Students’ conceptual understanding of a function and its derivative in an experimental calculus course

Calculus has been witnessing fundamental changes in its curriculum, with an increased emphasis on visualization. This mode for representing mathematical concepts is gaining more strength due to the advances in computer technology and the development of dynamical mathematical software. This paper focuses on the understanding of the function and its derivative as viewed by students of a reformed Calculus 1 course offered in two experimental sections at the Lebanese American University in Beirut, Lebanon. Results have shown that the general approach adopted in the course proved to be unpopular for a great majority of the students, but rewarding for others. Interviews conducted with some students and a study of their performance on very specific exam questions reveal that for most students, the algebraic representation of a function still dominated their thinking; however, these students showed an almost complete understanding of the derivative, particularly the idea of the instantaneous rate of change and/or the slope of a curve at a given point. Furthermore, very few of these students referred to the mechanical methods for finding derivatives.     

The fractional knowledge and algebraic reasoning of students with the first multiplicative concept

To understand relationships between students’ quantitative reasoning with fractions and their algebraic reasoning, a clinical interview study was conducted with 18 middle and high school students. Six students with each of three different multiplicative concepts participated. This paper reports on the fractional knowledge and algebraic reasoning of six students with the most basic multiplicative concept. The fractional knowledge of these students was found to be consistent with prior research, in that the students had constructed partitioning and iteration operations but not disembedding operations, and that the students conceived of fractions as parts within wholes. The students’ iterating operations facilitated their work on algebra problems, but the lack of disembedding operations was a significant constraint in writing algebraic equations and expressions, as well as in generalizing relationships. Implications for teaching these students are discussed.     

Students’ understanding of mathematical objects in the context of transformational geometry: Implications for constructing and understanding proofs

This research explored students’ views of geometric objects through the implementation of a curriculum module that allowed them to explore the relationships between transformational geometry and linear algebra. The majority of the students were middle and secondary mathematics education majors enrolled in a one-semester geometry course that is aimed at prospective teachers. A preponderance of the evidence suggests that the participating students, for the most part, viewed isometries operationally and viewed geometric objects (triangle, etc.) as “perceived.” Results also suggest that these two views influenced the students’ abilities to understand and to construct geometric proofs in transformational geometry.     

Folding back and the dynamical growth of mathematical understanding: Elaborating the Pirie-Kieren Theory

The study reported here extends the work of Pirie and Kieren on the nature and growth of mathematical understanding. The research examines in detail a key aspect of their theory, the process of ‘folding back’, and develops a theoretical framework of categories and sub-categories that more fully describe the phenomenon. This paper presents an overview of this ‘framework for folding back’, illustrates it with extracts of video data and elaborates on its key features. The paper also considers the implications of the study for the teaching and learning of mathematics, and for future research in the field.     

The role of operating premises and reasoning paths in upper elementary students’ problem solving

The validity of students’ reasoning is central to problem solving. However, equally important are the operating premises from which students’ reason about problems. These premises are based on students’ interpretations of the problem information. This paper describes various premises that 11- and 12-year-old students derived from the information in a particular problem, and the way in which these premises formed part of their reasoning during a lesson. The teacher’s identification of differences in students’ premises for reasoning in this problem shifted the emphasis in a class discussion from the reconciliation of the various problem solutions and a focus on a sole correct reasoning path, to the identification of the students’ premises and the appropriateness of their various reasoning paths. Problem information that can be interpreted ambiguously creates rich mathematical opportunities because students are required to articulate their assumptions, and, thereby identify the origin of their reasoning, and to evaluate the assumptions and reasoning of their peers.     

Students’ reflections on their learning experiences: lessons from a longitudinal study on the development of mathematical ideas and reasoning

This paper characterizes the views on mathematical learning of five high school students based on the students’ reflections on their mathematical experiences in a longitudinal study that focused on the development of mathematical ideas and reasoning in particular research conditions. The students’ views are presented according to five themes about learning which describe the students’ views on the nature of knowledge and what it means to know, source of knowledge, motivation to engage in learning, certainty in knowing, and how the students’ views vary with particular areas of mathematical activity. The study addresses the need for more research on epistemological beliefs of students below college age. In particular, the results provide evidence that challenge the existing assumption that, prior to college, students exhibit naïve epistemological beliefs.     

Reinventing the formal definition of limit: The case of Amy and Mike

Relatively little is known about how students come to reason coherently about the formal definition of limit. While some have conjectured how students might think about limits formally, there is insufficient empirical evidence of students making sense of the conventional ?-δ definition. This paper provides a detailed account of a teaching experiment designed to produce such empirical data. In a ten-week teaching experiment, two students, neither of whom had previously seen the conventional ?-δ definition of limit, reinvented a formal definition of limit capturing the intended meaning of the conventional definition. This paper focuses on the evolution of the students’ definition, and serves not only as an existence proof that students can reinvent a coherent definition of limit, but also as an illustration of how students might reason as they reinvent such a definition.     

复杂网络演化中的“熵减点”研究:以微博传播网络的演化为例

基于标准网络结构熵这一度量"有序性"的指标,以微博传播网络为例,首次将这一宏观指标运用到网络的微观演化研究中.首先,根据"网络大V"参与情况的不同,将微博传播网络分为"无V型网络"、"单V型网络"和"多V型网络".其次,依托实际数据,在标准结构熵的视角下,分别运用枚举法、跟踪法、介入法探讨了这三种网络的演化特征,发现了网络中不同节点的微观演化对宏观"有序性"的差异性影响,其中"大V"节点具备明显的"熵减"效应.最后,归纳提出"熵减点"的相关概念,并提出一套用来识别复杂网络重要节点的"熵减点判别法".方法相较于传统方法具有一些优点且适用性较强,可以扩展到其他复杂系统与复杂网络研究中,为网络结构熵理论的实际应用提供了一些思路.     

Exploring the use of narrative analysis as an operational research method: A case study in voluntary sector evaluation

Operational research frequently has to deal with situations where the perceptions and views of the various stakeholders involved may be quite different. One such situation is provided by the case of the evaluation of the work of voluntary sector groups, where concepts such as quality are frequently held to be the desired objectives whose achievement should be evaluated, yet where quite different perceptions of these concepts are held by the various parties to the evaluation. Through the use of a case study, this paper illustrates how approaches to narrative analysis like actant analysis and deconstruction can be used alongside other soft OR methods to enable negotiation of common understandings of important concepts like quality.     

Understanding multidigit whole numbers: The role of knowledge components,connections, and context in understanding regrouping 3+-digit numbers

This case study of a PST's understanding of regrouping with multidigit whole numbers in base-10 and non-base-10 contexts shows that although she seems to have all the knowledge elements necessary to give a conceptually based explanation of regrouping in the context of 3-digit numbers, she is unable to do so. This inability may be due to a lack of connections among various knowledge components (conceptual knowledge) or a lack of connections between knowledge components and context (strategic knowledge). Although she exhibited both conceptual and strategic knowledge of numbers while regrouping 2-digit numbers, her struggles in explaining regrouping 3-digit numbers in the context of the standard algorithms indicate that explaining regrouping with 3-digit is not a mere extension of doing so for 2-digit numbers. She also accepts an overgeneralization of the standard algorithms for subtraction to a time (mixed-base) context, indicating a lack of recognition of the connections between the base-10 contexts and the standard algorithms. Implications for instruction are discussed.     

The art and the science of British algebra: A study in the perception of mathematical truth

This paper investigates the origins of the concept of mathematical truth by focusing on the development of algebra in England in the early 19th century. In particular, it investigates the reasons why the English, despite their attention to the elements of abstract algebra, never produced a system comparable to modern algebra. Special consideration is given to the works of George Peacock, Augustus DeMorgan, William Whewell, and John Herschel. It is argued that what separated the early development of English algebra from modern algebra is a fundamental difference between 19th- and 20th-century views of truth.     

Exploring the VCG mechanism in combinatorial auctions: The threshold revenue and the threshold-price rule

We explore interesting potential extensions of the Vickrey–Clarke–Groves (VCG) rule under the assumption of players with independent and private valuations and no budget constraints. First, we apply the VCG rule to a coalition of bidders in order to compute the second price of the coalition. Then, we introduce and formulate the problem of determining that partition of players into coalitions which maximize the auctioneer’s revenue in the case whereby such coalitions take part to a VCG auction each one as a single agent; in particular, we provide an integer linear formulation of this problem. We also generalize this issue by allowing players to simultaneously belong to distinct coalitions in the case that players’ valuation functions are separable. Finally, we propose some applications of these theoretical results. For instance, we exploit them to provide a class of new payment rules and to decide which bids should be defined as the highest losing ones in combinatorial auctions.     

基于遗传算法优化BP神经网络在原油产量预测中的应用:以大庆油田BED试验区为例

油田产量预测工作一直是油田开发中的一项重要工作,许多传统的回归模型以及智能算法都已经在油田产量预测中有了应用.虽然神经网络以其较强的非线性拟合能力.而得到广泛应用,但是传统BP神经网络容易陷入局部最优值而影响预测结果.将利用遗传算法同时优化BP神经网络连接权值和阈值的算法应用到大庆油田BED试验区高含水阶段的油田产量预测,结果表明在面对高含水阶段更加复杂的地质条件和数据波动更强的情况下优化后的神经网络收敛速度更快而且预测精度更高.     

The role of non-Archimedean epsilon in finding the most efficient unit: With an application of professional tennis players

The determination of a single efficient decision making unit (DMU) as the most efficient unit has been attracted by decision makers in some situations. Some integrated mixed integer linear programming (MILP) and mixed integer nonlinear programming (MINLP) data envelopment analysis (DEA) models have been proposed to find a single efficient unit by the optimal common set of weights. In conventional DEA models, the non-Archimedean infinitesimal epsilon, which forestalls weights from being zero, is useless if one utilizes the well-known two-phase method. Nevertheless, this approach is inapplicable to integrated DEA models. Unfortunately, in some proposed integrated DEA models, the epsilon is neither considered nor determined. More importantly, based on this lack some approaches have been developed which will raise this drawback.In this paper, first of all some drawbacks of these models are discussed. Indeed, it is shown that, if the non-Archimedean epsilon is ignored, then these models can neither find the most efficient unit nor rank the extreme efficient units. Next, we formulate some new models to capture these drawbacks and hence attain assurance regions. Finally, a real data set of 53 professional tennis players is applied to illustrate the applicability of the suggested models.     

The role of coherent supply chain strategy and performance management in achieving competitive advantage: an international survey

The results of a major international survey into the relationship between corporate strategy, supply chain strategy and supply chain performance management are reported. Five clearly defined groups are identified: Supply Chain Leaders, Strong and Weak Players, Lagging Players and Non-players. Those business units that report a close link between their supply chain strategy and their supply chain technology, in comparison with those that report a weaker link, displayed a consistent set of characteristics. They rate supply chain strategy as more important for corporate strategy. They have a relatively sophisticated definition of their supply chain strategy. They think their supply chain is more important in achieving competitive advantage, they have invested more in supply chain infrastructure and IT support and they have more formal means of assessing their supply chain performance. There is evidence of inconsistency in the way many businesses relate their supply chain, corporate, and investment strategy.     

Graph coloring in the estimation of sparse derivative matrices: Instances and applications

We describe a graph coloring problem associated with the determination of mathematical derivatives. The coloring instances are obtained as intersection graphs of row partitioned sparse derivative matrices. The size of the graph is dependent on the partition and can be varied between the number of columns and the number of nonzero entries. If solved exactly our proposal will yield a significant reduction in computational cost of the derivative matrices. The effectiveness of our approach is demonstrated via a practical problem from computational molecular biology. We also remark on the hardness of the generated coloring instances.     

Complex variables in junior high school: the role and potential impact of an outreach mathematician

Outreach mathematicians are college faculty who are trainedin mathematics but who undertake an active role in improvingprimary and secondary education. This role is examined througha study where an outreach mathematician introduced the conceptof complex variables to junior high school students in the UnitedStates with the goal of stimulating their interest in mathematicsand improving their algebra skills. Comparison of pre- and post-testresults showed that ninth-grade students displayed a significantchange in algebraic skills while the eighth-grade students madelittle progress. The outreach mathematician lacked some awarenessof the eighth-grade students’ foundational backgroundand motivation. This illustrates the importance of working moreclosely with the participating teacher, who understands betterthe curriculum and the students’ background knowledge,levels of maturity and levels of motivation.     

On the importance of behavioral operational research: The case of understanding and communicating about dynamic systems

We point out the need for Behavioral Operational Research (BOR) in advancing the practice of OR. So far, in OR behavioral phenomena have been acknowledged only in behavioral decision theory but behavioral issues are always present when supporting human problem solving by modeling. Behavioral effects can relate to the group interaction and communication when facilitating with OR models as well as to the possibility of procedural mistakes and cognitive biases. As an illustrative example we use well known system dynamics studies related to the understanding of accumulation. We show that one gets completely opposite results depending on the way the phenomenon is described and how the questions are phrased and graphs used. The results suggest that OR processes are highly sensitive to various behavioral effects. As a result, we need to pay attention to the way we communicate about models as they are being increasingly used in addressing important problems like climate change.