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CALCULUS STUDENTS’ AND INSTRUCTORS’ CONCEPTUALIZATIONS OF SLOPE: A COMPARISON ACROSS ACADEMIC LEVELS
Courtney Nagle Deborah Moore-Russo Janine Viglietti Kristi Martin 《International Journal of Science and Mathematics Education》2013,11(6):1491-1515
This study considers tertiary calculus students’ and instructors’ conceptualizations of slope. Qualitative techniques were employed to classify responses to 5 items using conceptualizations of slope identified across various research settings. Students’ responses suggest that they rely on procedurally based conceptualizations of slope, showing little evidence of covariational reasoning. In contrast, instructors’ responses demonstrated a multi-dimensional understanding of slope as a functional property, which applies to real-world situations and plays an integral role in the development of key calculus concepts. While relatively diverse, the instructors’ responses seldom reported determining increasing or decreasing trends of a line from its slope. This conceptualization was used frequently by students and could help them better understand how slope ties to positive and negative derivatives. The most frequently used conceptualizations for students in this study align with past research findings on the emphasis of the secondary mathematics curriculum, supporting the possibility of cultural influences (academic and geographic) on individuals’ conceptualizations of slope. Thus, this study provides valuable insight into conceptualizations of slope and provides direction for future research on slope and the broader topic of cultural influences on mathematical meaning. 相似文献
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许文超 《洛阳师范学院学报》1996,(2)
本文从积分理论的几个主要方面（可积范围、收敛条件等）对黎曼积分进行了评析，认真地分析了黎曼积分相对于牛顿积分和柯西积分所表现出来的优势，以及相对于勒贝格积分所暴露出来的局限性． 相似文献
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高等数学中的积分包含不定积分和定积分（R(黎曼)积分）两类,不定积分是从逆运算的角度,把积分看作微分运算的逆运算,定积分则是从求极限的角度,把积分看作是一类特殊形式的和数极限。从两种积分的概念入手,通过例题分析来揭示这两种积分的内在关系。 相似文献
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Daniel Lee McGee Rafael Martinez-Planell 《International Journal of Science and Mathematics Education》2014,12(4):883-916
Tracing the path from a numerical Riemann sum approximating the area under a curve to a definite integral representing the precise area in various texts and online presentations, we found 3 semiotic registers that are used: the geometric register, the numerical register, and the symbolic register. The symbolic register had 3 representations: an expanded sum, a sum in sigma notation, and the definite integral. Reviewing the same texts, we found that in the presentation of double and triple integrals, not a single textbook continues to present the numerical register and the expanded sum representation of the symbolic register. They are implied and the expectation appears to be that students no longer need them. The omission of these representations is quite ubiquitous and correspondingly affects millions of students. Materials that present the missing numerical register representation and the expanded sum representation of the symbolic register throughout topics associated with double and triple integrals have been created. This paper presents the results of a clinical study on the improvement of student comprehension of multivariable integral topics when these representations are included. 相似文献
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情景模拟法在高校微积分教学中的应用——以模拟运动过程为例 总被引：1，自引：0，他引：1
丁海峰 《北京城市学院学报》2011,(6):66-68
本文针对高校微积分教学的对象主要为高校低年级学生的特点,探讨情景模拟法在高校微积分教学中的应用。本文重点选取物理中的运动过程作为基本情景对微积分的基本概念进行分析,分别选取匀速直线运动、匀加速直线运动和变加速直线运动等,通过从微积分角度对它们重新分析,很自然地得出微积分理论中的定积分、不定积分、二重积分等几个问题的基本原理,以此表明情景模拟法在微积分教学中的有效性。 相似文献
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研究了Riemann积分与Lebesgue之间的关系，在给出了正常Riemann积分与Lebesgue积分的联系的同时，重点研究了广义Riemann积分与Lebesgue积分的关系，即函数f(x)在[a,b]上Riemann可积时，f(x)在[a,b]上也Lebesgue可积，并且两积分分值相等；但广义Riemann积分与Lebesgue积分之间的关系则不尽然．当无穷积分或瑕积分在区间绝对收敛时，则函数f(x)在此区间也Lebesgue可积，并且两积分分值相等，当无穷积分或瑕积分在区间条件收敛时，则函数f(x)在此区间不Lebesgue可积． 相似文献
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Núria Planas 《Educational Studies in Mathematics》2018,97(3):215-234
Many mathematical concepts may have prototypical images associated with them. While prototypes can be beneficial for efficient thinking or reasoning, they may also have self-attributes that may impact reasoning about the concept. It is essential that mathematics educators understand these prototype images in order to fully recognize their benefits and limitations. In this paper, I examine prototypes in a context in which they seem to play an important role: graphical representations of the calculus concept of the definite integral. I use student data to empirically describe the makeup of the definite integral prototype image, and I report on the frequency of its appearance among student, instructor, and textbook image data. I end by discussing the possible benefits and drawbacks of this particular prototype, as well as what the results of this study may say about prototypes more generally. 相似文献
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Students are commonly asked to learn declarative concepts in many courses. One strategy students report using involves generating concrete examples of abstract concepts. If students have difficulties evaluating the quality of their generated examples, then instructors will need to provide students with appropriate scaffolds or feedback to improve judgmentaccuracy. No prior research has investigated if students can accurately evaluate the quality of the examples they generate, which was the first aim of the current research. The second aim of this research was to investigate the extent to which providing feedback while students evaluate their generated examples can improve the accuracy of their example-quality judgments. In two experiments, students generated examples for declarative concepts from social psychology and then judged the quality of their examples. When making judgments, students received no feedback (in which they were only given the key term), full definition feedback (in which they were shown the definition of the declarative concept) or idea unit feedback (in which they first evaluated if they represented each idea unit of the definition within their example). Outcomes showed that students were overconfident when judging the quality of their examples, specifically for commission errors (i.e., examples that were entirely incorrect). Surprisingly, full definition and idea unit feedback did not help students improve the accuracy of their example-quality judgments. Thus, until scaffolds are discovered to reduce student overconfidence, instructors will need to assist in evaluating generated examples as students use this strategy to learn declarative concepts. 相似文献
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D. P. Patil 《Resonance》2006,11(11):61-80
Riemann had revolutionized the fields of analysis, geometry and mathematical physics. His ideas concerning geometry of space
had a profound effect on the development of modern theoretical physics. Riemannian manifolds, Riemann surfaces, the Cauchy—Riemann
equations, the Riemann hypothesis-all these and more are packed into his one-volume collected works. Riemann clarified the
notion of integration by defining, a little over 135 years ago, what we now call theRiemann integral. This article is devoted to a study of the Riemann integral. 相似文献
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高等数学中积分学是一个复杂的知识体系,学生在学习的过程中,各种积分的定义、性质及计算经常混淆。为了方便学生学习,将定积分,二重积分,三重积分,第一型曲线积分,第一型曲面积分统一定义为几何形体上的积分,给出统一的性质,然后针对不同的几何形体研究计算方法。 相似文献
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李伟 《韩山师范学院学报》2006,27(3):6-9,13
利用函数的局部小黎曼和性质(LSRS)和函数列的一致局部小黎曼和性质(ULSRS), 建立了Mcshane积分的局部小黎曼和收敛定理(定理1)．然后,通过定理2证明了Lebesgue积分的控制收敛定理为该定理的一个推论． 相似文献
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This study aimed to determine the effect of a computer-assisted instruction method using GeoGebra on achievement of prospective secondary mathematics teachers in the definite integral topic and to determine their opinions about this method. The study group consisted of 35 prospective secondary mathematics teachers studying in the mathematics education program at a state university in Turkey. The study was carried out using an embedded design, and the Definite Integral Knowledge Test and an opinion form were used for data collection. Upon analyzing the data, the computer-assisted instruction method using GeoGebra was found to positively contribute to the success of teaching the definite integral topic. Prospective teachers stated that this method should be used in math courses as it creates a fun and interesting environment with dynamic learning elements, provides visualization and opportunities to learn mathematics through practice and exercises, enables thorough understanding and explication of skills, and makes way for conceptual learning instead of memorizing. Furthermore, this study was found to facilitate conceptual learning of the relationship between the lower sum, upper sum and Riemann sum. 相似文献
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对MCMC方法及其收敛进行了简要回顾，利用黎曼和方法模拟解决了高维复杂积分的近似问题。然后设计了一种收敛诊断估计量，用它来保证得到的样本是收敛的，这就确保了近似积分的精度。最后给出一个实例。 相似文献
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周克元 《牡丹江教育学院学报》2007,(5):143
通过函数极限的连续性定义与函数黎曼积分的连续性定义之间的关系,利用函数极限的离散型定义给出了函数黎曼积分的离散型定义,并证明了函数黎曼型定义与连续型定义是等价的。 相似文献
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在微积分课程中，极限作为一个极为重要的基本概念，贯穿于该学科始终。微积分中其他一些重要概念，如导数、微分、积分、级数等都用极限来定义，理解和掌握极限对微积分课程的学习至关重要。因此，培养学生的极限思维，对于学生准确把握该课程的概念与体系，运用极限的思维方法解决相关问题，都有非常重要的作用。 相似文献
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本文给出测度函数的定义,并得到如下结果:可测函数f(x)在可测集E上的勒贝格积分等于f(x)在E上的测度函数的黎曼积分。从而给出了证明勒贝格积分性质的一种新方法。 相似文献
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李景菊 《十堰职业技术学院学报》2010,23(3):9-11
当前沟通无处不在,且沟通在学生管理工作中显得越来越重要,它涵盖了学生管理工作的方方面面和整个过程,沟通育人观念应运而生。学生教育管理工作者应以"沟通育人观"作为工作理念,掌握适当的沟通方法和技巧并应用于工作中。 相似文献