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1.
Proof is an important topic in the area of mathematics curriculum and an essential aspect of mathematical competence. However, recent studies have revealed wide gaps in student's understanding of proof. Furthermore, effective teaching to prove, for example, by Schoenfeld's approach, is a real challenge for teachers. A very powerful and empirically well founded method of learning mathematics, which is also relatively easy to implement in the classroom, is learning through worked-out examples. It is, however, primarily suited for algorithmic content areas. We propose the concept of using heuristic worked-out examples, which do not provide an algorithmic problem solution but offer instead heuristic steps that lead towards finding a proof. We rely on Boero's model of proving in designing the single sub-steps of a heuristic example. We illustrate our instructional idea by presenting an heuristic example for proving that the interior angles in any triangle add up to 180°. 相似文献
2.
Keiki Takadama Takao Terano Katsunori Shimohara Koichi Hori Shinichi Nakasuka 《Computational & Mathematical Organization Theory》1999,5(3):229-252
The concepts of organizational learning in organization and management science cover a very wide range of organization-related activities in organization. Since socially situated intelligence is one of such activities, this paper makes the concept of organizational learning operational from the computational viewpoint for investigating socially situated intelligence. In particular, this paper focuses on the characteristics of multiagent learning as one kind of socially situated intelligence, and analyzes them using four operationalized learning mechanisms in organizational learning. A careful investigation on the characteristics of multiagent learning has revealed the following implications: (1) there are two levels in the learning mechanisms for multiagent learning (the individual level and organizational level) and each mechanism is divided into two types (single- and double-loop learning). The integration of these four learning mechanisms improves socially situated intelligence; and (2) the following properties support socially situated intelligence: (a) different dimensions in learning mechanisms, (b) interaction among various levels and types of learning mechanisms in addition to interaction among agents, and (c) combination of exploration at an individual level and exploitation at an organizational level. 相似文献
3.
在工科数学分析课程的教学中,对一些经典内容进行了针对性处理,加强了课程知识的前后联系,提高了学生的学习兴趣和效果. 相似文献
4.
5.
Sergiy Klymchuk 《International Journal of Mathematical Education in Science & Technology》2013,44(8):1260-1265
The paper describes a personal experience with using counterexamples as a pedagogical strategy in the teaching and learning of an introductory calculus course at a university of technology. 相似文献
6.
Sven Trenholm Lara Alcock Carol Robinson 《International Journal of Mathematical Education in Science & Technology》2013,44(8):1197-1221
Research suggests it is difficult to learn mathematics in the fully asynchronous online (FAO) instructional modality, yet little is known about associated teaching and assessment practices. In this study, we investigate FAO mathematics assessment and feedback practices in particular consideration of both claims and findings that these practices have a powerful influence on learning.A survey questionnaire was constructed and completed by 70 FAO undergraduate mathematics instructors, mostly from the USA, who were each asked to detail their assessment and feedback practices in a single FAO mathematics course. Alongside these questions, participants also answered the 16-item version of the Approaches to Teaching Inventory. In addition, a novel feedback framework was also created and used to examine how feedback practices may be related to participants' approaches to teaching.Results show that assessment and feedback practices are varied and complex: in particular, we found there was not a simple emphasis on summative assessment instruments, nor a concomitant expectation these would always be invigilated. Though richer assessment feedback appears to be emphasized, evidence suggests this feedback may not be primarily directed at advancing student learning. Moreover, we found evidence of a reliance on computer--human interactions (e.g. via computer-assisted assessment systems) and further evidence of a decline in human interactions, suggesting a dynamic that is both consistent with current online learning theory and claims FAO mathematics courses are becoming commodified. Several avenues for further research are suggested. 相似文献
7.
建构主义主张学生是教学活动的认知主体,教师由传统的知识传授者转变为学生意义上建构的协助者和促进者,教学策略也发生显著变化.从教育者的角度出发,探讨了建构主义下微积分教师的教学策略. 相似文献
8.
Muniru Aderemi Asiru 《International Journal of Mathematical Education in Science & Technology》2013,44(6):944-949
The Sumudu transform is an integral transform introduced to solve differential equations and control engineering problems. The transform possesses many interesting properties that make visualization easier and application has been demonstrated in the solution of partial differential equations, integral equations, integro-differential equations and dynamic/dynamic-differential systems. In this note, application of the method of Sumudu transform to the solution of discrete dynamic systems is described. Derivations find ready application in recurrence relations. Solutions are obtained to certain first degree, second degree and third degree dynamic systems. 相似文献
9.
The classification problem statement of multicriteria decision analysis is to model the classification of the alternatives/actions according to the decision maker's preferences. These models are based on outranking relations, utility functions or (linear) discriminant functions. Model parameters can be given explicitly or learnt from a preclassified set of alternatives/actions.In this paper we propose a novel approach, the Continuous Decision (CD) method, to learn parameters of a discriminant function, and we also introduce its extension, the Continuous Decision Tree (CDT) method, which describes the classification more accurately.The proposed methods are results of integration of Machine Learning methods in Decision Analysis. From a Machine Learning point of view, the CDT method can be considered as an extension of the C4.5 decision tree building algorithm that handles only numeric criteria but applies more complex tests in the inner nodes of the tree. For the sake of easier interpretation, the decision trees are transformed to rules. 相似文献
10.
Tony Gardiner 《ZDM》2004,36(2):67-76
It is generally accepted that proof is central to mathematics. There is less agreement about how proof should be introduced
at school level. We propose an approach—based on the systematic exploitation of structured calculation—which builds the notion
of objective mathematical proof into the curriculum for all pupils from the earliest years. To underline the urgent need for
such a change we analyse the current situation in England—including explicit evidence of the extent to which current instruction
fails even the best students. 相似文献
11.
Andrea Young Kathryn Bruhns 《International Journal of Mathematical Education in Science & Technology》2016,47(6):961-971
In this article, we give two examples of creating portable chalkboards using chalkboard paint for students to use during cooperative learning. This provides a creative method for professors to facilitate active learning in the undergraduate mathematics classroom. 相似文献
12.
More than a decade of research and innovation in using computer-based graphing and simulation environments has encouraged
many of us in the research community to believe important dimensions of calculus-related reasoning can be successfully understood
by young learners. This paper attempts to address what kinds of calculus-related insights seem to typify this early form of
calculus reasoning. The phrase “qualitative calculus” is introduced to frame the analysis of this “other” calculus. The learning
of qualitative calculus is the focus of the synthesis. The central claim is that qualitative calculus is a cognitive structure
in its own right and that qualitative calculus develops or evolves in ways that seem to fit with important general features
of Piaget's analyses of the development of operational thought. In particular, the intensification of rate and two kinds of
reversibility between what are called “how much” (amount) and “how fast” (rate) quantities are what interactively, and collectively,characterize
and help to define understanding qualitative calculus. Although sharing a family resemblance with traditional expectations
of what it might mean to learn calculus, qualitative calculus does not build from ratio- or proportion-based ideas of slope
as they are typically associated with defining rate. The paper does close, however, with a discussion of how understanding
qualitative calculus can support and link to the rate-related literature of slope, ratio and proportion. Additionally, curricular
connections and implications are discussed throughout to help illustrate and explore the significance of learning qualitative
calculus.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
13.
We demonstrate the power of Variation Theory as an analytical tool used to understand the underlying conceptual structure of mathematics lessons taught by English primary school teachers. We study excerpts of three lessons that are posted on a professional website. We show how lesson analysis using variation allows us to focus on what is made available to be learnt in the lesson excerpts. We identify some differences in their use of dimensions of variation and the associated ranges of change and discuss how suitable patterns of variation and invariance might differ according to the nature of the learning focus. We reflect on the value of our analytical approach. 相似文献
14.
Melva R. Grant William Crombie Mary Enderson Nell Cobb 《International Journal of Mathematical Education in Science & Technology》2016,47(6):823-836
Access to advanced study in mathematics, in general, and to calculus, in particular, depends in part on the conceptual architecture of these knowledge domains. In this paper, we outline an alternative conceptual architecture for elementary calculus. Our general strategy is to separate basic concepts from the particular advanced techniques used in their definition and exposition. We develop the beginning concepts of differential and integral calculus using only concepts and skills found in secondary algebra and geometry. It is our underlining objective to strengthen students' knowledge of these topics in an effort to prepare them for advanced mathematics study. The purpose of this reconstruction is not to alter the teaching of limit-based calculus but rather to affect students' learning and understanding of mathematics in general by introducing key concepts during secondary mathematics courses. This approach holds the promise of strengthening more students' understanding of limit-based calculus and enhancing their potential for success in post-secondary mathematics. 相似文献
15.
详细介绍了我们在主讲《工科数学分析》课程中以问题驱动的研究性微积分教学模式和培养学生创新能力的探索与实践. 相似文献
16.
Peter Bock 《Annals of Operations Research》1988,16(1):33-52
The classical approach to the acquisition of knowledge in artificial intelligence has been to program the intelligence into the machine in the form of specific rules for the application of the knowledge: expert systems. Unfortunately, the amount of time and resources required to program an expert system with sufficient knowledge for non-trivial problem-solving is prohibitively large. An alternative approach is to allow the machine tolearn the rules based upon trial-and-error interaction with the environment, much as humans do. This will require endowing the machine with a sophisticated set of sensors for the perception of the external world, the ability to generate trial actions based upon this perceived information, and a dynamic evaluation policy to allow it to measure the effectiveness of its trial actions and modify its repertoire accordingly. The principles underlying this paradigm, known ascollective learning systems theory, have already been applied to sophisticated gaming problems, demonstrating robust learning and dynamic adaptivity.The fundamental building block of a collective learning system is thelearning cell, which may be embedded in a massively parallel, hierarchical data communications network. Such a network comprising 100 million learning cells will approach the intelligence capacity of the human cortex. In the not-too-distant future, it may be possible to build a race of robotic slaves to perform a wide variety of tasks in our culture. This goal, while irresistibly attractive, is most certainly fraught with severe social, political, moral, and economic difficulties.This paper was given as an invited talk on the 12th Symposium on Operations Research, University of Passau, September 1987. 相似文献
17.
建构主义学习理论下的数学教学原则与策略 总被引:1,自引:0,他引:1
本文从建构主义学习理论出发,分析了传统数学教学中存在的弊端。提出了建构主义学习理论下的数学教学原则和策略,对于充分认识建构主义学习、教学理论,对促进数学教学改革,提高数学教学质量具有重要的意义. 相似文献
18.
A nonequivalent, control group design was used to investigate student achievement in secondary chemistry. This study investigated the effect of process‐oriented guided inquiry learning (POGIL) in high school chemistry to reduce alternate conceptions related to the particulate nature of matter versus traditional lecture pedagogy. Data were collected from chemistry students in four large high schools and were analyzed using analysis of covariance. The results show that POGIL pedagogy, as opposed to traditional lecture pedagogy, resulted in fewer alternate conceptions related to the particulate nature of matter. Male and female students in the POGIL group posted better posttest scores than their traditional group peers. African‐American and Hispanic students in the POGIL group exhibited achievement gains consistent with Caucasian and Asian students. Further studies are needed to determine the value of POGIL to address achievement gap concerns in chemistry. 相似文献
19.
Richard A. Hoban 《International Journal of Mathematical Education in Science & Technology》2019,50(4):603-616
Many students do not have a deep understanding of the integral concept. This article defines what a deep understanding of the integral is in respect to integration involving one independent variable; briefly discusses factors which may inhibit such an understanding; and then describes the design of a mathematical resource for introducing students to the integral concept. The resource addresses a number of challenges when introducing the integral: (1) choosing an appropriate, intuitive context which gives meaning to the symbols in the integral expression; (2) aiding the transfer of the integral expression to different contexts via using the Riemann sum in an informal way so that students can see and interpret the rectangles which are inherent in this sum; and (3) the gradual formalizing of the Riemann sum and its linkage with the Fundamental Theorem of Calculus. The resource has been used over a number of years at this university amongst first-year undergraduate science and engineering students. Anecdotal evidence would suggest that the resource is beneficial. 相似文献
20.
Estrella Johnson Jessica Ellis Chris Rasmussen 《International Journal of Mathematical Education in Science & Technology》2016,47(4):491-504
This paper is based on a large-scale empirical study designed to investigate Calculus I programmes across the United States to better understand the relationship between instructors' concerns about coverage, instructional practices, and the nature of the material covered. We found that there was no association between instructors feeling pressured to go through material quickly to cover all the required topics and intended pacing. Furthermore, our results suggest that both intended pacing and feelings of pressure are poor indicators of instructional practices. 相似文献