Mathematics in the making: Mapping verbal discourse in Pólya's “Let Us Teach Guessing” lesson

This paper describes a detailed analysis of verbal discourse within an exemplary mathematics lesson—that is, George Pólya teaching in the Mathematics Association of America [MAA] video classic, “Let Us Teach Guessing” (1966). The results of the analysis reveal an inductive model of teaching that represents recursive cycles rather than linear steps. The lesson begins with a frame of reference and builds meaning cyclically/recursively through inductive processes—that is, moving from specific cases, through recursive cycles, toward more general hypotheses and rules. Additionally, connections to specific forms of talk and verbal assessment, as well as to univocal (conveying meaning) and dialogic (new meaning through dialogue) discourse, are made.     

Control decisions and personal beliefs: their effect on solving mathematical problems

The main purpose of this paper is to discuss how college students enrolled in a college level elementary algebra course exercised control decisions while working on routine and non-routine problems, and how their personal belief systems shaped those control decisions. In order to prepare students for success in mathematics we as educators need to understand the process steps they use to solve homework or examination questions, in other words, understand how they “do” mathematics. The findings in this study suggest that an individual’s belief system impacts how they approach a problem. Lack of confidence and previous lack of success combined to prompt swift decisions to stop working. Further findings indicate that students continue with unsuccessful strategies when working on unfamiliar problems due to a perceived dependence of solution strategies to specific problem types. In this situation, the students persisted in an inappropriate solution strategy, never reaching a correct solution. Control decisions concerning the pursuit of alternative strategies are not an issue if the students are unaware that they might need to make different choices during their solutions. More successful control decisions were made when working with familiar problems.     

Independent sets and non-augmentable paths in generalizations of tournaments

We study different classes of digraphs, which are generalizations of tournaments, to have the property of possessing a maximal independent set intersecting every non-augmentable path (in particular, every longest path). The classes are the arc-local tournament, quasi-transitive, locally in-semicomplete (out-semicomplete), and semicomplete k-partite digraphs. We present results on strongly internally and finally non-augmentable paths as well as a result that relates the degree of vertices and the length of longest paths. A short survey is included in the introduction.     

Supervisory control of hybrid systems based on model abstraction and guided search

This paper addresses the task of computing supervisory controllers by which hybrid systems with nonlinear continuous dynamics are driven into goal sets while safety specifications are met. For this class of systems, the (conservatively approximative) determination of reachable states is an important but also a computationally expensive step of the controller synthesis. This contribution proposes a technique aiming at reducing the reach set computation by using abstract models and guided search. For a discrete abstraction of the hybrid model, candidate paths are determined as possible controlled evolutions which fulfill the given specifications. A validation scheme comprising three different techniques is applied to determine whether the candidate path represents a feasible control strategy for the hybrid system. If the specification is violated, the abstract model is refined according to the validation result. The iterative application of the determination of candidate paths, the path validation and the model refinement steers the search for a control strategy such that often only a relatively small part of the reachable state space has to be explored. The synthesis procedure is illustrated for two examples.     

Understanding the complexities of student motivations in mathematics learning

Student motivation has long been a concern of mathematics educators. However, commonly held distinctions between intrinsic and extrinsic motivations may be insufficient to inform our understandings of student motivations in learning mathematics or to appropriately shape pedagogical decisions. Here, motivation is defined, in general, as an individual's desire, power, and tendency to act in particular ways. We characterize details of motivation in mathematical learning through qualitative analysis of honors calculus students’ extended, collaborative problem solving efforts within a longitudinal research project in learning and teaching. Contextual Motivation Theory emerges as an interpretive means for understanding the complexities of student motivations. Students chose to act upon intellectual-mathematical motivations and social-personal motivations that manifested simultaneously. Students exhibited intellectual passion in persisting beyond obtaining correct answers to build understandings of mathematical ideas. Conceptually driven conditions that encourage mathematical necessity are shown to support the growth of intellectual passion in mathematics learning.     

Analyzing the word-problem performance and strategies of students experiencing mathematics difficulty

The purpose of this study was to examine the word-problem performance and strategies utilized by 3rd-grade students experiencing mathematics difficulty (MD). We assessed the efficacy of a word-problem intervention and compared the word-problem performance of students with MD who received intervention (n = 51) to students with MD who received general education classroom word-problem instruction (n = 60). Intervention occurred for 16 weeks, 3 times per week, 30 min per session and focused on helping students understand the schemas of word problems. Results demonstrated that students with MD who received the word-problem intervention outperformed students with MD who received general education classroom word-problem instruction. We also analyzed the word-problem strategies of 30 randomly-selected students from the study to understand how students set up and solve word problems. Students who received intervention demonstrated more sophisticated word-problem strategies than students who only received general education classroom word-problem instruction. Findings suggest students with MD benefit from use of meta-cognitive strategies and explicit schema instruction to solve word problems.     

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.     

How do students’ behaviors relate to the growth of their mathematical ideas?

The purpose of this study is to analyze the relationship between student behaviors and the growth of mathematical ideas (using the Pirie-Kieren model). This analysis was accomplished through a series of case studies, involving middle school students of varying ability levels, who were investigating a combinatorics problem in after-school problem-solving sessions. The results suggest that certain types of student behaviors appear to be associated with the growth of ideas and emerge in specific patterns. More specifically, as understanding grows, there is a general shift from behaviors such as students questioning each other, explaining and using their own and others’ ideas toward behaviors involving the setting up of hypothetical situations, linking of representations and connecting of contexts. Recognizing that certain types of student behaviors tend to emerge in specific layers of the Pirie-Kieren model can be important in helping us to understand the development of mathematical ideas in children.     

An investigation of preservice teachers’ use of guess and check in solving a semi open-ended mathematics problem

Open-ended problems have been regarded as powerful tools for teaching mathematics. This study examined the problem solving of eight mathematics/science middle-school teachers. A semi-structured interview was conducted with (PTs) after completing an open-ended triangle task with four unique solutions. Of particular emphasis was how the PTs used a specific heuristic strategy. The results showed that the primary strategy PTs employed in attempting to solve the triangle problem task was guess and check; however, from the PTs’ reflections, we found there existed misapplications of guess and check as a systematic problem-solving strategy. In order to prepare prospective teachers to effectively teach, teacher educators should pay more attention to the mathematical proficiency of PTs, particularly their abilities to systematically and efficiently use guess and check while solving problems and explain their solutions and reasoning to middle-school students.     

Generalization: One technique of computational and applied mathematical methodology

In many researches and applications in applied mathematics and engineering, we need to generalize the existing results, algorithms or methods in order to apply them to different situations or considerations. Generalization will allow certain particular problems to be solved more efficiently. Here we use some examples in scientific computing to demonstrate the importance of generalization technique for some researches, and how to generalize results or to improve conditions.     

A quadratic assignment formulation of the molecular conformation problem

The molecular conformation problem is discussed, and a concave quadratic global minimization approach for solving it is described. This approach is based on a quadratic assignment formulation of a discrete approximation to the original problem.     

Structures in decision making: On the subjective geometry of hierarchies and networks

In the field of decision making, creating a structure is the first step in organizing, representing and solving a problem. A structure is a model, an abstraction of a problem. It helps us visualize and understand the relevant elements within it that we know from the real world and then use our understanding to solve the problem represented in the structure with greater confidence. In general, there are two kinds of structures used to represent problems: hierarchies and networks. Both rely to a varying degree on the interactions. Some examples are given followed by a discussion about how to structure the problem. At a minimum, a structure must satisfy two requirements: that it be logical in identifying and grouping similar things together, and that it relates them accurately according to the flow of influence among them. It must be complete with nothing left out that has an important influence. The structure is then tested as to whether it helps solve the problem to one’s satisfaction.     

A new upper bound in the second Kershaw's double inequality and its generalizations

In the paper, a new upper bound in the second Kershaw's double inequality involving ratio of gamma functions is established, and, as generalizations of the second Kershaw's double inequality, the divided differences of the psi and polygamma functions are bounded.     

A Skorokhod Problem formulation and large deviation analysis of a processor sharing model

Generalized processor sharing has been proposed as a policy for distributing processing in a fair manner between different data classes in high-speed networks. In this paper we show how recent results on the Skorokhod Problem can be used to construct and analyze the mapping that takes the input processes into the buffer content. More precisely, we show how to represent the map in terms of a Skorokhod Problem, and from this infer that the mapping is well defined (existence and uniqueness) and well behaved (Lipschitz continuity). As an elementary application we present some large deviation estimates for a many data source model. This revised version was published online in June 2006 with corrections to the Cover Date.     

“Playing the game” of story problems: Coordinating situation-based reasoning with algebraic representation

This study critically examines a key justification used by educational stakeholders for placing mathematics in context –the idea that contextualization provides students with access to mathematical ideas. We present interviews of 24 ninth grade students from a low-performing urban school solving algebra story problems, some of which were personalized to their experiences. Using a situated cognition framework, we discuss how students use informal strategies and situational knowledge when solving story problems, as well how they engage in non-coordinative reasoning where situation-based reasoning is disconnected from symbol-based reasoning and other problem-solving actions. Results suggest that if contextualization is going to provide students with access to algebraic ideas, supports need to be put in place for students to make connections between formal algebraic representation, informal arithmetic-based reasoning, and situational knowledge.     

Integer linear programming formulation of the material requirements planning problem

Lot sizing procedures for discrete and dynamic demand form a distinct class of inventory control problems, usually referred to asmaterial requirements planning. A general integer programming formulation is presented, covering an extensive range of problems: single-item, multi-item, and multi-level optimization; conditions on lot sizes and time phasing; conditions on storage and production capacities; and changes in production and storage costs per unit. The formulation serves as a uniform framework for presenting a problem and a starting point for developing and evaluating heuristic and tailor-made optimum-seeking techniques.     

研究四色问题的意义及理论构想

四色问题又称四色猜想,是世界近代三大数学难题之一．1976年两位美国数学家Appel与Haken借助计算机给出了一个证明．时至今日,四色问题的正确性早已得到数学界所承认．但是围绕它的非计算机证明,在近几十年来涌现出了各种不同的研究成果．一方面丰富了图论的内容,另一方面又促进了图的染色理论的发展．本文从研究四色问题的意义出发;揭示了四色问题所隐藏的深刻规律,在此基础上提出了一个比四色问题更具有广泛意义的理论构想．主要目地为四色问题的非计算机证明提供一个研究方向．     

Portfolio separation properties of the skew-elliptical distributions, with generalizations

The two-fund separation property of the elliptical distributions is extended to the skew-elliptical case by adding a number of funds equaling the rank of the skewness matrix. The singular extended skew-elliptical distributions are covered, as is a further generalization to the case where the set conditioned upon is not an orthant.