Mathematical Thinking Involved in U.S. and Chinese Students' Solving of Process-Constrained and Process-Open Problems

This study examined U.S. and Chinese 6th-grade students' mathematical thinking and reasoning involved in solving 6 process-constrained and 6 process-open problems. The Chinese sample (from Guiyang, Guizhou) had a significantly higher mean score than the U.S. sample (from Milwaukee, Wisconsin) on the process-constrained tasks, but the sample of U.S. students had a significantly higher mean score than the sample of the Chinese students on the process-open tasks. A qualitative analysis of students' responses was conducted to understand the mathematical thinking and reasoning involved in solving these problems. The qualitative results indicate that the Chinese sample preferred to use routine algorithms and symbolic representations, whereas the U.S. sample preferred to use concrete visual representations. Such a qualitative analysis of students' responses provided insights into U.S. and Chinese students' mathematical thinking, thereby facilitating interpretation of the cross-national differences in solving the process-constrained and process-open problems.     

中美数学研究人员交流计划申请指南

为了加强中美青年科学家在数学领域交流,营造有利于中国科研人员参与国际(地区)合作与竞争的良好环境,鼓励基金承担者开展积极而富有成效的国际合作与交流.中国国家自然科学基金委员会和美国基金会就数学领域的合作签署合作备忘录,共同支持中美数学研究人员.     

国家自然科学基金数学动态报道 中美数学研究人员交流计划申请指南

为了加强中美青年科学家在数学领域交流，营造有利于中国科研人员参与国际（地区）合作与竞争的良好环境，鼓励基金承担者开展积极而富有成效的国际合作与交流。中国国家自然科学基金委员会和美国基金会签署合作协议，共同支持中美数学研究人员。     

Economical Iterative Algorithms for Solving Stationary Problems of Mathematical Physics

We construct and investigate additive iterative methods of complete approximation for solving stationary problems of mathematical physics. We prove the convergence of the proposed methods and obtain error estimates without the requirement of commutativity of the decomposition operators. We provide the results of a computational experiment for a three-dimensional boundary-value problem. We consider possible generalizations of algorithms for equations with mixed derivatives and Navier–Stokes equation systems.     

浅谈微积分所蕴含的特征值计算思想方法

数学思想方法是数学的灵魂.主要阐述微积分中泰勒公式的应用,泰勒公式不但能用来近似计算数学常数π,而且能用来理论证明两个粗糙的近似值通过简单的组合加工技术(现代人称之为"外推法")产生更准确的近似值,也把外推法推广用于求解Steklov特征值问题,数值算例表明非协调Crouzeix-Raviart有限元外推法既能提高解的精度又能提供准确特征值的下界.     

An Interactive Method as an Aid in Solving Bicriterion Mathematical Programming Problems

In this paper, a man-machine interactive method is presented as an aid in solving the bicriterion mathematical programming problem. It is assumed that the two objective functions are real-valued functions of the decision variables which are themselves constrained to some compact and nonempty set. The overall utility function is assumed to be unknown explicitly to the decision-maker but is assumed to be a real-valued function defined on the pairs of feasible values of the objective functions and monotone non-decreasing in each argument. The decision-maker is required only to provide yes or no answers to questions regarding the desirability of increase or decrease in objective function values of solutions that he will not accept as optimal. Convergence of the method is indicated and a numerical example is presented in order to demonstrate its applicability.     

Assessing and understanding U.S. and Chinese students' mathematical thinking

If the main goal of educational research and refinement of instructional program is to improve students' learning, it is necessary to assess students' emerging understandings and to see how they arise. The purpose of this paper is to address issues related to assessments of students' mathematical thinking in cross-national studies and then to discuss the lessons we may learn from these studies to assess and improve students' learning. In particular, the issues related to assessing U.S. and Chinese students' mathematical thinking were discussed. Then, this paper discussed the findings from two studies examining the impact of early algebra learning and teachers' beliefs on U.S. and Chinese students' mathematical thinking. Lastly, the issues related to interpreting and understanding the differences between U.S. and Chinese students' thinking were discussed.     

Opportunities to Learn: Inverse Relations in U.S. and Chinese Textbooks

This study, focusing on inverse relations, examines how representative U.S. and Chinese elementary textbooks may provide opportunities to learn fundamental mathematical ideas. Findings from this study indicate that both of the U.S. textbook series (grades K-6) in comparison to the Chinese textbook samples (grades 1–6), presented more instances of inverse relations, while also containing more unique types of problems; yet, the Chinese textbooks provided more opportunities supporting meaningful and explicit learning. In particular, before presenting corresponding practice problems, Chinese textbooks contextualized worked examples of inverse relations in real-world situations to aid in sense making of computational or checking procedures. The Chinese worked examples also differed in representation uses especially through concreteness fading. Finally, the Chinese textbooks spaced learning over time, systematically stressing structural relations including the inverse quantities relationships. These findings shed light on ways to support students’ meaningful and explicit learning of fundamental mathematical ideas in elementary school.     

The Gender Differential Effects of a Procedural Plan For Solving Mathematical Word Problems

This research investigated potential gender-related effects of an explicitly stated problem solving plan for sixth graders. Females and males did equally well on both forms of the test. Students scored significantly higher (P = .023) on the second occasion of testing that included the problem solving plan. A mixed, two factor repeated measures Analysis of Variance found a significant effect for occasion of testing (P = .001) and a significant interaction effect of occasion of testing, test sequence, and gender (P = .006). It was concluded that females benefited from free exploration of problem situations followed by an organized exploration, while males appeared to function equally well regardless of the problem solving plan.     

An Exploratory Study of College Students' Views of Mathematical Thinking in a Historical Approach Calculus Course

Heuristic training alone is not enough for developing one's mathematical thinking. One missing component is a mathematical point of view. This study reports findings regarding outcomes of a historical approach calculus course to foster Taiwanese college students' views of mathematical thinking. This study consisted of 3 stages. During the initial phase, 44 engineering majors' views on mathematical thinking were tabulated by an open-ended questionnaire, and 9 randomly selected students were invited to participate in follow-up interviews. Students then received an 18-week historical approach calculus course in which mathematical concepts were problematized to challenge their intuition-based empirical beliefs about doing mathematics. Near the end of the semester, all participants answered the identical questionnaire, and we interviewed the same students to pinpoint any shifts in their views on mathematical thinking. We found that participants were more likely to value logical sense, creativity, and imagination in doing mathematics. Further, students were leaning toward a conservative attitude toward certainty of mathematical knowledge. Participants' focus seemingly shifted from mathematics as a product to mathematics as a process.     

An Exploratory Study of College Students' Views of Mathematical Thinking in a Historical Approach Calculus Course

Heuristic training alone is not enough for developing one's mathematical thinking. One missing component is a mathematical point of view. This study reports findings regarding outcomes of a historical approach calculus course to foster Taiwanese college students' views of mathematical thinking. This study consisted of 3 stages. During the initial phase, 44 engineering majors' views on mathematical thinking were tabulated by an open-ended questionnaire, and 9 randomly selected students were invited to participate in follow-up interviews. Students then received an 18-week historical approach calculus course in which mathematical concepts were problematized to challenge their intuition-based empirical beliefs about doing mathematics. Near the end of the semester, all participants answered the identical questionnaire, and we interviewed the same students to pinpoint any shifts in their views on mathematical thinking. We found that participants were more likely to value logical sense, creativity, and imagination in doing mathematics. Further, students were leaning toward a conservative attitude toward certainty of mathematical knowledge. Participants' focus seemingly shifted from mathematics as a product to mathematics as a process.     

A Brief Comparison of the U.S. and Chinese Middle School Mathematics Programs

The U.S. generally has a less intense mathematics curriculum in the middle school grades than China. Some factors contributing to the lower intensity in the U.S. mathematics curriculum are textbooks with extensive drill, repetition of content, lack of challenging problem solving, lower curricular and cultural expectations, and ability grouping. In comparison, China utilizes challenging problem solving, sequential development of content without repetition, expectations of hard work, high values for mathematics by the curriculum and culture, and a common curriculum for all as aspects of mathematics instruction. The U.S. is taking a positive direction in its mathematics curriculum with the use of technology and reform while compulsory education is mandating that the theoretical depth of middle school curriculums in China be lowered for all of its students in grades 1–9.     

Affective Pathways and Representation in Mathematical Problem Solving

An affective system of representation is joined with several cognitive representational systems in endeavoring to construct a realistic model for problem-solving competence. The affective states described are not global attitudes or traits, but local changing states of feeling that the solver experiences and can utilize during problem solving-to store and provide useful information, facilitate monitoring, and evoke heuristic processes. Thus affect, like language, is seen as fundamentally representational as well as communicative. Two major affective pathways-one favorable and one unfavorable-are discussed, together with conjectured relationships between affective states and useful or counterproductive heuristic configurations. Implications of the model include local affective goals for mathematics teaching related to problem-solving heuristics.