teachers showed growth
in the ways that they consider tasks, and that some of the teachers changed their patterns of task choice. Further, this study
provides a new research instrument for measuring teachers’ growth in pedagogical content knowledge.
An earlier version of this paper was presented at the American Educational Research Association Annual Meeting, New Orleans,
LA, April 2002. 相似文献