Mathematics for All: Perspectives and Promising Practices

The Core-Plus Mathematics Project (CPMP, 1995) is one of four comprehensive curriculum development projects that, in 1992, were awarded 5-year grants by the National Science Foundation to design, evaluate, and disseminate innovative high school curricula that interpret and implement the recommendations of the National Council of Teachers of Mathematics (NCTM) Curriculum and Evaluation Standards for School Mathematics ( NCTM, 1989 ) and Professional Standards for Teaching Mathematics ( NCTM, 1991 ). This article describes CPMP perspectives on a new curriculum organization for high school mathematics, identifies implications of these perspectives for promoting access and equity for all students, and reports some of the supporting oral data from an ongoing formative evaluation of the curriculum.     

A Look at Project AIMS

Project AIMS (Activities that Integrate Mathematics and Science) began in 1981 as a National Science Foundation grant. Twelve years later it is a flourishing program which relies on materials developed and written by teachers, and one which uses inexpensive, easily acquired equipment and materials. It is particularly noteworthy that the focus of AIMS-integration through hands-on mathematics and science-supports such recent initiatives as the NCTM Standards and Project 2061 of the AAAS, even though it predates both of them.     

Middle‐School Mathematics Teachers' Beliefs in NCTM's Vision

This study examined the extent to which seventh‐ and eighth‐grade mathematics teachers are aware of National Council of Teachers of Mathematics (NCTM) standards documents, Curriculum and Evaluation Standards for School Mathematics and Principles and Standards for School Mathematics and agree with NCTM's vision of school mathematics as expressed in these documents. Quantitative data were collected through the Mathematics Standards Belief Survey (MSBS), a survey specifically designed to measure teachers' overall belief in NCTM's vision as well as in certain philosophical tenets of NCTM. Of the 82 seventh‐ and eighth‐grade mathematics teachers in the identified school district of Nevada, 73 (89.0%) participated in this study. The data revealed that, among seventh‐ and eighth‐grade mathematics teachers, secondary‐certified teachers had significantly higher MSBS scores than elementary‐certified teachers. A number of other findings were made, including significant differences among mean belief scores in the philosophical tenets of NCTM.     

The Relationship Between Teacher and Student Beliefs About Mathematics

In this study the relationship between teacher (n = 7) beliefs about mathematics, the learning and teaching of mathematics and their respective students' beliefs about mathematics (n = 158) are examined. The data were collected by means of two instruments specifically designed to measure belief systems about mathematics. Teacher scores were adjusted so that a higher score reflected beliefs in aligntnent with the National Council of Teachers of Mathematics (NCTM) Standards. Results indicated that the students of teachers whose beliefs were in alignment with the NCTM Standards had significantly different beliefs about factors that lead to success in mathematics than did other students. Specifically these students felt that working hard to solve problems and striving for understanding would lead to success. No student differences were found for subscales of ego orientation, competitiveness, interest and extrinsic factors such as neatness and cooperation. These findings suggest that this group of teachers practiced what they believed and that these practices affected what their students believed about mathematics. We suggest that using these two assessments in tandem give a clearer picture of the mathematical environment within a classroom and can be used in professional development workshops to initiate teacher reflection about classroom practices.     

Alignment of Elementary Geometry Curriculum With Current Standards

The subject of geometry in the curriculum continues to be an area of concern among mathematics educators. The gap between standards expectations and actual textbook content continues to reflect the need for substantial change. In this study, the geometry language introduced in K‐6 textbooks was carefully examined and compared to the language used in the Curriculum and Evaluation Standards for School Mathematics ( National Council of Teachers of Mathematics, 1989 ), Principles and Standards for School Mathematics ( NCTM, 1999 ), and New Standards Elementary School Mathematics Performance Standards ( National Center for Education and the Economy, 1997 ). The. logic followed was that the language in textbooks reflects the content presented in the textbook, and the language in the standards reflects what should be presented. The results of the study clearly indicate that there is a substantial misalignment between the geometry presented in the textbooks, the. geometry expected to be taught by groups such as NCTM, and the geometry being assessed in student performance measures as suggested by NCEE. Specific areas of misalignment are identified, along with implications, and some areas in which the curriculum should be revised are identified.     

Balancing mathematics education research and the NCTM standards

The release of the Principles and Standards for School Mathematics in the United States by the National Council of Teachers of Mathematics (NCTM) brought to the forefront the debate of whether research should determine the validity of the espoused Standards? Or conversely whether the Standards should influence the research agenda of the mathematics education community? How should university teacher educators address this issue? Should pre-service and practicing teachers blindly, accept the Standards as well as the research, or do we cultivate the critical thinking skills that will allow preparing teachers to resolve this dilemma? In this article a university mathematics educator and an idealistic pre-service elementary teacher try to resolve the dilemma of balancing the Standards with research and personal beliefs about the teaching and learning of mathematics.     

The Special Issue Is Dedicated to Prof Li Tatsien on the Occasion of His 70th Birthday

Professor Li has published more than 200 papers and 16 monographs and textbooks, among which 4 monographs are printed in English in U.S.A., U.K. and France respectively. He has received domestic prizes and awards, including: Shanghai Top Science and Technology Award, Hua Loo-Keng Prize of Mathematics, one Second Prize and one Third Prize of National Natural Sciences from the State, one First Prize of Scientific and Technological Progress from the State Education Commission, and one First Prize of Scientific and Technological Progress from Shanghai Municipality. Two of his works are awarded as Excellent University Textbooks from the State. One Exceptional Shanghai Award for Teaching Achievements in Higher Education and one First National Award for Teaching Achievements in Higher Education. Moreover, he has received the Science and Technology Progress Award of Ho Leug Ho Lee Foundation.     

Supervisor and Teacher Educator Perceived Relevance of Recommendations in the NCTM Curriculum Standards

Based on judgments of mathematics supervisors and teacher educators, the findings of this empirical study confirm that the Curriculum and Evaluation Standards (National Council of Teachers of Mathematics [NCTM], 1989) reflect content that is relevant to mathematics instruction. However, some concern did exist about the relevancy of teaching statistics and probability at the elementary level. Despite the general consensus about the Standards, total agreement was not evident. Respondents with asecondary-school focus rated many of the standards and criteria across grade levels significantly lower in importance than did other respondents; elementary- and middle-school respondents judged several K-8 standards and criteria to be more important than did some higher-education and secondary respondents, while secondary-and middle-school respondents judged the 9–12 standards and critera to be more important than did elementary and some higher education respondents. Respondents with 30 or more years of experience or who were less familiar with the standards rated the K-8 standards and criteria to be lower in importance than did other respondents.     

Crutch or Catalyst: Teachers' Beliefs and Practices Regarding Calculator use in Mathematics Instruction

This study investigated K‐12 teachers' beliefs and reported teaching practices regarding calculator use in their mathematics instruction. A survey was administered to more than 800 elementary, middle and high school teachers in a large metropolitan area to address the following questions: (a) what are the beliefs and practices of mathematics teachers regarding calculator use? and (b) how do these beliefs and practices differ among teachers in three grade bands? Factor analysis of 20 Likert scale items revealed four factors that accounted for 54% of the variance in the ratings. These factors were named Catalyst Beliefs, Teacher Knowledge, Crutch Beliefs, and Teacher Practices. Compared to elementary teachers, high school teachers were significantly higher in their perception of calculator use as a catalyst in mathematics instruction. However, the higher the grade level of the teacher, the higher the mean score on the perception that calculator use may be a way of getting answers without understanding mathematical processes. The mean scores for teachers in all three grade bands indicated agreement that students can learn mathematics through calculator use and using calculators in instruction will lead to better student understanding and make mathematics more interesting. The survey results shed light on teachers' self reported beliefs, knowledge, and practices in regard to consistency with elements of the National Council of Teachers of Mathematics Principles and Standards for School Mathematics (2000) technology principle and the NCTM use of technology position paper (2003). This study extended previous research on teachers' beliefs regarding calculator use in classrooms by examining and comparing the results of teacher surveys across three grade bands.     

A Vision for What Science Education Should Be Like for the First 25 Years of a New Millennium

Robert Yager is a native Iowan who was worked at the University of Iowa since 1956. He has served as president of seven national associations, including the National Science Teachers Association, School Science and Mathematics Association, the National Association of Biology Teachers, the Association for the Education of Teachers of Science, the National Association for Research in Science Teaching, Section Q of the American Association for the Advancement of Science, and the National Association of Science‐Technology‐Society. Yager has directed over 100 projects supported by the National Science Foundation, several research projects supported by the U.S. Department of Education, and several project evaluations. The University of Iowa has one of the largest graduate programs in the U.S., while also focusing on in‐service programs for science teachers and a model preservice program (which enjoyed NSF support for a 10‐year period). Yager has published over 500 books, monographs, and research reports. Currently, he is active with several research projects in Korea, Japan, Thailand, Malaysia, Israel, and Estonia.     

Mathematical Literacy and Abstraction in the 21st Century

Ed Dubinsky received his Ph.D. in mathematics from the University of Michigan in 1962 and for the next 25 years engaged in research in theoretical mathematics. In the mid‐80′s he became interested in mathematics education and has worked exclusively in the area since then. In his research, he tries to understand how a person's mind might be working when he or she tries to understand mathematical concepts at the postsecondary level. Based on this research, he has conducted large‐scale curriculum development projects in calculus, discrete mathematics, abstract algebra and cooperative learning. He has been editor or co‐editor of UME Trends, Research in Collegiate Mathematics Education, and the Journal of Computers in Mathematics and Science Teaching. He has held faculty positions at 8 universities in 5 countries on 3 continents: Fourah Bey College (Sierra Leone), University of Ghana, Tulane University, McMaster University, Polish Academy of Sciences, Clarkson University, Purdue University, and Georgia State University. Dr. Dubinsky is presently retired and consults with several universities on education matters.     

Principles and Standards for School Mathematics and Teacher Education: Preparing and Empowering Teachers

Teachers and teacher educators play a critical role in the improvement of mathematics education. Recommendations for appropriate teacher support can be gained from examining the Teaching Principle in the document, Principles and Standards for School Mathematics ( NCTM, 2000 ), as well as earlier recommendations from the Professional Standards for Teaching Mathematics ( NCTM, 1991 ). This paper discusses implications of the recommendations for preservice teacher preparation, the continuing professional development of teachers, and mathematics teacher certification policy.     

Identifying an optimal basis in linear programming

We propose a sufficient condition that allows an optimal basis to be identified from a central path point in a linear programming problem. This condition can be applied when there is a gap in the sorted list of slack values. Unlike previously known conditions, this condition is valid for real-number data and does not involve the number of bits in the data.This work is supported in part by the National Science Foundation, the Air Force Office of Scientific Research, and the Office of Naval Research, through NSF Grant DMS-8920550. Also supported in part by an NSF Presidential Young Investigator Award with matching funds received from AT&T and the Xerox Corporation. Part of this work was carried out while the author was visiting the Sandia National Laboratories, supported by the U.S. Department of Energy under Contract DE-AC04-76DP00789.The author is supported in part by NSF Grant DDM-9207347. Part of this work was carried out while the author was on a sabbatical leave from the University of Iowa and visiting the Cornell Theory Center, Cornell University, Ithaca, NY 14853, supported in part by the Cornell Center for Applied Mathematics and by the Advanced Computing Research Institute, a unit of the Cornell Theory Center, which receives major funding from the National Science Foundation and the IBM Corporation, with additional support from New York State and members of its Corporate Research Institute.     

The Effects of a Mathematics Infusion Curriculum on Middle School Student Mathematics Achievement

Increasing mathematical competencies of American students has been a focus for educators, researchers, and policy makers alike. One purported approach to increase student learning is through connecting mathematics and science curricula. Yet there is a lack of research examining the impact of making these connections. The Mathematics Infusion into Science Project, funded by the National Science Foundation, developed a middle school mathematics‐infused science curriculum. Twenty teachers utilized this curriculum with over 1,200 students. The current research evaluated the effects of this curriculum on students' mathematics learning and compared effects to students who did not receive the curriculum. Students who were taught the infusion curriculum showed a significant increase in mathematical content scores when compared with the control students.     

Solving combinatorial optimization problems using Karmarkar's algorithm

We describe a cutting plane algorithm for solving combinatorial optimization problems. The primal projective standard-form variant of Karmarkar's algorithm for linear programming is applied to the duals of a sequence of linear programming relaxations of the combinatorial optimization problem.Computational facilities provided by the Cornell Computational Optimization Project supported by NSF Grant DMS-8706133 and by the Cornell National Supercomputer Facility. The Cornell National Supercomputer Facility is a resource of the Center for Theory and Simulation in Science and Engineering at Cornell Unversity, which is funded in part by the National Science Foundation, New York State, and the IBM Corporation. The research of both authors was partially supported by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University.Research partially supported by ONR Grant N00014-90-J-1714.Research partially supported by NSF Grant ECS-8602534 and by ONR Contract N00014-87-K-0212.     

Asymptotic expansions and Richardson extrapolation of approximate solutions for integro-differential equations by mixed finite element methods

In this paper asymptotic error expansions for mixed finite element approximations of the integro-differential equation are derived, and Richardson extrapolation is applied to improve the accuracy of the approximations by two different schemes with the help of an interpolation post-processing technique. The results of this paper provide new asymptotic expansions. As a by-product, we illustrate that all the approximations of higher accuracy can be used to form a class of a-posteriori error estimators for this mixed finite element method. Finally, a numerical example is provided to validate the theoretical results. This project was supported in part by the Special Funds for Major State Basic Research Project (2007CB8149), the National Natural Science Foundation of China (10471103 and 10771158), Social Science Foundation of the Ministry of Education of China (Numerical Methods for Convertible Bonds, 06JA630047), the NSERC, Tianjin Natural Science Foundation (07JCYBJC14300), Tianjin Educational Committee, Liu Hui Center for Applied Mathematics of Nankai University and Tianjin University, and Tianjin University of Finance and Economics.     

A lower bound on the number of iterations of long-step primal-dual linear programming algorithms

Recently, Todd has analyzed in detail the primal-dual affine-scaling method for linear programming, which is close to what is implemented in practice, and proved that it may take at leastn ^1/3 iterations to improve the initial duality gap by a constant factor. He also showed that this lower bound holds for some polynomial variants of primal-dual interior-point methods, which restrict all iterates to certain neighborhoods of the central path. In this paper, we further extend his result to long-step primal-dual variants that restrict the iterates to a wider neighborhood. This neigh-borhood seems the least restrictive one to guarantee polynomiality for primal-dual path-following methods, and the variants are also even closer to what is implemented in practice.Research supported in part by NSF, AFOSR and ONR through NSF Grant DMS-8920550.This author is supported in part by NSF Grant DDM-9207347. Part of thiw work was done while the author was on a sabbatical leave from the University of Iowa and visiting the Cornell Theory Center, Cornell University, Ithaca, NY 14853, supported in part by the Cornell Center for Applied Mathematics and by the Advanced Computing Research Institute, a unit of the Cornell Theory Center, which receives major funding from the National Science Foundation and IBM Corporation, with additional support from New York State and members of its Corporate Research Institute.     

Rotating Charge Coupled to the Maxwell Field: Scattering Theory and Adiabatic Limit

We consider a spinning charge coupled to the Maxwell field. Through the appropriate symmetry in the initial conditions the charge remains at rest. We establish that any time-dependent finite energy solution converges to a sum of a soliton wave and an outgoing free wave. The convergence holds in global energy norm. Under a small constant external magnetic field the soliton manifold is stable in local energy seminorms and the evolution of the angular velocity is guided by an effective finite-dimensional dynamics. The proof uses a non-autonomous integral inequality method.Supported partly by the Wittgenstein 2000 Award of Peter Markowich, funded by the Austrian Science Foundation (FWF), research grants of DFG (436 RUS 113/615/0-1(R)) and RFBR (01-01-04002).On leave Department of Mechanics and Mathematics, Moscow State University, Moscow 119899, Russia. Supported partly by Max Planck Institute for the Mathematics in Sciences (Leipzig) and the Austrian Science Foundation (FWF) START Project (Y-137-TEC) of Norbert Mauser.     

The NCTM Process Standards and the Five Es of Science: Connecting Math and Science

This study investigates defining characteristics among the process standards of the Principles and Standards for School Mathematics and the 5 Es from the National Science Education Standards and the Inquiry and the National Science Education Standards. These characteristics are used to demonstrate similarities and differences between the learning of mathematics and science, discuss implications from such, and argue for the integration of mathematics and science curriculum and instruction.